# Distributions¶

## PyTorch Distributions¶

Most distributions in Pyro are thin wrappers around PyTorch distributions. For details on the PyTorch distribution interface, see torch.distributions.distribution.Distribution. For differences between the Pyro and PyTorch interfaces, see TorchDistributionMixin.

### Bernoulli¶

class Bernoulli(probs=None, logits=None, validate_args=None)

Creates a Bernoulli distribution parameterized by probs or logits (but not both).

Samples are binary (0 or 1). They take the value 1 with probability p and 0 with probability 1 - p.

Example:

>>> m = Bernoulli(torch.tensor([0.3]))
>>> m.sample()  # 30% chance 1; 70% chance 0
tensor([ 0.])

Parameters
• probs (Number, Tensor) – the probability of sampling 1

• logits (Number, Tensor) – the log-odds of sampling 1

### Beta¶

class Beta(concentration1, concentration0, validate_args=None)[source]

Beta distribution parameterized by concentration1 and concentration0.

Example:

>>> m = Beta(torch.tensor([0.5]), torch.tensor([0.5]))
>>> m.sample()  # Beta distributed with concentration concentration1 and concentration0
tensor([ 0.1046])

Parameters
• concentration1 (float or Tensor) – 1st concentration parameter of the distribution (often referred to as alpha)

• concentration0 (float or Tensor) – 2nd concentration parameter of the distribution (often referred to as beta)

### Binomial¶

class Binomial(total_count=1, probs=None, logits=None, validate_args=None)[source]

Creates a Binomial distribution parameterized by total_count and either probs or logits (but not both). total_count must be broadcastable with probs/logits.

Example:

>>> m = Binomial(100, torch.tensor([0 , .2, .8, 1]))
>>> x = m.sample()
tensor([   0.,   22.,   71.,  100.])

>>> m = Binomial(torch.tensor([[5.], [10.]]), torch.tensor([0.5, 0.8]))
>>> x = m.sample()
tensor([[ 4.,  5.],
[ 7.,  6.]])

Parameters
• total_count (int or Tensor) – number of Bernoulli trials

• probs (Tensor) – Event probabilities

• logits (Tensor) – Event log-odds

### Categorical¶

class Categorical(probs=None, logits=None, validate_args=None)[source]

Creates a categorical distribution parameterized by either probs or logits (but not both).

Note

It is equivalent to the distribution that torch.multinomial() samples from.

Samples are integers from $$\{0, \ldots, K-1\}$$ where K is probs.size(-1).

If probs is 1-dimensional with length-K, each element is the relative probability of sampling the class at that index.

If probs is N-dimensional, the first N-1 dimensions are treated as a batch of relative probability vectors.

Note

The probs argument must be non-negative, finite and have a non-zero sum, and it will be normalized to sum to 1 along the last dimension. probs will return this normalized value. The logits argument will be interpreted as unnormalized log probabilities and can therefore be any real number. It will likewise be normalized so that the resulting probabilities sum to 1 along the last dimension. logits will return this normalized value.

See also: torch.multinomial()

Example:

>>> m = Categorical(torch.tensor([ 0.25, 0.25, 0.25, 0.25 ]))
>>> m.sample()  # equal probability of 0, 1, 2, 3
tensor(3)

Parameters
• probs (Tensor) – event probabilities

• logits (Tensor) – event log probabilities (unnormalized)

### Cauchy¶

class Cauchy(loc, scale, validate_args=None)

Samples from a Cauchy (Lorentz) distribution. The distribution of the ratio of independent normally distributed random variables with means 0 follows a Cauchy distribution.

Example:

>>> m = Cauchy(torch.tensor([0.0]), torch.tensor([1.0]))
>>> m.sample()  # sample from a Cauchy distribution with loc=0 and scale=1
tensor([ 2.3214])

Parameters
• loc (float or Tensor) – mode or median of the distribution.

• scale (float or Tensor) – half width at half maximum.

### Chi2¶

class Chi2(df, validate_args=None)

Creates a Chi-squared distribution parameterized by shape parameter df. This is exactly equivalent to Gamma(alpha=0.5*df, beta=0.5)

Example:

>>> m = Chi2(torch.tensor([1.0]))
>>> m.sample()  # Chi2 distributed with shape df=1
tensor([ 0.1046])

Parameters

df (float or Tensor) – shape parameter of the distribution

### ContinuousBernoulli¶

class ContinuousBernoulli(probs=None, logits=None, lims=(0.499, 0.501), validate_args=None)

Creates a continuous Bernoulli distribution parameterized by probs or logits (but not both).

The distribution is supported in [0, 1] and parameterized by ‘probs’ (in (0,1)) or ‘logits’ (real-valued). Note that, unlike the Bernoulli, ‘probs’ does not correspond to a probability and ‘logits’ does not correspond to log-odds, but the same names are used due to the similarity with the Bernoulli. See  for more details.

Example:

>>> m = ContinuousBernoulli(torch.tensor([0.3]))
>>> m.sample()
tensor([ 0.2538])

Parameters
• probs (Number, Tensor) – (0,1) valued parameters

• logits (Number, Tensor) – real valued parameters whose sigmoid matches ‘probs’

 The continuous Bernoulli: fixing a pervasive error in variational autoencoders, Loaiza-Ganem G and Cunningham JP, NeurIPS 2019. https://arxiv.org/abs/1907.06845

### Dirichlet¶

class Dirichlet(concentration, validate_args=None)[source]

Creates a Dirichlet distribution parameterized by concentration concentration.

Example:

>>> m = Dirichlet(torch.tensor([0.5, 0.5]))
>>> m.sample()  # Dirichlet distributed with concentrarion concentration
tensor([ 0.1046,  0.8954])

Parameters

concentration (Tensor) – concentration parameter of the distribution (often referred to as alpha)

### Exponential¶

class Exponential(rate, validate_args=None)

Creates a Exponential distribution parameterized by rate.

Example:

>>> m = Exponential(torch.tensor([1.0]))
>>> m.sample()  # Exponential distributed with rate=1
tensor([ 0.1046])

Parameters

rate (float or Tensor) – rate = 1 / scale of the distribution

### ExponentialFamily¶

class ExponentialFamily(batch_shape=torch.Size([]), event_shape=torch.Size([]), validate_args=None)

ExponentialFamily is the abstract base class for probability distributions belonging to an exponential family, whose probability mass/density function has the form is defined below

$p_{F}(x; \theta) = \exp(\langle t(x), \theta\rangle - F(\theta) + k(x))$

where $$\theta$$ denotes the natural parameters, $$t(x)$$ denotes the sufficient statistic, $$F(\theta)$$ is the log normalizer function for a given family and $$k(x)$$ is the carrier measure.

Note

This class is an intermediary between the Distribution class and distributions which belong to an exponential family mainly to check the correctness of the .entropy() and analytic KL divergence methods. We use this class to compute the entropy and KL divergence using the AD framework and Bregman divergences (courtesy of: Frank Nielsen and Richard Nock, Entropies and Cross-entropies of Exponential Families).

### FisherSnedecor¶

class FisherSnedecor(df1, df2, validate_args=None)

Creates a Fisher-Snedecor distribution parameterized by df1 and df2.

Example:

>>> m = FisherSnedecor(torch.tensor([1.0]), torch.tensor([2.0]))
>>> m.sample()  # Fisher-Snedecor-distributed with df1=1 and df2=2
tensor([ 0.2453])

Parameters
• df1 (float or Tensor) – degrees of freedom parameter 1

• df2 (float or Tensor) – degrees of freedom parameter 2

### Gamma¶

class Gamma(concentration, rate, validate_args=None)[source]

Creates a Gamma distribution parameterized by shape concentration and rate.

Example:

>>> m = Gamma(torch.tensor([1.0]), torch.tensor([1.0]))
>>> m.sample()  # Gamma distributed with concentration=1 and rate=1
tensor([ 0.1046])

Parameters
• concentration (float or Tensor) – shape parameter of the distribution (often referred to as alpha)

• rate (float or Tensor) – rate = 1 / scale of the distribution (often referred to as beta)

### Geometric¶

class Geometric(probs=None, logits=None, validate_args=None)[source]

Creates a Geometric distribution parameterized by probs, where probs is the probability of success of Bernoulli trials. It represents the probability that in $$k + 1$$ Bernoulli trials, the first $$k$$ trials failed, before seeing a success.

Samples are non-negative integers [0, $$\inf$$).

Example:

>>> m = Geometric(torch.tensor([0.3]))
>>> m.sample()  # underlying Bernoulli has 30% chance 1; 70% chance 0
tensor([ 2.])

Parameters
• probs (Number, Tensor) – the probability of sampling 1. Must be in range (0, 1]

• logits (Number, Tensor) – the log-odds of sampling 1.

### Gumbel¶

class Gumbel(loc, scale, validate_args=None)

Samples from a Gumbel Distribution.

Examples:

>>> m = Gumbel(torch.tensor([1.0]), torch.tensor([2.0]))
>>> m.sample()  # sample from Gumbel distribution with loc=1, scale=2
tensor([ 1.0124])

Parameters
• loc (float or Tensor) – Location parameter of the distribution

• scale (float or Tensor) – Scale parameter of the distribution

### HalfCauchy¶

class HalfCauchy(scale, validate_args=None)

Creates a half-Cauchy distribution parameterized by scale where:

X ~ Cauchy(0, scale)
Y = |X| ~ HalfCauchy(scale)


Example:

>>> m = HalfCauchy(torch.tensor([1.0]))
>>> m.sample()  # half-cauchy distributed with scale=1
tensor([ 2.3214])

Parameters

scale (float or Tensor) – scale of the full Cauchy distribution

### HalfNormal¶

class HalfNormal(scale, validate_args=None)

Creates a half-normal distribution parameterized by scale where:

X ~ Normal(0, scale)
Y = |X| ~ HalfNormal(scale)


Example:

>>> m = HalfNormal(torch.tensor([1.0]))
>>> m.sample()  # half-normal distributed with scale=1
tensor([ 0.1046])

Parameters

scale (float or Tensor) – scale of the full Normal distribution

### Independent¶

class Independent(base_distribution, reinterpreted_batch_ndims, validate_args=None)[source]

Reinterprets some of the batch dims of a distribution as event dims.

This is mainly useful for changing the shape of the result of log_prob(). For example to create a diagonal Normal distribution with the same shape as a Multivariate Normal distribution (so they are interchangeable), you can:

>>> loc = torch.zeros(3)
>>> scale = torch.ones(3)
>>> mvn = MultivariateNormal(loc, scale_tril=torch.diag(scale))
>>> [mvn.batch_shape, mvn.event_shape]
[torch.Size(()), torch.Size((3,))]
>>> normal = Normal(loc, scale)
>>> [normal.batch_shape, normal.event_shape]
[torch.Size((3,)), torch.Size(())]
>>> diagn = Independent(normal, 1)
>>> [diagn.batch_shape, diagn.event_shape]
[torch.Size(()), torch.Size((3,))]

Parameters

### Kumaraswamy¶

class Kumaraswamy(concentration1, concentration0, validate_args=None)

Samples from a Kumaraswamy distribution.

Example:

>>> m = Kumaraswamy(torch.tensor([1.0]), torch.tensor([1.0]))
>>> m.sample()  # sample from a Kumaraswamy distribution with concentration alpha=1 and beta=1
tensor([ 0.1729])

Parameters
• concentration1 (float or Tensor) – 1st concentration parameter of the distribution (often referred to as alpha)

• concentration0 (float or Tensor) – 2nd concentration parameter of the distribution (often referred to as beta)

### LKJCholesky¶

class LKJCholesky(dim, concentration=1.0, validate_args=None)

LKJ distribution for lower Cholesky factor of correlation matrices. The distribution is controlled by concentration parameter $$\eta$$ to make the probability of the correlation matrix $$M$$ generated from a Cholesky factor propotional to $$\det(M)^{\eta - 1}$$. Because of that, when concentration == 1, we have a uniform distribution over Cholesky factors of correlation matrices. Note that this distribution samples the Cholesky factor of correlation matrices and not the correlation matrices themselves and thereby differs slightly from the derivations in  for the LKJCorr distribution. For sampling, this uses the Onion method from  Section 3.

L ~ LKJCholesky(dim, concentration) X = L @ L’ ~ LKJCorr(dim, concentration)

Example:

>>> l = LKJCholesky(3, 0.5)
>>> l.sample()  # l @ l.T is a sample of a correlation 3x3 matrix
tensor([[ 1.0000,  0.0000,  0.0000],
[ 0.3516,  0.9361,  0.0000],
[-0.1899,  0.4748,  0.8593]])

Parameters
• dimension (dim) – dimension of the matrices

• concentration (float or Tensor) – concentration/shape parameter of the distribution (often referred to as eta)

References

 Generating random correlation matrices based on vines and extended onion method, Daniel Lewandowski, Dorota Kurowicka, Harry Joe.

### Laplace¶

class Laplace(loc, scale, validate_args=None)

Creates a Laplace distribution parameterized by loc and scale.

Example:

>>> m = Laplace(torch.tensor([0.0]), torch.tensor([1.0]))
>>> m.sample()  # Laplace distributed with loc=0, scale=1
tensor([ 0.1046])

Parameters
• loc (float or Tensor) – mean of the distribution

• scale (float or Tensor) – scale of the distribution

### LogNormal¶

class LogNormal(loc, scale, validate_args=None)[source]

Creates a log-normal distribution parameterized by loc and scale where:

X ~ Normal(loc, scale)
Y = exp(X) ~ LogNormal(loc, scale)


Example:

>>> m = LogNormal(torch.tensor([0.0]), torch.tensor([1.0]))
>>> m.sample()  # log-normal distributed with mean=0 and stddev=1
tensor([ 0.1046])

Parameters
• loc (float or Tensor) – mean of log of distribution

• scale (float or Tensor) – standard deviation of log of the distribution

### LogisticNormal¶

class LogisticNormal(loc, scale, validate_args=None)

Wraps torch.distributions.logistic_normal.LogisticNormal with TorchDistributionMixin.

Creates a logistic-normal distribution parameterized by loc and scale that define the base Normal distribution transformed with the StickBreakingTransform such that:

X ~ LogisticNormal(loc, scale)
Y = log(X / (1 - X.cumsum(-1)))[..., :-1] ~ Normal(loc, scale)

Parameters
• loc (float or Tensor) – mean of the base distribution

• scale (float or Tensor) – standard deviation of the base distribution

Example:

>>> # logistic-normal distributed with mean=(0, 0, 0) and stddev=(1, 1, 1)
>>> # of the base Normal distribution
>>> m = distributions.LogisticNormal(torch.tensor([0.0] * 3), torch.tensor([1.0] * 3))
>>> m.sample()
tensor([ 0.7653,  0.0341,  0.0579,  0.1427])


### LowRankMultivariateNormal¶

class LowRankMultivariateNormal(loc, cov_factor, cov_diag, validate_args=None)[source]

Creates a multivariate normal distribution with covariance matrix having a low-rank form parameterized by cov_factor and cov_diag:

covariance_matrix = cov_factor @ cov_factor.T + cov_diag


Example

>>> m = LowRankMultivariateNormal(torch.zeros(2), torch.tensor([[1.], [0.]]), torch.ones(2))
>>> m.sample()  # normally distributed with mean=[0,0], cov_factor=[,], cov_diag=[1,1]
tensor([-0.2102, -0.5429])

Parameters
• loc (Tensor) – mean of the distribution with shape batch_shape + event_shape

• cov_factor (Tensor) – factor part of low-rank form of covariance matrix with shape batch_shape + event_shape + (rank,)

• cov_diag (Tensor) – diagonal part of low-rank form of covariance matrix with shape batch_shape + event_shape

Note

The computation for determinant and inverse of covariance matrix is avoided when cov_factor.shape << cov_factor.shape thanks to Woodbury matrix identity and matrix determinant lemma. Thanks to these formulas, we just need to compute the determinant and inverse of the small size “capacitance” matrix:

capacitance = I + cov_factor.T @ inv(cov_diag) @ cov_factor


### MixtureSameFamily¶

class MixtureSameFamily(mixture_distribution, component_distribution, validate_args=None)

The MixtureSameFamily distribution implements a (batch of) mixture distribution where all component are from different parameterizations of the same distribution type. It is parameterized by a Categorical “selecting distribution” (over k component) and a component distribution, i.e., a Distribution with a rightmost batch shape (equal to [k]) which indexes each (batch of) component.

Examples:

# Construct Gaussian Mixture Model in 1D consisting of 5 equally
# weighted normal distributions
>>> mix = D.Categorical(torch.ones(5,))
>>> comp = D.Normal(torch.randn(5,), torch.rand(5,))
>>> gmm = MixtureSameFamily(mix, comp)

# Construct Gaussian Mixture Modle in 2D consisting of 5 equally
# weighted bivariate normal distributions
>>> mix = D.Categorical(torch.ones(5,))
>>> comp = D.Independent(D.Normal(
torch.randn(5,2), torch.rand(5,2)), 1)
>>> gmm = MixtureSameFamily(mix, comp)

# Construct a batch of 3 Gaussian Mixture Models in 2D each
# consisting of 5 random weighted bivariate normal distributions
>>> mix = D.Categorical(torch.rand(3,5))
>>> comp = D.Independent(D.Normal(
torch.randn(3,5,2), torch.rand(3,5,2)), 1)
>>> gmm = MixtureSameFamily(mix, comp)

Parameters
• mixture_distributiontorch.distributions.Categorical-like instance. Manages the probability of selecting component. The number of categories must match the rightmost batch dimension of the component_distribution. Must have either scalar batch_shape or batch_shape matching component_distribution.batch_shape[:-1]

• component_distributiontorch.distributions.Distribution-like instance. Right-most batch dimension indexes component.

### Multinomial¶

class Multinomial(total_count=1, probs=None, logits=None, validate_args=None)[source]

Creates a Multinomial distribution parameterized by total_count and either probs or logits (but not both). The innermost dimension of probs indexes over categories. All other dimensions index over batches.

Note that total_count need not be specified if only log_prob() is called (see example below)

Note

The probs argument must be non-negative, finite and have a non-zero sum, and it will be normalized to sum to 1 along the last dimension. probs will return this normalized value. The logits argument will be interpreted as unnormalized log probabilities and can therefore be any real number. It will likewise be normalized so that the resulting probabilities sum to 1 along the last dimension. logits will return this normalized value.

• sample() requires a single shared total_count for all parameters and samples.

• log_prob() allows different total_count for each parameter and sample.

Example:

>>> m = Multinomial(100, torch.tensor([ 1., 1., 1., 1.]))
>>> x = m.sample()  # equal probability of 0, 1, 2, 3
tensor([ 21.,  24.,  30.,  25.])

>>> Multinomial(probs=torch.tensor([1., 1., 1., 1.])).log_prob(x)
tensor([-4.1338])

Parameters
• total_count (int) – number of trials

• probs (Tensor) – event probabilities

• logits (Tensor) – event log probabilities (unnormalized)

### MultivariateNormal¶

class MultivariateNormal(loc, covariance_matrix=None, precision_matrix=None, scale_tril=None, validate_args=None)[source]

Creates a multivariate normal (also called Gaussian) distribution parameterized by a mean vector and a covariance matrix.

The multivariate normal distribution can be parameterized either in terms of a positive definite covariance matrix $$\mathbf{\Sigma}$$ or a positive definite precision matrix $$\mathbf{\Sigma}^{-1}$$ or a lower-triangular matrix $$\mathbf{L}$$ with positive-valued diagonal entries, such that $$\mathbf{\Sigma} = \mathbf{L}\mathbf{L}^\top$$. This triangular matrix can be obtained via e.g. Cholesky decomposition of the covariance.

Example

>>> m = MultivariateNormal(torch.zeros(2), torch.eye(2))
>>> m.sample()  # normally distributed with mean=[0,0] and covariance_matrix=I
tensor([-0.2102, -0.5429])

Parameters
• loc (Tensor) – mean of the distribution

• covariance_matrix (Tensor) – positive-definite covariance matrix

• precision_matrix (Tensor) – positive-definite precision matrix

• scale_tril (Tensor) – lower-triangular factor of covariance, with positive-valued diagonal

Note

Only one of covariance_matrix or precision_matrix or scale_tril can be specified.

Using scale_tril will be more efficient: all computations internally are based on scale_tril. If covariance_matrix or precision_matrix is passed instead, it is only used to compute the corresponding lower triangular matrices using a Cholesky decomposition.

### NegativeBinomial¶

class NegativeBinomial(total_count, probs=None, logits=None, validate_args=None)

Creates a Negative Binomial distribution, i.e. distribution of the number of successful independent and identical Bernoulli trials before total_count failures are achieved. The probability of success of each Bernoulli trial is probs.

Parameters
• total_count (float or Tensor) – non-negative number of negative Bernoulli trials to stop, although the distribution is still valid for real valued count

• probs (Tensor) – Event probabilities of success in the half open interval [0, 1)

• logits (Tensor) – Event log-odds for probabilities of success

### Normal¶

class Normal(loc, scale, validate_args=None)[source]

Creates a normal (also called Gaussian) distribution parameterized by loc and scale.

Example:

>>> m = Normal(torch.tensor([0.0]), torch.tensor([1.0]))
>>> m.sample()  # normally distributed with loc=0 and scale=1
tensor([ 0.1046])

Parameters
• loc (float or Tensor) – mean of the distribution (often referred to as mu)

• scale (float or Tensor) – standard deviation of the distribution (often referred to as sigma)

### OneHotCategorical¶

class OneHotCategorical(probs=None, logits=None, validate_args=None)[source]

Creates a one-hot categorical distribution parameterized by probs or logits.

Samples are one-hot coded vectors of size probs.size(-1).

Note

The probs argument must be non-negative, finite and have a non-zero sum, and it will be normalized to sum to 1 along the last dimension. probs will return this normalized value. The logits argument will be interpreted as unnormalized log probabilities and can therefore be any real number. It will likewise be normalized so that the resulting probabilities sum to 1 along the last dimension. logits will return this normalized value.

See also: torch.distributions.Categorical() for specifications of probs and logits.

Example:

>>> m = OneHotCategorical(torch.tensor([ 0.25, 0.25, 0.25, 0.25 ]))
>>> m.sample()  # equal probability of 0, 1, 2, 3
tensor([ 0.,  0.,  0.,  1.])

Parameters
• probs (Tensor) – event probabilities

• logits (Tensor) – event log probabilities (unnormalized)

### OneHotCategoricalStraightThrough¶

class OneHotCategoricalStraightThrough(probs=None, logits=None, validate_args=None)

Wraps torch.distributions.one_hot_categorical.OneHotCategoricalStraightThrough with TorchDistributionMixin.

Creates a reparameterizable OneHotCategorical distribution based on the straight- through gradient estimator from .

 Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation (Bengio et al, 2013)

### Pareto¶

class Pareto(scale, alpha, validate_args=None)

Samples from a Pareto Type 1 distribution.

Example:

>>> m = Pareto(torch.tensor([1.0]), torch.tensor([1.0]))
>>> m.sample()  # sample from a Pareto distribution with scale=1 and alpha=1
tensor([ 1.5623])

Parameters
• scale (float or Tensor) – Scale parameter of the distribution

• alpha (float or Tensor) – Shape parameter of the distribution

### Poisson¶

class Poisson(rate, *, is_sparse=False, validate_args=None)[source]

Creates a Poisson distribution parameterized by rate, the rate parameter.

Samples are nonnegative integers, with a pmf given by

$\mathrm{rate}^k \frac{e^{-\mathrm{rate}}}{k!}$

Example:

>>> m = Poisson(torch.tensor())
>>> m.sample()
tensor([ 3.])

Parameters

rate (Number, Tensor) – the rate parameter

### RelaxedBernoulli¶

class RelaxedBernoulli(temperature, probs=None, logits=None, validate_args=None)

Creates a RelaxedBernoulli distribution, parametrized by temperature, and either probs or logits (but not both). This is a relaxed version of the Bernoulli distribution, so the values are in (0, 1), and has reparametrizable samples.

Example:

>>> m = RelaxedBernoulli(torch.tensor([2.2]),
torch.tensor([0.1, 0.2, 0.3, 0.99]))
>>> m.sample()
tensor([ 0.2951,  0.3442,  0.8918,  0.9021])

Parameters
• temperature (Tensor) – relaxation temperature

• probs (Number, Tensor) – the probability of sampling 1

• logits (Number, Tensor) – the log-odds of sampling 1

### RelaxedOneHotCategorical¶

class RelaxedOneHotCategorical(temperature, probs=None, logits=None, validate_args=None)

Creates a RelaxedOneHotCategorical distribution parametrized by temperature, and either probs or logits. This is a relaxed version of the OneHotCategorical distribution, so its samples are on simplex, and are reparametrizable.

Example:

>>> m = RelaxedOneHotCategorical(torch.tensor([2.2]),
torch.tensor([0.1, 0.2, 0.3, 0.4]))
>>> m.sample()
tensor([ 0.1294,  0.2324,  0.3859,  0.2523])

Parameters
• temperature (Tensor) – relaxation temperature

• probs (Tensor) – event probabilities

• logits (Tensor) – unnormalized log probability for each event

### StudentT¶

class StudentT(df, loc=0.0, scale=1.0, validate_args=None)

Creates a Student’s t-distribution parameterized by degree of freedom df, mean loc and scale scale.

Example:

>>> m = StudentT(torch.tensor([2.0]))
>>> m.sample()  # Student's t-distributed with degrees of freedom=2
tensor([ 0.1046])

Parameters
• df (float or Tensor) – degrees of freedom

• loc (float or Tensor) – mean of the distribution

• scale (float or Tensor) – scale of the distribution

### TransformedDistribution¶

class TransformedDistribution(base_distribution, transforms, validate_args=None)

Extension of the Distribution class, which applies a sequence of Transforms to a base distribution. Let f be the composition of transforms applied:

X ~ BaseDistribution
Y = f(X) ~ TransformedDistribution(BaseDistribution, f)
log p(Y) = log p(X) + log |det (dX/dY)|


Note that the .event_shape of a TransformedDistribution is the maximum shape of its base distribution and its transforms, since transforms can introduce correlations among events.

An example for the usage of TransformedDistribution would be:

# Building a Logistic Distribution
# X ~ Uniform(0, 1)
# f = a + b * logit(X)
# Y ~ f(X) ~ Logistic(a, b)
base_distribution = Uniform(0, 1)
transforms = [SigmoidTransform().inv, AffineTransform(loc=a, scale=b)]
logistic = TransformedDistribution(base_distribution, transforms)


For more examples, please look at the implementations of Gumbel, HalfCauchy, HalfNormal, LogNormal, Pareto, Weibull, RelaxedBernoulli and RelaxedOneHotCategorical

### Uniform¶

class Uniform(low, high, validate_args=None)[source]

Generates uniformly distributed random samples from the half-open interval [low, high).

Example:

>>> m = Uniform(torch.tensor([0.0]), torch.tensor([5.0]))
>>> m.sample()  # uniformly distributed in the range [0.0, 5.0)
tensor([ 2.3418])

Parameters
• low (float or Tensor) – lower range (inclusive).

• high (float or Tensor) – upper range (exclusive).

### VonMises¶

class VonMises(loc, concentration, validate_args=None)

A circular von Mises distribution.

This implementation uses polar coordinates. The loc and value args can be any real number (to facilitate unconstrained optimization), but are interpreted as angles modulo 2 pi.

Example::
>>> m = dist.VonMises(torch.tensor([1.0]), torch.tensor([1.0]))
>>> m.sample() # von Mises distributed with loc=1 and concentration=1
tensor([1.9777])

Parameters

### Weibull¶

class Weibull(scale, concentration, validate_args=None)

Samples from a two-parameter Weibull distribution.

Example

>>> m = Weibull(torch.tensor([1.0]), torch.tensor([1.0]))
>>> m.sample()  # sample from a Weibull distribution with scale=1, concentration=1
tensor([ 0.4784])

Parameters
• scale (float or Tensor) – Scale parameter of distribution (lambda).

• concentration (float or Tensor) – Concentration parameter of distribution (k/shape).

### Wishart¶

class Wishart(df: Union[torch.Tensor, numbers.Number], covariance_matrix: torch.Tensor = None, precision_matrix: torch.Tensor = None, scale_tril: torch.Tensor = None, validate_args=None)

Creates a Wishart distribution parameterized by a symmetric positive definite matrix $$\Sigma$$, or its Cholesky decomposition $$\mathbf{\Sigma} = \mathbf{L}\mathbf{L}^\top$$

Example

>>> m = Wishart(torch.eye(2), torch.Tensor())
>>> m.sample()  #Wishart distributed with mean=df * I and
#variance(x_ij)=df for i != j and variance(x_ij)=2 * df for i == j

Parameters
• covariance_matrix (Tensor) – positive-definite covariance matrix

• precision_matrix (Tensor) – positive-definite precision matrix

• scale_tril (Tensor) – lower-triangular factor of covariance, with positive-valued diagonal

• df (float or Tensor) – real-valued parameter larger than the (dimension of Square matrix) - 1

Note

Only one of covariance_matrix or precision_matrix or scale_tril can be specified. Using scale_tril will be more efficient: all computations internally are based on scale_tril. If covariance_matrix or precision_matrix is passed instead, it is only used to compute the corresponding lower triangular matrices using a Cholesky decomposition. ‘torch.distributions.LKJCholesky’ is a restricted Wishart distribution.

References

 On equivalence of the LKJ distribution and the restricted Wishart distribution, Zhenxun Wang, Yunan Wu, Haitao Chu.

## Pyro Distributions¶

### Abstract Distribution¶

class Distribution(*args, **kwargs)[source]

Bases: object

Base class for parameterized probability distributions.

Distributions in Pyro are stochastic function objects with sample() and log_prob() methods. Distribution are stochastic functions with fixed parameters:

d = dist.Bernoulli(param)
x = d()                                # Draws a random sample.
p = d.log_prob(x)                      # Evaluates log probability of x.


Implementing New Distributions:

Derived classes must implement the methods: sample(), log_prob().

Examples:

Take a look at the examples to see how they interact with inference algorithms.

has_rsample = False
has_enumerate_support = False
__call__(*args, **kwargs)[source]

Samples a random value (just an alias for .sample(*args, **kwargs)).

For tensor distributions, the returned tensor should have the same .shape as the parameters.

Returns

A random value.

Return type

torch.Tensor

abstract sample(*args, **kwargs)[source]

Samples a random value.

For tensor distributions, the returned tensor should have the same .shape as the parameters, unless otherwise noted.

Parameters

sample_shape (torch.Size) – the size of the iid batch to be drawn from the distribution.

Returns

A random value or batch of random values (if parameters are batched). The shape of the result should be self.shape().

Return type

torch.Tensor

abstract log_prob(x, *args, **kwargs)[source]

Evaluates log probability densities for each of a batch of samples.

Parameters

x (torch.Tensor) – A single value or a batch of values batched along axis 0.

Returns

log probability densities as a one-dimensional Tensor with same batch size as value and params. The shape of the result should be self.batch_size.

Return type

torch.Tensor

score_parts(x, *args, **kwargs)[source]

Computes ingredients for stochastic gradient estimators of ELBO.

The default implementation is correct both for non-reparameterized and for fully reparameterized distributions. Partially reparameterized distributions should override this method to compute correct .score_function and .entropy_term parts.

Setting .has_rsample on a distribution instance will determine whether inference engines like SVI use reparameterized samplers or the score function estimator.

Parameters

x (torch.Tensor) – A single value or batch of values.

Returns

A ScoreParts object containing parts of the ELBO estimator.

Return type

ScoreParts

enumerate_support(expand=True)[source]

Returns a representation of the parametrized distribution’s support, along the first dimension. This is implemented only by discrete distributions.

Note that this returns support values of all the batched RVs in lock-step, rather than the full cartesian product.

Parameters

expand (bool) – whether to expand the result to a tensor of shape (n,) + batch_shape + event_shape. If false, the return value has unexpanded shape (n,) + (1,)*len(batch_shape) + event_shape which can be broadcasted to the full shape.

Returns

An iterator over the distribution’s discrete support.

Return type

iterator

conjugate_update(other)[source]

EXPERIMENTAL Creates an updated distribution fusing information from another compatible distribution. This is supported by only a few conjugate distributions.

This should satisfy the equation:

fg, log_normalizer = f.conjugate_update(g)
assert f.log_prob(x) + g.log_prob(x) == fg.log_prob(x) + log_normalizer


Note this is equivalent to funsor.ops.add on Funsor distributions, but we return a lazy sum (updated, log_normalizer) because PyTorch distributions must be normalized. Thus conjugate_update() should commute with dist_to_funsor() and tensor_to_funsor()

dist_to_funsor(f) + dist_to_funsor(g)
== dist_to_funsor(fg) + tensor_to_funsor(log_normalizer)

Parameters

other – A distribution representing p(data|latent) but normalized over latent rather than data. Here latent is a candidate sample from self and data is a ground observation of unrelated type.

Returns

a pair (updated,log_normalizer) where updated is an updated distribution of type type(self), and log_normalizer is a Tensor representing the normalization factor.

has_rsample_(value)[source]

Force reparameterized or detached sampling on a single distribution instance. This sets the .has_rsample attribute in-place.

This is useful to instruct inference algorithms to avoid reparameterized gradients for variables that discontinuously determine downstream control flow.

Parameters

value (bool) – Whether samples will be pathwise differentiable.

Returns

self

Return type

Distribution

property rv

EXPERIMENTAL Switch to the Random Variable DSL for applying transformations to random variables. Supports either chaining operations or arithmetic operator overloading.

Example usage:

# This should be equivalent to an Exponential distribution.
Uniform(0, 1).rv.log().neg().dist

# These two distributions Y1, Y2 should be the same
X = Uniform(0, 1).rv
Y1 = X.mul(4).pow(0.5).sub(1).abs().neg().dist
Y2 = (-abs((4*X)**(0.5) - 1)).dist

Returns

A :class: ~pyro.contrib.randomvariable.random_variable.RandomVariable object wrapping this distribution.

Return type

RandomVariable

### TorchDistributionMixin¶

class TorchDistributionMixin(*args, **kwargs)[source]

Mixin to provide Pyro compatibility for PyTorch distributions.

You should instead use TorchDistribution for new distribution classes.

This is mainly useful for wrapping existing PyTorch distributions for use in Pyro. Derived classes must first inherit from torch.distributions.distribution.Distribution and then inherit from TorchDistributionMixin.

__call__(sample_shape=torch.Size([]))[source]

Samples a random value.

This is reparameterized whenever possible, calling rsample() for reparameterized distributions and sample() for non-reparameterized distributions.

Parameters

sample_shape (torch.Size) – the size of the iid batch to be drawn from the distribution.

Returns

A random value or batch of random values (if parameters are batched). The shape of the result should be self.shape().

Return type

torch.Tensor

property event_dim

Number of dimensions of individual events. :rtype: int

Type

return

shape(sample_shape=torch.Size([]))[source]

The tensor shape of samples from this distribution.

Samples are of shape:

d.shape(sample_shape) == sample_shape + d.batch_shape + d.event_shape

Parameters

sample_shape (torch.Size) – the size of the iid batch to be drawn from the distribution.

Returns

Tensor shape of samples.

Return type

torch.Size

classmethod infer_shapes(**arg_shapes)[source]

Infers batch_shape and event_shape given shapes of args to __init__().

Note

This assumes distribution shape depends only on the shapes of tensor inputs, not in the data contained in those inputs.

Parameters

**arg_shapes – Keywords mapping name of input arg to torch.Size or tuple representing the sizes of each tensor input.

Returns

A pair (batch_shape, event_shape) of the shapes of a distribution that would be created with input args of the given shapes.

Return type

tuple

expand(batch_shape, _instance=None)[source]

Returns a new ExpandedDistribution instance with batch dimensions expanded to batch_shape.

Parameters
• batch_shape (tuple) – batch shape to expand to.

• _instance – unused argument for compatibility with torch.distributions.Distribution.expand()

Returns

an instance of ExpandedDistribution.

Return type

ExpandedDistribution

expand_by(sample_shape)[source]

Expands a distribution by adding sample_shape to the left side of its batch_shape.

To expand internal dims of self.batch_shape from 1 to something larger, use expand() instead.

Parameters

sample_shape (torch.Size) – The size of the iid batch to be drawn from the distribution.

Returns

An expanded version of this distribution.

Return type

ExpandedDistribution

reshape(sample_shape=None, extra_event_dims=None)[source]
to_event(reinterpreted_batch_ndims=None)[source]

Reinterprets the n rightmost dimensions of this distributions batch_shape as event dims, adding them to the left side of event_shape.

Example

>>> [d1.batch_shape, d1.event_shape]
[torch.Size([2, 3]), torch.Size([4, 5])]
>>> d2 = d1.to_event(1)
>>> [d2.batch_shape, d2.event_shape]
[torch.Size(), torch.Size([3, 4, 5])]
>>> d3 = d1.to_event(2)
>>> [d3.batch_shape, d3.event_shape]
[torch.Size([]), torch.Size([2, 3, 4, 5])]

Parameters

reinterpreted_batch_ndims (int) – The number of batch dimensions to reinterpret as event dimensions. May be negative to remove dimensions from an pyro.distributions.torch.Independent . If None, convert all dimensions to event dimensions.

Returns

A reshaped version of this distribution.

Return type

pyro.distributions.torch.Independent

independent(reinterpreted_batch_ndims=None)[source]

Masks a distribution by a boolean or boolean-valued tensor that is broadcastable to the distributions batch_shape .

Parameters

mask (bool or torch.Tensor) – A boolean or boolean valued tensor.

Returns

A masked copy of this distribution.

Return type

MaskedDistribution

### TorchDistribution¶

class TorchDistribution(batch_shape=torch.Size([]), event_shape=torch.Size([]), validate_args=None)[source]

Base class for PyTorch-compatible distributions with Pyro support.

This should be the base class for almost all new Pyro distributions.

Note

Parameters and data should be of type Tensor and all methods return type Tensor unless otherwise noted.

Tensor Shapes:

TorchDistributions provide a method .shape() for the tensor shape of samples:

x = d.sample(sample_shape)
assert x.shape == d.shape(sample_shape)


Pyro follows the same distribution shape semantics as PyTorch. It distinguishes between three different roles for tensor shapes of samples:

• sample shape corresponds to the shape of the iid samples drawn from the distribution. This is taken as an argument by the distribution’s sample method.

• batch shape corresponds to non-identical (independent) parameterizations of the distribution, inferred from the distribution’s parameter shapes. This is fixed for a distribution instance.

• event shape corresponds to the event dimensions of the distribution, which is fixed for a distribution class. These are collapsed when we try to score a sample from the distribution via d.log_prob(x).

These shapes are related by the equation:

assert d.shape(sample_shape) == sample_shape + d.batch_shape + d.event_shape


Distributions provide a vectorized log_prob() method that evaluates the log probability density of each event in a batch independently, returning a tensor of shape sample_shape + d.batch_shape:

x = d.sample(sample_shape)
assert x.shape == d.shape(sample_shape)
log_p = d.log_prob(x)
assert log_p.shape == sample_shape + d.batch_shape


Implementing New Distributions:

Derived classes must implement the methods sample() (or rsample() if .has_rsample == True) and log_prob(), and must implement the properties batch_shape, and event_shape. Discrete classes may also implement the enumerate_support() method to improve gradient estimates and set .has_enumerate_support = True.

expand(batch_shape, _instance=None)

Returns a new ExpandedDistribution instance with batch dimensions expanded to batch_shape.

Parameters
• batch_shape (tuple) – batch shape to expand to.

• _instance – unused argument for compatibility with torch.distributions.Distribution.expand()

Returns

an instance of ExpandedDistribution.

Return type

ExpandedDistribution

### AffineBeta¶

class AffineBeta(concentration1, concentration0, loc, scale, validate_args=None)[source]

Beta distribution scaled by scale and shifted by loc:

X ~ Beta(concentration1, concentration0)
f(X) = loc + scale * X
Y = f(X) ~ AffineBeta(concentration1, concentration0, loc, scale)

Parameters
arg_constraints: Dict[str, torch.distributions.constraints.Constraint] = {'concentration0': GreaterThan(lower_bound=0.0), 'concentration1': GreaterThan(lower_bound=0.0), 'loc': Real(), 'scale': GreaterThan(lower_bound=0.0)}
property concentration0
property concentration1
expand(batch_shape, _instance=None)[source]
property high
static infer_shapes(concentration1, concentration0, loc, scale)[source]
property loc
property low
property mean
rsample(sample_shape=torch.Size([]))[source]

Generates a sample from Beta distribution and applies AffineTransform. Additionally clamps the output in order to avoid NaN and Inf values in the gradients.

sample(sample_shape=torch.Size([]))[source]

Generates a sample from Beta distribution and applies AffineTransform. Additionally clamps the output in order to avoid NaN and Inf values in the gradients.

property sample_size
property scale
property support
property variance

### AsymmetricLaplace¶

class AsymmetricLaplace(loc, scale, asymmetry, *, validate_args=None)[source]

Asymmetric version of the Laplace distribution.

To the left of loc this acts like an -Exponential(1/(asymmetry*scale)); to the right of loc this acts like an Exponential(asymmetry/scale). The density is continuous so the left and right densities at loc agree.

Parameters
• loc – Location parameter, i.e. the mode.

• scale – Scale parameter = geometric mean of left and right scales.

• asymmetry – Square of ratio of left to right scales.

arg_constraints = {'asymmetry': GreaterThan(lower_bound=0.0), 'loc': Real(), 'scale': GreaterThan(lower_bound=0.0)}
expand(batch_shape, _instance=None)[source]
has_rsample = True
property left_scale
log_prob(value)[source]
property mean
property right_scale
rsample(sample_shape=torch.Size([]))[source]
support = Real()
property variance

### AVFMultivariateNormal¶

class AVFMultivariateNormal(loc, scale_tril, control_var)[source]

Multivariate normal (Gaussian) distribution with transport equation inspired control variates (adaptive velocity fields).

A distribution over vectors in which all the elements have a joint Gaussian density.

Parameters
• loc (torch.Tensor) – D-dimensional mean vector.

• scale_tril (torch.Tensor) – Cholesky of Covariance matrix; D x D matrix.

• control_var (torch.Tensor) – 2 x L x D tensor that parameterizes the control variate; L is an arbitrary positive integer. This parameter needs to be learned (i.e. adapted) to achieve lower variance gradients. In a typical use case this parameter will be adapted concurrently with the loc and scale_tril that define the distribution.

Example usage:

control_var = torch.tensor(0.1 * torch.ones(2, 1, D), requires_grad=True)
opt_cv = torch.optim.Adam([control_var], lr=0.1, betas=(0.5, 0.999))

for _ in range(1000):
d = AVFMultivariateNormal(loc, scale_tril, control_var)
z = d.rsample()
cost = torch.pow(z, 2.0).sum()
cost.backward()
opt_cv.step()

arg_constraints = {'control_var': Real(), 'loc': Real(), 'scale_tril': LowerTriangular()}
rsample(sample_shape=torch.Size([]))[source]

### BetaBinomial¶

class BetaBinomial(concentration1, concentration0, total_count=1, validate_args=None)[source]

Compound distribution comprising of a beta-binomial pair. The probability of success (probs for the Binomial distribution) is unknown and randomly drawn from a Beta distribution prior to a certain number of Bernoulli trials given by total_count.

Parameters
• concentration1 (float or torch.Tensor) – 1st concentration parameter (alpha) for the Beta distribution.

• concentration0 (float or torch.Tensor) – 2nd concentration parameter (beta) for the Beta distribution.

• total_count (float or torch.Tensor) – Number of Bernoulli trials.

approx_log_prob_tol = 0.0
arg_constraints = {'concentration0': GreaterThan(lower_bound=0.0), 'concentration1': GreaterThan(lower_bound=0.0), 'total_count': IntegerGreaterThan(lower_bound=0)}
property concentration0
property concentration1
enumerate_support(expand=True)[source]
expand(batch_shape, _instance=None)[source]
has_enumerate_support = True
log_prob(value)[source]
property mean
sample(sample_shape=())[source]
property support
property variance

### CoalescentTimes¶

class CoalescentTimes(leaf_times, rate=1.0, *, validate_args=None)[source]

Distribution over sorted coalescent times given irregular sampled leaf_times and constant population size.

Sample values will be sorted sets of binary coalescent times. Each sample value will have cardinality value.size(-1) = leaf_times.size(-1) - 1, so that phylogenies are complete binary trees. This distribution can thus be batched over multiple samples of phylogenies given fixed (number of) leaf times, e.g. over phylogeny samples from BEAST or MrBayes.

References

 J.F.C. Kingman (1982)

“On the Genealogy of Large Populations” Journal of Applied Probability

 J.F.C. Kingman (1982)

“The Coalescent” Stochastic Processes and their Applications

Parameters
• leaf_times (torch.Tensor) – Vector of times of sampling events, i.e. leaf nodes in the phylogeny. These can be arbitrary real numbers with arbitrary order and duplicates.

• rate (torch.Tensor) – Base coalescent rate (pairwise rate of coalescence) under a constant population size model. Defaults to 1.

arg_constraints = {'leaf_times': Real(), 'rate': GreaterThan(lower_bound=0.0)}
log_prob(value)[source]
sample(sample_shape=torch.Size([]))[source]
property support

### CoalescentTimesWithRate¶

class CoalescentTimesWithRate(leaf_times, rate_grid, *, validate_args=None)[source]

Distribution over coalescent times given irregular sampled leaf_times and piecewise constant coalescent rates defined on a regular time grid.

This assumes a piecewise constant base coalescent rate specified on time intervals (-inf,1], [1,2], …, [T-1,inf), where T = rate_grid.size(-1). Leaves may be sampled at arbitrary real times, but are commonly sampled in the interval [0, T].

Sample values will be sorted sets of binary coalescent times. Each sample value will have cardinality value.size(-1) = leaf_times.size(-1) - 1, so that phylogenies are complete binary trees. This distribution can thus be batched over multiple samples of phylogenies given fixed (number of) leaf times, e.g. over phylogeny samples from BEAST or MrBayes.

This distribution implements log_prob() but not .sample().

See also CoalescentRateLikelihood.

References

 J.F.C. Kingman (1982)

“On the Genealogy of Large Populations” Journal of Applied Probability

 J.F.C. Kingman (1982)

“The Coalescent” Stochastic Processes and their Applications

 A. Popinga, T. Vaughan, T. Statler, A.J. Drummond (2014)

“Inferring epidemiological dynamics with Bayesian coalescent inference: The merits of deterministic and stochastic models” https://arxiv.org/pdf/1407.1792.pdf

Parameters
• leaf_times (torch.Tensor) – Tensor of times of sampling events, i.e. leaf nodes in the phylogeny. These can be arbitrary real numbers with arbitrary order and duplicates.

• rate_grid (torch.Tensor) – Tensor of base coalescent rates (pairwise rate of coalescence). For example in a simple SIR model this might be beta S / I. The rightmost dimension is time, and this tensor represents a (batch of) rates that are piecewise constant in time.

arg_constraints = {'leaf_times': Real(), 'rate_grid': GreaterThan(lower_bound=0.0)}
property duration
expand(batch_shape, _instance=None)[source]
log_prob(value)[source]

Computes likelihood as in equations 7-8 of .

This has time complexity O(T + S N log(N)) where T is the number of time steps, N is the number of leaves, and S = sample_shape.numel() is the number of samples of value.

Parameters

value (torch.Tensor) – A tensor of coalescent times. These denote sets of size leaf_times.size(-1) - 1 along the trailing dimension and should be sorted along that dimension.

Returns

Likelihood p(coal_times | leaf_times, rate_grid)

Return type

torch.Tensor

property support

### ConditionalDistribution¶

class ConditionalDistribution[source]

Bases: abc.ABC

abstract condition(context)[source]
Return type

torch.distributions.Distribution

### ConditionalTransformedDistribution¶

class ConditionalTransformedDistribution(base_dist, transforms)[source]
clear_cache()[source]
condition(context)[source]

### Delta¶

class Delta(v, log_density=0.0, event_dim=0, validate_args=None)[source]

Degenerate discrete distribution (a single point).

Discrete distribution that assigns probability one to the single element in its support. Delta distribution parameterized by a random choice should not be used with MCMC based inference, as doing so produces incorrect results.

Parameters
• v (torch.Tensor) – The single support element.

• log_density (torch.Tensor) – An optional density for this Delta. This is useful to keep the class of Delta distributions closed under differentiable transformation.

• event_dim (int) – Optional event dimension, defaults to zero.

arg_constraints = {'log_density': Real(), 'v': Dependent()}
expand(batch_shape, _instance=None)[source]
has_rsample = True
log_prob(x)[source]
property mean
rsample(sample_shape=torch.Size([]))[source]
property support
property variance

### DirichletMultinomial¶

class DirichletMultinomial(concentration, total_count=1, is_sparse=False, validate_args=None)[source]

Compound distribution comprising of a dirichlet-multinomial pair. The probability of classes (probs for the Multinomial distribution) is unknown and randomly drawn from a Dirichlet distribution prior to a certain number of Categorical trials given by total_count.

Parameters
• concentration (float or torch.Tensor) – concentration parameter (alpha) for the Dirichlet distribution.

• total_count (int or torch.Tensor) – number of Categorical trials.

• is_sparse (bool) – Whether to assume value is mostly zero when computing log_prob(), which can speed up computation when data is sparse.

arg_constraints = {'concentration': IndependentConstraint(GreaterThan(lower_bound=0.0), 1), 'total_count': IntegerGreaterThan(lower_bound=0)}
property concentration
expand(batch_shape, _instance=None)[source]
static infer_shapes(concentration, total_count=())[source]
log_prob(value)[source]
property mean
sample(sample_shape=())[source]
property support
property variance

### DiscreteHMM¶

class DiscreteHMM(initial_logits, transition_logits, observation_dist, validate_args=None, duration=None)[source]

Bases: pyro.distributions.hmm.HiddenMarkovModel

Hidden Markov Model with discrete latent state and arbitrary observation distribution.

This uses  to parallelize over time, achieving O(log(time)) parallel complexity for computing log_prob(), filter(), and sample().

The event_shape of this distribution includes time on the left:

event_shape = (num_steps,) + observation_dist.event_shape


This distribution supports any combination of homogeneous/heterogeneous time dependency of transition_logits and observation_dist. However, because time is included in this distribution’s event_shape, the homogeneous+homogeneous case will have a broadcastable event_shape with num_steps = 1, allowing log_prob() to work with arbitrary length data:

# homogeneous + homogeneous case:
event_shape = (1,) + observation_dist.event_shape


References:

 Simo Sarkka, Angel F. Garcia-Fernandez (2019)

“Temporal Parallelization of Bayesian Filters and Smoothers” https://arxiv.org/pdf/1905.13002.pdf

Parameters
• initial_logits (Tensor) – A logits tensor for an initial categorical distribution over latent states. Should have rightmost size state_dim and be broadcastable to batch_shape + (state_dim,).

• transition_logits (Tensor) – A logits tensor for transition conditional distributions between latent states. Should have rightmost shape (state_dim, state_dim) (old, new), and be broadcastable to batch_shape + (num_steps, state_dim, state_dim).

• observation_dist (Distribution) – A conditional distribution of observed data conditioned on latent state. The .batch_shape should have rightmost size state_dim and be broadcastable to batch_shape + (num_steps, state_dim). The .event_shape may be arbitrary.

• duration (int) – Optional size of the time axis event_shape. This is required when sampling from homogeneous HMMs whose parameters are not expanded along the time axis.

arg_constraints = {'initial_logits': Real(), 'transition_logits': Real()}
expand(batch_shape, _instance=None)[source]
filter(value)[source]

Compute posterior over final state given a sequence of observations.

Parameters

value (Tensor) – A sequence of observations.

Returns

A posterior distribution over latent states at the final time step. result.logits can then be used as initial_logits in a sequential Pyro model for prediction.

Return type

Categorical

log_prob(value)[source]
sample(sample_shape=torch.Size([]))[source]
property support

### EmpiricalDistribution¶

class Empirical(samples, log_weights, validate_args=None)[source]

Empirical distribution associated with the sampled data. Note that the shape requirement for log_weights is that its shape must match the leftmost shape of samples. Samples are aggregated along the aggregation_dim, which is the rightmost dim of log_weights.

Example:

>>> emp_dist = Empirical(torch.randn(2, 3, 10), torch.ones(2, 3))
>>> emp_dist.batch_shape
torch.Size()
>>> emp_dist.event_shape
torch.Size()

>>> single_sample = emp_dist.sample()
>>> single_sample.shape
torch.Size([2, 10])
>>> batch_sample = emp_dist.sample((100,))
>>> batch_sample.shape
torch.Size([100, 2, 10])

>>> emp_dist.log_prob(single_sample).shape
torch.Size()
>>> # Vectorized samples cannot be scored by log_prob.
>>> with pyro.validation_enabled():
...     emp_dist.log_prob(batch_sample).shape
Traceback (most recent call last):
...
ValueError: value.shape must be torch.Size([2, 10])

Parameters
• samples (torch.Tensor) – samples from the empirical distribution.

• log_weights (torch.Tensor) – log weights (optional) corresponding to the samples.

arg_constraints = {}
enumerate_support(expand=True)[source]

See pyro.distributions.torch_distribution.TorchDistribution.enumerate_support()

property event_shape

See pyro.distributions.torch_distribution.TorchDistribution.event_shape()

has_enumerate_support = True
log_prob(value)[source]

Returns the log of the probability mass function evaluated at value. Note that this currently only supports scoring values with empty sample_shape.

Parameters

value (torch.Tensor) – scalar or tensor value to be scored.

property log_weights
property mean

See pyro.distributions.torch_distribution.TorchDistribution.mean()

sample(sample_shape=torch.Size([]))[source]

See pyro.distributions.torch_distribution.TorchDistribution.sample()

property sample_size

Number of samples that constitute the empirical distribution.

Return int

number of samples collected.

support = Real()
property variance

See pyro.distributions.torch_distribution.TorchDistribution.variance()

### ExtendedBetaBinomial¶

class ExtendedBetaBinomial(concentration1, concentration0, total_count=1, validate_args=None)[source]

EXPERIMENTAL BetaBinomial distribution extended to have logical support the entire integers and to allow arbitrary integer total_count. Numerical support is still the integer interval [0, total_count].

arg_constraints = {'concentration0': GreaterThan(lower_bound=0.0), 'concentration1': GreaterThan(lower_bound=0.0), 'total_count': Integer}
log_prob(value)[source]
support = Integer

### ExtendedBinomial¶

class ExtendedBinomial(total_count=1, probs=None, logits=None, validate_args=None)[source]

EXPERIMENTAL Binomial distribution extended to have logical support the entire integers and to allow arbitrary integer total_count. Numerical support is still the integer interval [0, total_count].

arg_constraints = {'logits': Real(), 'probs': Interval(lower_bound=0.0, upper_bound=1.0), 'total_count': Integer}
log_prob(value)[source]
support = Integer

### FoldedDistribution¶

class FoldedDistribution(base_dist, validate_args=None)[source]

Equivalent to TransformedDistribution(base_dist, AbsTransform()), but additionally supports log_prob() .

Parameters

base_dist (Distribution) – The distribution to reflect.

expand(batch_shape, _instance=None)[source]
log_prob(value)[source]
support = GreaterThan(lower_bound=0.0)

### GammaGaussianHMM¶

class GammaGaussianHMM(scale_dist, initial_dist, transition_matrix, transition_dist, observation_matrix, observation_dist, validate_args=None, duration=None)[source]

Bases: pyro.distributions.hmm.HiddenMarkovModel

Hidden Markov Model with the joint distribution of initial state, hidden state, and observed state is a MultivariateStudentT distribution along the line of references  and . This adapts  to parallelize over time to achieve O(log(time)) parallel complexity.

This GammaGaussianHMM class corresponds to the generative model:

s = Gamma(df/2, df/2).sample()
z = scale(initial_dist, s).sample()
x = []
for t in range(num_events):
z = z @ transition_matrix + scale(transition_dist, s).sample()
x.append(z @ observation_matrix + scale(observation_dist, s).sample())


where scale(mvn(loc, precision), s) := mvn(loc, s * precision).

The event_shape of this distribution includes time on the left:

event_shape = (num_steps,) + observation_dist.event_shape


This distribution supports any combination of homogeneous/heterogeneous time dependency of transition_dist and observation_dist. However, because time is included in this distribution’s event_shape, the homogeneous+homogeneous case will have a broadcastable event_shape with num_steps = 1, allowing log_prob() to work with arbitrary length data:

event_shape = (1, obs_dim)  # homogeneous + homogeneous case


References:

 Simo Sarkka, Angel F. Garcia-Fernandez (2019)

“Temporal Parallelization of Bayesian Filters and Smoothers” https://arxiv.org/pdf/1905.13002.pdf

 F. J. Giron and J. C. Rojano (1994)

“Bayesian Kalman filtering with elliptically contoured errors”

 Filip Tronarp, Toni Karvonen, and Simo Sarkka (2019)

“Student’s t-filters for noise scale estimation” https://users.aalto.fi/~ssarkka/pub/SPL2019.pdf

Variables
• hidden_dim (int) – The dimension of the hidden state.

• obs_dim (int) – The dimension of the observed state.

Parameters
• scale_dist (Gamma) – Prior of the mixing distribution.

• initial_dist (MultivariateNormal) – A distribution with unit scale mixing over initial states. This should have batch_shape broadcastable to self.batch_shape. This should have event_shape (hidden_dim,).

• transition_matrix (Tensor) – A linear transformation of hidden state. This should have shape broadcastable to self.batch_shape + (num_steps, hidden_dim, hidden_dim) where the rightmost dims are ordered (old, new).

• transition_dist (MultivariateNormal) – A process noise distribution with unit scale mixing. This should have batch_shape broadcastable to self.batch_shape + (num_steps,). This should have event_shape (hidden_dim,).

• observation_matrix (Tensor) – A linear transformation from hidden to observed state. This should have shape broadcastable to self.batch_shape + (num_steps, hidden_dim, obs_dim).

• observation_dist (MultivariateNormal) – An observation noise distribution with unit scale mixing. This should have batch_shape broadcastable to self.batch_shape + (num_steps,). This should have event_shape (obs_dim,).

• duration (int) – Optional size of the time axis event_shape. This is required when sampling from homogeneous HMMs whose parameters are not expanded along the time axis.

arg_constraints = {}
expand(batch_shape, _instance=None)[source]
filter(value)[source]

Compute posteriors over the multiplier and the final state given a sequence of observations. The posterior is a pair of Gamma and MultivariateNormal distributions (i.e. a GammaGaussian instance).

Parameters

value (Tensor) – A sequence of observations.

Returns

A pair of posterior distributions over the mixing and the latent state at the final time step.

Return type

a tuple of ~pyro.distributions.Gamma and ~pyro.distributions.MultivariateNormal

log_prob(value)[source]
support = IndependentConstraint(Real(), 2)

### GammaPoisson¶

class GammaPoisson(concentration, rate, validate_args=None)[source]

Compound distribution comprising of a gamma-poisson pair, also referred to as a gamma-poisson mixture. The rate parameter for the Poisson distribution is unknown and randomly drawn from a Gamma distribution.

Note

This can be treated as an alternate parametrization of the NegativeBinomial (total_count, probs) distribution, with concentration = total_count and rate = (1 - probs) / probs.

Parameters
• concentration (float or torch.Tensor) – shape parameter (alpha) of the Gamma distribution.

• rate (float or torch.Tensor) – rate parameter (beta) for the Gamma distribution.

arg_constraints = {'concentration': GreaterThan(lower_bound=0.0), 'rate': GreaterThan(lower_bound=0.0)}
property concentration
expand(batch_shape, _instance=None)[source]
log_prob(value)[source]
property mean
property rate
sample(sample_shape=())[source]
support = IntegerGreaterThan(lower_bound=0)
property variance

### GaussianHMM¶

class GaussianHMM(initial_dist, transition_matrix, transition_dist, observation_matrix, observation_dist, validate_args=None, duration=None)[source]

Bases: pyro.distributions.hmm.HiddenMarkovModel

Hidden Markov Model with Gaussians for initial, transition, and observation distributions. This adapts  to parallelize over time to achieve O(log(time)) parallel complexity, however it differs in that it tracks the log normalizer to ensure log_prob() is differentiable.

This corresponds to the generative model:

z = initial_distribution.sample()
x = []
for t in range(num_events):
z = z @ transition_matrix + transition_dist.sample()
x.append(z @ observation_matrix + observation_dist.sample())


The event_shape of this distribution includes time on the left:

event_shape = (num_steps,) + observation_dist.event_shape


This distribution supports any combination of homogeneous/heterogeneous time dependency of transition_dist and observation_dist. However, because time is included in this distribution’s event_shape, the homogeneous+homogeneous case will have a broadcastable event_shape with num_steps = 1, allowing log_prob() to work with arbitrary length data:

event_shape = (1, obs_dim)  # homogeneous + homogeneous case


References:

 Simo Sarkka, Angel F. Garcia-Fernandez (2019)

“Temporal Parallelization of Bayesian Filters and Smoothers” https://arxiv.org/pdf/1905.13002.pdf

Variables
• hidden_dim (int) – The dimension of the hidden state.

• obs_dim (int) – The dimension of the observed state.

Parameters
• initial_dist (MultivariateNormal) – A distribution over initial states. This should have batch_shape broadcastable to self.batch_shape. This should have event_shape (hidden_dim,).

• transition_matrix (Tensor) – A linear transformation of hidden state. This should have shape broadcastable to self.batch_shape + (num_steps, hidden_dim, hidden_dim) where the rightmost dims are ordered (old, new).

• transition_dist (MultivariateNormal) – A process noise distribution. This should have batch_shape broadcastable to self.batch_shape + (num_steps,). This should have event_shape (hidden_dim,).

• observation_matrix (Tensor) – A linear transformation from hidden to observed state. This should have shape broadcastable to self.batch_shape + (num_steps, hidden_dim, obs_dim).

• observation_dist (MultivariateNormal or Normal) – An observation noise distribution. This should have batch_shape broadcastable to self.batch_shape + (num_steps,). This should have event_shape (obs_dim,).

• duration (int) – Optional size of the time axis event_shape. This is required when sampling from homogeneous HMMs whose parameters are not expanded along the time axis.

arg_constraints = {}
conjugate_update(other)[source]

EXPERIMENTAL Creates an updated GaussianHMM fusing information from another compatible distribution.

This should satisfy:

fg, log_normalizer = f.conjugate_update(g)
assert f.log_prob(x) + g.log_prob(x) == fg.log_prob(x) + log_normalizer

Parameters

other (MultivariateNormal or Normal) – A distribution representing p(data|self.probs) but normalized over self.probs rather than data.

Returns

a pair (updated,log_normalizer) where updated is an updated GaussianHMM , and log_normalizer is a Tensor representing the normalization factor.

expand(batch_shape, _instance=None)[source]
filter(value)[source]

Compute posterior over final state given a sequence of observations.

Parameters

value (Tensor) – A sequence of observations.

Returns

A posterior distribution over latent states at the final time step. result can then be used as initial_dist in a sequential Pyro model for prediction.

Return type

MultivariateNormal

has_rsample = True
log_prob(value)[source]
prefix_condition(data)[source]

EXPERIMENTAL Given self has event_shape == (t+f, d) and data x of shape batch_shape + (t, d), compute a conditional distribution of event_shape (f, d). Typically t is the number of training time steps, f is the number of forecast time steps, and d is the data dimension.

Parameters

data (Tensor) – data of dimension at least 2.

rsample(sample_shape=torch.Size([]))[source]
rsample_posterior(value, sample_shape=torch.Size([]))[source]

EXPERIMENTAL Sample from the latent state conditioned on observation.

support = IndependentConstraint(Real(), 2)

### GaussianMRF¶

class GaussianMRF(initial_dist, transition_dist, observation_dist, validate_args=None)[source]

Temporal Markov Random Field with Gaussian factors for initial, transition, and observation distributions. This adapts  to parallelize over time to achieve O(log(time)) parallel complexity, however it differs in that it tracks the log normalizer to ensure log_prob() is differentiable.

The event_shape of this distribution includes time on the left:

event_shape = (num_steps,) + observation_dist.event_shape


This distribution supports any combination of homogeneous/heterogeneous time dependency of transition_dist and observation_dist. However, because time is included in this distribution’s event_shape, the homogeneous+homogeneous case will have a broadcastable event_shape with num_steps = 1, allowing log_prob() to work with arbitrary length data:

event_shape = (1, obs_dim)  # homogeneous + homogeneous case


References:

 Simo Sarkka, Angel F. Garcia-Fernandez (2019)

“Temporal Parallelization of Bayesian Filters and Smoothers” https://arxiv.org/pdf/1905.13002.pdf

Variables
• hidden_dim (int) – The dimension of the hidden state.

• obs_dim (int) – The dimension of the observed state.

Parameters
• initial_dist (MultivariateNormal) – A distribution over initial states. This should have batch_shape broadcastable to self.batch_shape. This should have event_shape (hidden_dim,).

• transition_dist (MultivariateNormal) – A joint distribution factor over a pair of successive time steps. This should have batch_shape broadcastable to self.batch_shape + (num_steps,). This should have event_shape (hidden_dim + hidden_dim,) (old+new).

• observation_dist (MultivariateNormal) – A joint distribution factor over a hidden and an observed state. This should have batch_shape broadcastable to self.batch_shape + (num_steps,). This should have event_shape (hidden_dim + obs_dim,).

arg_constraints = {}
expand(batch_shape, _instance=None)[source]
log_prob(value)[source]
property support

### GaussianScaleMixture¶

class GaussianScaleMixture(coord_scale, component_logits, component_scale)[source]

Mixture of Normal distributions with zero mean and diagonal covariance matrices.

That is, this distribution is a mixture with K components, where each component distribution is a D-dimensional Normal distribution with zero mean and a D-dimensional diagonal covariance matrix. The K different covariance matrices are controlled by the parameters coord_scale and component_scale. That is, the covariance matrix of the k’th component is given by

Sigma_ii = (component_scale_k * coord_scale_i) ** 2 (i = 1, …, D)

where component_scale_k is a positive scale factor and coord_scale_i are positive scale parameters shared between all K components. The mixture weights are controlled by a K-dimensional vector of softmax logits, component_logits. This distribution implements pathwise derivatives for samples from the distribution. This distribution does not currently support batched parameters.

See reference  for details on the implementations of the pathwise derivative. Please consider citing this reference if you use the pathwise derivative in your research.

 Pathwise Derivatives for Multivariate Distributions, Martin Jankowiak & Theofanis Karaletsos. arXiv:1806.01856

Note that this distribution supports both even and odd dimensions, but the former should be more a bit higher precision, since it doesn’t use any erfs in the backward call. Also note that this distribution does not support D = 1.

Parameters
• coord_scale (torch.tensor) – D-dimensional vector of scales

• component_logits (torch.tensor) – K-dimensional vector of logits

• component_scale (torch.tensor) – K-dimensional vector of scale multipliers

arg_constraints = {'component_logits': Real(), 'component_scale': GreaterThan(lower_bound=0.0), 'coord_scale': GreaterThan(lower_bound=0.0)}
has_rsample = True
log_prob(value)[source]
rsample(sample_shape=torch.Size([]))[source]

### GroupedNormalNormal¶

class GroupedNormalNormal(prior_loc, prior_scale, obs_scale, group_idx, validate_args=None)[source]

This likelihood, which operates on groups of real-valued scalar observations, is obtained by integrating out a latent mean for each group. Both the prior on each latent mean as well as the observation likelihood for each data point are univariate Normal distributions. The prior means are controlled by prior_loc and prior_scale. The observation noise of the Normal likelihood is controlled by obs_scale, which is allowed to vary from observation to observation. The tensor of indices group_idx connects each observation to one of the groups specified by prior_loc and prior_scale.

See e.g. Eqn. (55) in ref.  for relevant expressions in a simpler case with scalar obs_scale.

Example:

>>> num_groups = 3
>>> num_data = 4
>>> prior_loc = torch.randn(num_groups)
>>> prior_scale = torch.rand(num_groups)
>>> obs_scale = torch.rand(num_data)
>>> group_idx = torch.tensor([1, 0, 2, 1]).long()
>>> values = torch.randn(num_data)
>>> gnn = GroupedNormalNormal(prior_loc, prior_scale, obs_scale, group_idx)
>>> assert gnn.log_prob(values).shape == ()


References:  “Conjugate Bayesian analysis of the Gaussian distribution,” Kevin P. Murphy.

Parameters
• prior_loc (torch.Tensor) – Tensor of shape (num_groups,) specifying the prior mean of the latent of each group.

• prior_scale (torch.Tensor) – Tensor of shape (num_groups,) specifying the prior scale of the latent of each group.

• obs_scale (torch.Tensor) – Tensor of shape (num_data,) specifying the scale of the observation noise of each observation.

• group_idx (torch.LongTensor) – Tensor of indices of shape (num_data,) linking each observation to one of the num_groups groups that are specified in prior_loc and prior_scale.

arg_constraints = {'obs_scale': GreaterThan(lower_bound=0.0), 'prior_loc': Real(), 'prior_scale': GreaterThan(lower_bound=0.0)}
expand(batch_shape, _instance=None)[source]
get_posterior(value)[source]

Get a pyro.distributions.Normal distribution that encodes the posterior distribution over the vector of latents specified by prior_loc and prior_scale conditioned on the observed data specified by value.

log_prob(value)[source]
sample(sample_shape=())[source]
support = Real()

### ImproperUniform¶

class ImproperUniform(support, batch_shape, event_shape)[source]

Improper distribution with zero log_prob() and undefined sample().

This is useful for transforming a model from generative dag form to factor graph form for use in HMC. For example the following are equal in distribution:

# Version 1. a generative dag
x = pyro.sample("x", Normal(0, 1))
y = pyro.sample("y", Normal(x, 1))
z = pyro.sample("z", Normal(y, 1))

# Version 2. a factor graph
xyz = pyro.sample("xyz", ImproperUniform(constraints.real, (), (3,)))
x, y, z = xyz.unbind(-1)
pyro.sample("x", Normal(0, 1), obs=x)
pyro.sample("y", Normal(x, 1), obs=y)
pyro.sample("z", Normal(y, 1), obs=z)


Note this distribution errors when sample() is called. To create a similar distribution that instead samples from a specified distribution consider using .mask(False) as in:

xyz = dist.Normal(0, 1).expand().to_event(1).mask(False)

Parameters
• support (Constraint) – The support of the distribution.

• batch_shape (torch.Size) – The batch shape.

• event_shape (torch.Size) – The event shape.

arg_constraints = {}
expand(batch_shape, _instance=None)[source]
log_prob(value)[source]
sample(sample_shape=torch.Size([]))[source]
property support

### IndependentHMM¶

class IndependentHMM(base_dist)[source]

Wrapper class to treat a batch of independent univariate HMMs as a single multivariate distribution. This converts distribution shapes as follows:

.batch_shape

.event_shape

base_dist

shape + (obs_dim,)

(duration, 1)

result

shape

(duration, obs_dim)

Parameters

base_dist (HiddenMarkovModel) – A base hidden Markov model instance.

arg_constraints = {}
property duration
expand(batch_shape, _instance=None)[source]
property has_rsample
log_prob(value)[source]
rsample(sample_shape=torch.Size([]))[source]
property support

### InverseGamma¶

class InverseGamma(concentration, rate, validate_args=None)[source]

Creates an inverse-gamma distribution parameterized by concentration and rate.

X ~ Gamma(concentration, rate) Y = 1/X ~ InverseGamma(concentration, rate)

Parameters
• concentration (torch.Tensor) – the concentration parameter (i.e. alpha).

• rate (torch.Tensor) – the rate parameter (i.e. beta).

arg_constraints: Dict[str, torch.distributions.constraints.Constraint] = {'concentration': GreaterThan(lower_bound=0.0), 'rate': GreaterThan(lower_bound=0.0)}
property concentration
expand(batch_shape, _instance=None)[source]
has_rsample = True
property rate
support = GreaterThan(lower_bound=0.0)

### LinearHMM¶

class LinearHMM(initial_dist, transition_matrix, transition_dist, observation_matrix, observation_dist, validate_args=None, duration=None)[source]

Bases: pyro.distributions.hmm.HiddenMarkovModel

Hidden Markov Model with linear dynamics and observations and arbitrary noise for initial, transition, and observation distributions. Each of those distributions can be e.g. MultivariateNormal or Independent of Normal, StudentT, or Stable . Additionally the observation distribution may be constrained, e.g. LogNormal

This corresponds to the generative model:

z = initial_distribution.sample()
x = []
for t in range(num_events):
z = z @ transition_matrix + transition_dist.sample()
y = z @ observation_matrix + obs_base_dist.sample()
x.append(obs_transform(y))


where observation_dist is split into obs_base_dist and an optional obs_transform (defaulting to the identity).

This implements a reparameterized rsample() method but does not implement a log_prob() method. Derived classes may implement log_prob() .

Inference without log_prob() can be performed using either reparameterization with LinearHMMReparam or likelihood-free algorithms such as EnergyDistance . Note that while stable processes generally require a common shared stability parameter $$\alpha$$ , this distribution and the above inference algorithms allow heterogeneous stability parameters.

The event_shape of this distribution includes time on the left:

event_shape = (num_steps,) + observation_dist.event_shape


This distribution supports any combination of homogeneous/heterogeneous time dependency of transition_dist and observation_dist. However at least one of the distributions or matrices must be expanded to contain the time dimension.

Variables
• hidden_dim (int) – The dimension of the hidden state.

• obs_dim (int) – The dimension of the observed state.

Parameters
• initial_dist – A distribution over initial states. This should have batch_shape broadcastable to self.batch_shape. This should have event_shape (hidden_dim,).

• transition_matrix (Tensor) – A linear transformation of hidden state. This should have shape broadcastable to self.batch_shape + (num_steps, hidden_dim, hidden_dim) where the rightmost dims are ordered (old, new).

• transition_dist – A distribution over process noise. This should have batch_shape broadcastable to self.batch_shape + (num_steps,). This should have event_shape (hidden_dim,).

• observation_matrix (Tensor) – A linear transformation from hidden to observed state. This should have shape broadcastable to self.batch_shape + (num_steps, hidden_dim, obs_dim).

• observation_dist – A observation noise distribution. This should have batch_shape broadcastable to self.batch_shape + (num_steps,). This should have event_shape (obs_dim,).

• duration (int) – Optional size of the time axis event_shape. This is required when sampling from homogeneous HMMs whose parameters are not expanded along the time axis.

arg_constraints = {}
expand(batch_shape, _instance=None)[source]
has_rsample = True
log_prob(value)[source]
rsample(sample_shape=torch.Size([]))[source]
property support

### LKJ¶

class LKJ(dim, concentration=1.0, validate_args=None)[source]

LKJ distribution for correlation matrices. The distribution is controlled by concentration parameter $$\eta$$ to make the probability of the correlation matrix $$M$$ propotional to $$\det(M)^{\eta - 1}$$. Because of that, when concentration == 1, we have a uniform distribution over correlation matrices.

When concentration > 1, the distribution favors samples with large large determinent. This is useful when we know a priori that the underlying variables are not correlated. When concentration < 1, the distribution favors samples with small determinent. This is useful when we know a priori that some underlying variables are correlated.

Parameters
• dimension (int) – dimension of the matrices

• concentration (ndarray) – concentration/shape parameter of the distribution (often referred to as eta)

References

 Generating random correlation matrices based on vines and extended onion method, Daniel Lewandowski, Dorota Kurowicka, Harry Joe

arg_constraints: Dict[str, torch.distributions.constraints.Constraint] = {'concentration': GreaterThan(lower_bound=0.0)}
expand(batch_shape, _instance=None)[source]
property mean
support = CorrMatrix()

### LKJCorrCholesky¶

class LKJCorrCholesky(d, eta, validate_args=None)[source]

### LogNormalNegativeBinomial¶

class LogNormalNegativeBinomial(total_count, logits, multiplicative_noise_scale, *, num_quad_points=8, validate_args=None)[source]

A three-parameter generalization of the Negative Binomial distribution . It can be understood as a continuous mixture of Negative Binomial distributions in which we inject Normally-distributed noise into the logits of the Negative Binomial distribution:

$\begin{split}\begin{eqnarray} &\rm{LNNB}(y | \rm{total\_count}=\nu, \rm{logits}=\ell, \rm{multiplicative\_noise\_scale}=sigma) = \\ &\int d\epsilon \mathcal{N}(\epsilon | 0, \sigma) \rm{NB}(y | \rm{total\_count}=\nu, \rm{logits}=\ell + \epsilon) \end{eqnarray}\end{split}$

where $$y \ge 0$$ is a non-negative integer. Thus while a Negative Binomial distribution can be formulated as a Poisson distribution with a Gamma-distributed rate, this distribution adds an additional level of variability by also modulating the rate by Log Normally-distributed multiplicative noise.

This distribution has a mean given by

$\mathbb{E}[y] = \nu e^{\ell} = e^{\ell + \log \nu + \tfrac{1}{2}\sigma^2}$

and a variance given by

$\rm{Var}[y] = \mathbb{E}[y] + \left( e^{\sigma^2} (1 + 1/\nu) - 1 \right) \left( \mathbb{E}[y] \right)^2$

Thus while a given mean and variance together uniquely characterize a Negative Binomial distribution, there is a one-dimensional family of Log Normal Negative Binomial distributions with a given mean and variance.

Note that in some applications it may be useful to parameterize the logits as

$\ell = \ell^\prime - \log \nu - \tfrac{1}{2}\sigma^2$

so that the mean is given by $$\mathbb{E}[y] = e^{\ell^\prime}$$ and does not depend on $$\nu$$ and $$\sigma$$, which serve to determine the higher moments.

References:

 “Lognormal and Gamma Mixed Negative Binomial Regression,” Mingyuan Zhou, Lingbo Li, David Dunson, and Lawrence Carin.

Parameters
• total_count (float or torch.Tensor) – non-negative number of negative Bernoulli trials. The variance decreases as total_count increases.

• logits (torch.Tensor) – Event log-odds for probabilities of success for underlying Negative Binomial distribution.

• multiplicative_noise_scale (torch.Tensor) – Controls the level of the injected Normal logit noise.

• num_quad_points (int) – Number of quadrature points used to compute the (approximate) log_prob. Defaults to 8.

arg_constraints = {'logits': Real(), 'multiplicative_noise_scale': GreaterThan(lower_bound=0.0), 'total_count': GreaterThanEq(lower_bound=0)}
expand(batch_shape, _instance=None)[source]
log_prob(value)[source]
property mean
sample(sample_shape=torch.Size([]))[source]
support = IntegerGreaterThan(lower_bound=0)
property variance

### Logistic¶

class Logistic(loc, scale, *, validate_args=None)[source]

Logistic distribution.

This is a smooth distribution with symmetric asymptotically exponential tails and a concave log density. For standard loc=0, scale=1, the density is given by

$p(x) = \frac {e^{-x}} {(1 + e^{-x})^2}$

Like the Laplace density, this density has the heaviest possible tails (asymptotically) while still being log-convex. Unlike the Laplace distribution, this distribution is infinitely differentiable everywhere, and is thus suitable for constructing Laplace approximations.

Parameters
• loc – Location parameter.

• scale – Scale parameter.

arg_constraints = {'loc': Real(), 'scale': GreaterThan(lower_bound=0.0)}
cdf(value)[source]
entropy()[source]
expand(batch_shape, _instance=None)[source]
has_rsample = True
icdf(value)[source]
log_prob(value)[source]
property mean
rsample(sample_shape=torch.Size([]))[source]
support = Real()
property variance

Masks a distribution by a boolean tensor that is broadcastable to the distribution’s batch_shape.

In the special case mask is False, computation of log_prob() , score_parts() , and kl_divergence() is skipped, and constant zero values are returned instead.

Parameters

mask (torch.Tensor or bool) – A boolean or boolean-valued tensor.

arg_constraints = {}
conjugate_update(other)[source]

EXPERIMENTAL.

enumerate_support(expand=True)[source]
expand(batch_shape, _instance=None)[source]
property has_enumerate_support
property has_rsample
log_prob(value)[source]
property mean
rsample(sample_shape=torch.Size([]))[source]
sample(sample_shape=torch.Size([]))[source]
score_parts(value)[source]
property support
property variance

A masked deterministic mixture of two distributions.

This is useful when the mask is sampled from another distribution, possibly correlated across the batch. Often the mask can be marginalized out via enumeration.

Example:

change_point = pyro.sample("change_point",
dist.Categorical(torch.ones(len(data) + 1)),
infer={'enumerate': 'parallel'})
mask = torch.arange(len(data), dtype=torch.long) >= changepoint
with pyro.plate("data", len(data)):

Parameters
arg_constraints = {}
expand(batch_shape)[source]
property has_rsample
log_prob(value)[source]
property mean
rsample(sample_shape=torch.Size([]))[source]
sample(sample_shape=torch.Size([]))[source]
property support
property variance

### MixtureOfDiagNormals¶

class MixtureOfDiagNormals(locs, coord_scale, component_logits)[source]

Mixture of Normal distributions with arbitrary means and arbitrary diagonal covariance matrices.

That is, this distribution is a mixture with K components, where each component distribution is a D-dimensional Normal distribution with a D-dimensional mean parameter and a D-dimensional diagonal covariance matrix. The K different component means are gathered into the K x D dimensional parameter locs and the K different scale parameters are gathered into the K x D dimensional parameter coord_scale. The mixture weights are controlled by a K-dimensional vector of softmax logits, component_logits. This distribution implements pathwise derivatives for samples from the distribution.

See reference  for details on the implementations of the pathwise derivative. Please consider citing this reference if you use the pathwise derivative in your research. Note that this distribution does not support dimension D = 1.

 Pathwise Derivatives for Multivariate Distributions, Martin Jankowiak & Theofanis Karaletsos. arXiv:1806.01856

Parameters
arg_constraints = {'component_logits': Real(), 'coord_scale': GreaterThan(lower_bound=0.0), 'locs': Real()}
expand(batch_shape, _instance=None)[source]
has_rsample = True
log_prob(value)[source]
rsample(sample_shape=torch.Size([]))[source]

### MixtureOfDiagNormalsSharedCovariance¶

class MixtureOfDiagNormalsSharedCovariance(locs, coord_scale, component_logits)[source]

Mixture of Normal distributions with diagonal covariance matrices.

That is, this distribution is a mixture with K components, where each component distribution is a D-dimensional Normal distribution with a D-dimensional mean parameter loc and a D-dimensional diagonal covariance matrix specified by a scale parameter coord_scale. The K different component means are gathered into the parameter locs and the scale parameter is shared between all K components. The mixture weights are controlled by a K-dimensional vector of softmax logits, component_logits. This distribution implements pathwise derivatives for samples from the distribution.

See reference  for details on the implementations of the pathwise derivative. Please consider citing this reference if you use the pathwise derivative in your research. Note that this distribution does not support dimension D = 1.

 Pathwise Derivatives for Multivariate Distributions, Martin Jankowiak & Theofanis Karaletsos. arXiv:1806.01856

Parameters
• locs (torch.Tensor) – K x D mean matrix

• coord_scale (torch.Tensor) – shared D-dimensional scale vector

• component_logits (torch.Tensor) – K-dimensional vector of softmax logits

arg_constraints = {'component_logits': Real(), 'coord_scale': GreaterThan(lower_bound=0.0), 'locs': Real()}
expand(batch_shape, _instance=None)[source]
has_rsample = True
log_prob(value)[source]
rsample(sample_shape=torch.Size([]))[source]

### MultivariateStudentT¶

class MultivariateStudentT(df, loc, scale_tril, validate_args=None)[source]

Creates a multivariate Student’s t-distribution parameterized by degree of freedom df, mean loc and scale scale_tril.

Parameters
• df (Tensor) – degrees of freedom

• loc (Tensor) – mean of the distribution

• scale_tril (Tensor) – scale of the distribution, which is a lower triangular matrix with positive diagonal entries

arg_constraints = {'df': GreaterThan(lower_bound=0.0), 'loc': IndependentConstraint(Real(), 1), 'scale_tril': LowerCholesky()}
property covariance_matrix
expand(batch_shape, _instance=None)[source]
has_rsample = True
static infer_shapes(df, loc, scale_tril)[source]
log_prob(value)[source]
property mean
property precision_matrix
rsample(sample_shape=torch.Size([]))[source]
property scale_tril
support = IndependentConstraint(Real(), 1)
property variance

Wrapper around Normal to allow partially observed data as specified by NAN elements in log_prob(); the log_prob of these elements will be zero. This is useful for likelihoods with missing data.

Example:

from math import nan
data = torch.tensor([0.5, 0.1, nan, 0.9])
with pyro.plate("data", len(data)):

log_prob(value: torch.Tensor) [source]

class NanMaskedMultivariateNormal(loc, covariance_matrix=None, precision_matrix=None, scale_tril=None, validate_args=None)[source]

Wrapper around MultivariateNormal to allow partially observed data as specified by NAN elements in the argument to log_prob(). The log_prob of these events will marginalize over the NAN elements. This is useful for likelihoods with missing data.

Example:

from math import nan
data = torch.tensor([
[0.1, 0.2, 3.4],
[0.5, 0.1, nan],
[0.6, nan, nan],
[nan, 0.5, nan],
[nan, nan, nan],
])
with pyro.plate("data", len(data)):
pyro.sample(
"obs",
obs=data,
)

log_prob(value: torch.Tensor) [source]

### OMTMultivariateNormal¶

class OMTMultivariateNormal(loc, scale_tril)[source]

Multivariate normal (Gaussian) distribution with OMT gradients w.r.t. both parameters. Note the gradient computation w.r.t. the Cholesky factor has cost O(D^3), although the resulting gradient variance is generally expected to be lower.

A distribution over vectors in which all the elements have a joint Gaussian density.

Parameters
arg_constraints = {'loc': Real(), 'scale_tril': LowerTriangular()}
rsample(sample_shape=torch.Size([]))[source]

### OneOneMatching¶

class OneOneMatching(logits, *, bp_iters=None, validate_args=None)[source]

Random perfect matching from N sources to N destinations where each source matches exactly one destination and each destination matches exactly one source.

Samples are represented as long tensors of shape (N,) taking values in {0,...,N-1} and satisfying the above one-one constraint. The log probability of a sample v is the sum of edge logits, up to the log partition function log Z:

$\log p(v) = \sum_s \text{logits}[s, v[s]] - \log Z$

Exact computations are expensive. To enable tractable approximations, set a number of belief propagation iterations via the bp_iters argument. The log_partition_function() and log_prob() methods use a Bethe approximation [1,2,3,4].

References:

 Michael Chertkov, Lukas Kroc, Massimo Vergassola (2008)

“Belief propagation and beyond for particle tracking” https://arxiv.org/pdf/0806.1199.pdf

 Bert Huang, Tony Jebara (2009)

“Approximating the Permanent with Belief Propagation” https://arxiv.org/pdf/0908.1769.pdf

 Pascal O. Vontobel (2012)

“The Bethe Permanent of a Non-Negative Matrix” https://arxiv.org/pdf/1107.4196.pdf

 M Chertkov, AB Yedidia (2013)

“Approximating the permanent with fractional belief propagation” http://www.jmlr.org/papers/volume14/chertkov13a/chertkov13a.pdf

Parameters
• logits (Tensor) – An (N, N)-shaped tensor of edge logits.

• bp_iters (int) – Optional number of belief propagation iterations. If unspecified or None expensive exact algorithms will be used.

arg_constraints = {'logits': Real()}
enumerate_support(expand=True)[source]
has_enumerate_support = True
property log_partition_function
log_prob(value)[source]
mode()[source]

Computes a maximum probability matching.

sample(sample_shape=torch.Size([]))[source]
property support

### OneTwoMatching¶

class OneTwoMatching(logits, *, bp_iters=None, validate_args=None)[source]

Random matching from 2*N sources to N destinations where each source matches exactly one destination and each destination matches exactly two sources.

Samples are represented as long tensors of shape (2*N,) taking values in {0,...,N-1} and satisfying the above one-two constraint. The log probability of a sample v is the sum of edge logits, up to the log partition function log Z:

$\log p(v) = \sum_s \text{logits}[s, v[s]] - \log Z$

Exact computations are expensive. To enable tractable approximations, set a number of belief propagation iterations via the bp_iters argument. The log_partition_function() and log_prob() methods use a Bethe approximation [1,2,3,4].

References:

 Michael Chertkov, Lukas Kroc, Massimo Vergassola (2008)

“Belief propagation and beyond for particle tracking” https://arxiv.org/pdf/0806.1199.pdf

 Bert Huang, Tony Jebara (2009)

“Approximating the Permanent with Belief Propagation” https://arxiv.org/pdf/0908.1769.pdf

 Pascal O. Vontobel (2012)

“The Bethe Permanent of a Non-Negative Matrix” https://arxiv.org/pdf/1107.4196.pdf

 M Chertkov, AB Yedidia (2013)

“Approximating the permanent with fractional belief propagation” http://www.jmlr.org/papers/volume14/chertkov13a/chertkov13a.pdf

Parameters
• logits (Tensor) – An (2 * N, N)-shaped tensor of edge logits.

• bp_iters (int) – Optional number of belief propagation iterations. If unspecified or None expensive exact algorithms will be used.

arg_constraints = {'logits': Real()}
enumerate_support(expand=True)[source]
has_enumerate_support = True
property log_partition_function
log_prob(value)[source]
mode()[source]

Computes a maximum probability matching.

sample(sample_shape=torch.Size([]))[source]
property support

### OrderedLogistic¶

class OrderedLogistic(predictor, cutpoints, validate_args=None)[source]

Alternative parametrization of the distribution over a categorical variable.

Instead of the typical parametrization of a categorical variable in terms of the probability mass of the individual categories p, this provides an alternative that is useful in specifying ordered categorical models. This accepts a vector of cutpoints which are an ordered vector of real numbers denoting baseline cumulative log-odds of the individual categories, and a model vector predictor which modifies the baselines for each sample individually.

These cumulative log-odds are then transformed into a discrete cumulative probability distribution, that is finally differenced to return the probability mass matrix p that specifies the categorical distribution.

Parameters
• predictor (Tensor) – A tensor of predictor variables of arbitrary shape. The output shape of non-batched samples from this distribution will be the same shape as predictor.

• cutpoints (Tensor) – A tensor of cutpoints that are used to determine the cumulative probability of each entry in predictor belonging to a given category. The first cutpoints.ndim-1 dimensions must be broadcastable to predictor, and the -1 dimension is monotonically increasing.

arg_constraints = {'cutpoints': OrderedVector(), 'predictor': Real()}
expand(batch_shape, _instance=None)[source]

### ProjectedNormal¶

class ProjectedNormal(concentration, *, validate_args=None)[source]

Projected isotropic normal distribution of arbitrary dimension.

This distribution over directional data is qualitatively similar to the von Mises and von Mises-Fisher distributions, but permits tractable variational inference via reparametrized gradients.

To use this distribution with autoguides, use poutine.reparam with a ProjectedNormalReparam reparametrizer in the model, e.g.:

@poutine.reparam(config={"direction": ProjectedNormalReparam()})
def model():
direction = pyro.sample("direction",
ProjectedNormal(torch.zeros(3)))
...


or simply wrap in MinimalReparam or AutoReparam , e.g.:

@MinimalReparam()
def model():
...


Note

This implements log_prob() only for dimensions {2,3}.

 D. Hernandez-Stumpfhauser, F.J. Breidt, M.J. van der Woerd (2017)

“The General Projected Normal Distribution of Arbitrary Dimension: Modeling and Bayesian Inference” https://projecteuclid.org/euclid.ba/1453211962

Parameters

concentration (torch.Tensor) – A combined location-and-concentration vector. The direction of this vector is the location, and its magnitude is the concentration.

arg_constraints = {'concentration': IndependentConstraint(Real(), 1)}
expand(batch_shape, _instance=None)[source]
has_rsample = True
static infer_shapes(concentration)[source]
log_prob(value)[source]
property mean

Note this is the mean in the sense of a centroid in the submanifold that minimizes expected squared geodesic distance.

property mode
rsample(sample_shape=torch.Size([]))[source]
support = Sphere

### RelaxedBernoulliStraightThrough¶

class RelaxedBernoulliStraightThrough(temperature, probs=None, logits=None, validate_args=None)[source]

An implementation of RelaxedBernoulli with a straight-through gradient estimator.

This distribution has the following properties:

• The samples returned by the rsample() method are discrete/quantized.

• The log_prob() method returns the log probability of the relaxed/unquantized sample using the GumbelSoftmax distribution.

• In the backward pass the gradient of the sample with respect to the parameters of the distribution uses the relaxed/unquantized sample.

References:

 The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables,

Chris J. Maddison, Andriy Mnih, Yee Whye Teh

 Categorical Reparameterization with Gumbel-Softmax,

Eric Jang, Shixiang Gu, Ben Poole

log_prob(value)[source]

See pyro.distributions.torch.RelaxedBernoulli.log_prob()

rsample(sample_shape=torch.Size([]))[source]

See pyro.distributions.torch.RelaxedBernoulli.rsample()

### RelaxedOneHotCategoricalStraightThrough¶

class RelaxedOneHotCategoricalStraightThrough(temperature, probs=None, logits=None, validate_args=None)[source]

An implementation of RelaxedOneHotCategorical with a straight-through gradient estimator.

This distribution has the following properties:

• The samples returned by the rsample() method are discrete/quantized.

• The log_prob() method returns the log probability of the relaxed/unquantized sample using the GumbelSoftmax distribution.

• In the backward pass the gradient of the sample with respect to the parameters of the distribution uses the relaxed/unquantized sample.

References:

 The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables,

Chris J. Maddison, Andriy Mnih, Yee Whye Teh

 Categorical Reparameterization with Gumbel-Softmax,

Eric Jang, Shixiang Gu, Ben Poole

log_prob(value)[source]

See pyro.distributions.torch.RelaxedOneHotCategorical.log_prob()

rsample(sample_shape=torch.Size([]))[source]

See pyro.distributions.torch.RelaxedOneHotCategorical.rsample()

### Rejector¶

class Rejector(propose, log_prob_accept, log_scale, *, batch_shape=None, event_shape=None)[source]

Rejection sampled distribution given an acceptance rate function.

Parameters
• propose (Distribution) – A proposal distribution that samples batched proposals via propose(). rsample() supports a sample_shape arg only if propose() supports a sample_shape arg.

• log_prob_accept (callable) – A callable that inputs a batch of proposals and returns a batch of log acceptance probabilities.

• log_scale – Total log probability of acceptance.

arg_constraints = {}
has_rsample = True
log_prob(x)[source]
rsample(sample_shape=torch.Size([]))[source]
score_parts(x)[source]

### SineBivariateVonMises¶

class SineBivariateVonMises(phi_loc, psi_loc, phi_concentration, psi_concentration, correlation=None, weighted_correlation=None, validate_args=None)[source]

Unimodal distribution of two dependent angles on the 2-torus (S^1 ⨂ S^1) given by

$C^{-1}\exp(\kappa_1\cos(x-\mu_1) + \kappa_2\cos(x_2 -\mu_2) + \rho\sin(x_1 - \mu_1)\sin(x_2 - \mu_2))$

and

$C = (2\pi)^2 \sum_{i=0} {2i \choose i} \left(\frac{\rho^2}{4\kappa_1\kappa_2}\right)^i I_i(\kappa_1)I_i(\kappa_2),$

where I_i(cdot) is the modified bessel function of first kind, mu’s are the locations of the distribution, kappa’s are the concentration and rho gives the correlation between angles x_1 and x_2.

This distribution is a submodel of the Bivariate von Mises distribution, called the Sine Distribution  in directional statistics.

This distribution is helpful for modeling coupled angles such as torsion angles in peptide chains. To infer parameters, use NUTS or HMC with priors that avoid parameterizations where the distribution becomes bimodal; see note below.

Note

Sample efficiency drops as

$\frac{\rho}{\kappa_1\kappa_2} \rightarrow 1$

because the distribution becomes increasingly bimodal. To avoid bimodality use the weighted_correlation parameter with a skew away from one (e.g., Beta(1,3)). The weighted_correlation should be in [0,1].

Note

The correlation and weighted_correlation params are mutually exclusive.

Note

In the context of SVI, this distribution can be used as a likelihood but not for latent variables.

** References: **
1. Probabilistic model for two dependent circular variables Singh, H., Hnizdo, V., and Demchuck, E. (2002)

2. Protein Bioinformatics and Mixtures of Bivariate von Mises Distributions for Angular Data, Mardia, K. V, Taylor, T. C., and Subramaniam, G. (2007)

Parameters
• phi_loc (torch.Tensor) – location of first angle

• psi_loc (torch.Tensor) – location of second angle

• phi_concentration (torch.Tensor) – concentration of first angle

• psi_concentration (torch.Tensor) – concentration of second angle

• correlation (torch.Tensor) – correlation between the two angles

• weighted_correlation (torch.Tensor) – set correlation to weighted_corr * sqrt(phi_conc*psi_conc) to avoid bimodality (see note). The weighted_correlation should be in [0,1].

arg_constraints = {'correlation': Real(), 'phi_concentration': GreaterThan(lower_bound=0.0), 'phi_loc': Real(), 'psi_concentration': GreaterThan(lower_bound=0.0), 'psi_loc': Real()}
expand(batch_shape, _instance=None)[source]
classmethod infer_shapes(**arg_shapes)[source]
log_prob(value)[source]
max_sample_iter = 1000
property mean
property norm_const
sample(sample_shape=torch.Size([]))[source]
** References: **
1. A New Unified Approach for the Simulation of aWide Class of Directional Distributions John T. Kent, Asaad M. Ganeiber & Kanti V. Mardia (2018)

support = IndependentConstraint(Real(), 1)

### SineSkewed¶

class SineSkewed(base_dist: pyro.distributions.torch_distribution.TorchDistribution, skewness, validate_args=None)[source]

Sine Skewing  is a procedure for producing a distribution that breaks pointwise symmetry on a torus distribution. The new distribution is called the Sine Skewed X distribution, where X is the name of the (symmetric) base distribution.

Torus distributions are distributions with support on products of circles (i.e., ⨂^d S^1 where S^1=[-pi,pi) ). So, a 0-torus is a point, the 1-torus is a circle, and the 2-torus is commonly associated with the donut shape.

The Sine Skewed X distribution is parameterized by a weight parameter for each dimension of the event of X. For example with a von Mises distribution over a circle (1-torus), the Sine Skewed von Mises Distribution has one skew parameter. The skewness parameters can be inferred using HMC or NUTS. For example, the following will produce a uniform prior over skewness for the 2-torus,:

def model(obs):
# Sine priors
phi_loc = pyro.sample('phi_loc', VonMises(pi, 2.))
psi_loc = pyro.sample('psi_loc', VonMises(-pi / 2, 2.))
phi_conc = pyro.sample('phi_conc', Beta(halpha_phi, beta_prec_phi - halpha_phi))
psi_conc = pyro.sample('psi_conc', Beta(halpha_psi, beta_prec_psi - halpha_psi))
corr_scale = pyro.sample('corr_scale', Beta(2., 5.))

# SS prior
skew_phi = pyro.sample('skew_phi', Uniform(-1., 1.))
psi_bound = 1 - skew_phi.abs()
skew_psi = pyro.sample('skew_psi', Uniform(-1., 1.))
skewness = torch.stack((skew_phi, psi_bound * skew_psi), dim=-1)
assert skewness.shape == (num_mix_comp, 2)

with pyro.plate('obs_plate'):
sine = SineBivariateVonMises(phi_loc=phi_loc, psi_loc=psi_loc,
phi_concentration=1000 * phi_conc,
psi_concentration=1000 * psi_conc,
weighted_correlation=corr_scale)
return pyro.sample('phi_psi', SineSkewed(sine, skewness), obs=obs)


To ensure the skewing does not alter the normalization constant of the (Sine Bivaraite von Mises) base distribution the skewness parameters are constraint. The constraint requires the sum of the absolute values of skewness to be less than or equal to one. So for the above snippet it must hold that:

skew_phi.abs()+skew_psi.abs() <= 1


We handle this in the prior by computing psi_bound and use it to scale skew_psi. We do not use psi_bound as:

skew_psi = pyro.sample('skew_psi', Uniform(-psi_bound, psi_bound))


as it would make the support for the Uniform distribution dynamic.

In the context of SVI, this distribution can freely be used as a likelihood, but use as latent variables it will lead to slow inference for 2 and higher dim toruses. This is because the base_dist cannot be reparameterized.

Note

An event in the base distribution must be on a d-torus, so the event_shape must be (d,).

Note

For the skewness parameter, it must hold that the sum of the absolute value of its weights for an event must be less than or equal to one. See eq. 2.1 in .

** References: **
1. Sine-skewed toroidal distributions and their application in protein bioinformatics Ameijeiras-Alonso, J., Ley, C. (2019)

Parameters
arg_constraints = {'skewness': IndependentConstraint(Interval(lower_bound=-1.0, upper_bound=1.0), 1)}
expand(batch_shape, _instance=None)[source]
log_prob(value)[source]
sample(sample_shape=torch.Size([]))[source]
support = IndependentConstraint(Real(), 1)

### SkewLogistic¶

class SkewLogistic(loc, scale, asymmetry=1.0, *, validate_args=None)[source]

Skewed generalization of the Logistic distribution (Type I in ).

This is a smooth distribution with asymptotically exponential tails and a concave log density. For standard loc=0, scale=1, asymmetry=α the density is given by

$p(x;\alpha) = \frac {\alpha e^{-x}} {(1 + e^{-x})^{\alpha+1}}$

Like the AsymmetricLaplace density, this density has the heaviest possible tails (asymptotically) while still being log-convex. Unlike the AsymmetricLaplace distribution, this distribution is infinitely differentiable everywhere, and is thus suitable for constructing Laplace approximations.

References

 Generalized logistic distribution

https://en.wikipedia.org/wiki/Generalized_logistic_distribution

Parameters
• loc – Location parameter.

• scale – Scale parameter.

• asymmetry – Asymmetry parameter (positive). The distribution skews right when asymmetry > 1 and left when asymmetry < 1. Defaults to asymmetry = 1 corresponding to the standard Logistic distribution.

arg_constraints = {'asymmetry': GreaterThan(lower_bound=0.0), 'loc': Real(), 'scale': GreaterThan(lower_bound=0.0)}
cdf(value)[source]
expand(batch_shape, _instance=None)[source]
has_rsample = True
icdf(value)[source]
log_prob(value)[source]
rsample(sample_shape=torch.Size([]))[source]
support = Real()

### SoftAsymmetricLaplace¶

class SoftAsymmetricLaplace(loc, scale, asymmetry=1.0, softness=1.0, *, validate_args=None)[source]

Soft asymmetric version of the Laplace distribution.

This has a smooth (infinitely differentiable) density with two asymmetric asymptotically exponential tails, one on the left and one on the right. In the limit of softness → 0, this converges in distribution to the AsymmetricLaplace distribution.

This is equivalent to the sum of three random variables z - u + v where:

z ~ Normal(loc, scale * softness)
u ~ Exponential(1 / (scale * asymmetry))
v ~ Exponential(asymetry / scale)


This is also equivalent the sum of two random variables z + a where:

z ~ Normal(loc, scale * softness)
a ~ AsymmetricLaplace(0, scale, asymmetry)

Parameters
• loc – Location parameter, i.e. the mode.

• scale – Scale parameter = geometric mean of left and right scales.

• asymmetry – Square of ratio of left to right scales. Defaults to 1.

• softness – Scale parameter of the Gaussian smoother. Defaults to 1.

arg_constraints = {'asymmetry': GreaterThan(lower_bound=0.0), 'loc': Real(), 'scale': GreaterThan(lower_bound=0.0), 'softness': GreaterThan(lower_bound=0.0)}
expand(batch_shape, _instance=None)[source]
has_rsample = True
property left_scale
log_prob(value)[source]
property mean
property right_scale
rsample(sample_shape=torch.Size([]))[source]
property soft_scale
support = Real()
property variance

### SoftLaplace¶

class SoftLaplace(loc, scale, *, validate_args=None)[source]

Smooth distribution with Laplace-like tail behavior.

This distribution corresponds to the log-convex density:

z = (value - loc) / scale
log_prob = log(2 / pi) - log(scale) - logaddexp(z, -z)


Like the Laplace density, this density has the heaviest possible tails (asymptotically) while still being log-convex. Unlike the Laplace distribution, this distribution is infinitely differentiable everywhere, and is thus suitable for constructing Laplace approximations.

Parameters
• loc – Location parameter.

• scale – Scale parameter.

arg_constraints = {'loc': Real(), 'scale': GreaterThan(lower_bound=0.0)}
cdf(value)[source]
expand(batch_shape, _instance=None)[source]
has_rsample = True
icdf(value)[source]
log_prob(value)[source]
property mean
rsample(sample_shape=torch.Size([]))[source]
support = Real()
property variance

### SpanningTree¶

class SpanningTree(edge_logits, sampler_options=None, validate_args=None)[source]

Distribution over spanning trees on a fixed number V of vertices.

A tree is represented as torch.LongTensor edges of shape (V-1,2) satisfying the following properties:

1. The edges constitute a tree, i.e. are connected and cycle free.

2. Each edge (v1,v2) = edges[e] is sorted, i.e. v1 < v2.

3. The entire tensor is sorted in colexicographic order.

Use validate_edges() to verify edges are correctly formed.

The edge_logits tensor has one entry for each of the V*(V-1)//2 edges in the complete graph on V vertices, where edges are each sorted and the edge order is colexicographic:

(0,1), (0,2), (1,2), (0,3), (1,3), (2,3), (0,4), (1,4), (2,4), ...


This ordering corresponds to the size-independent pairing function:

k = v1 + v2 * (v2 - 1) // 2


where k is the rank of the edge (v1,v2) in the complete graph. To convert a matrix of edge logits to the linear representation used here:

assert my_matrix.shape == (V, V)
i, j = make_complete_graph(V)
edge_logits = my_matrix[i, j]

Parameters
• edge_logits (torch.Tensor) – A tensor of length V*(V-1)//2 containing logits (aka negative energies) of all edges in the complete graph on V vertices. See above comment for edge ordering.

• sampler_options (dict) – An optional dict of sampler options including: mcmc_steps defaulting to a single MCMC step (which is pretty good); initial_edges defaulting to a cheap approximate sample; backend one of “python” or “cpp”, defaulting to “python”.

arg_constraints = {'edge_logits': Real()}
property edge_mean

Computes marginal probabilities of each edge being active.

Note

This is similar to other distributions’ .mean() method, but with a different shape because this distribution’s values are not encoded as binary matrices.

Returns

A symmetric square (V,V)-shaped matrix with values in [0,1] denoting the marginal probability of each edge being in a sampled value.

Return type

Tensor

enumerate_support(expand=True)[source]

This is implemented for trees with up to 6 vertices (and 5 edges).

has_enumerate_support = True
property log_partition_function
log_prob(edges)[source]
property mode

The maximum weight spanning tree. :rtype: Tensor

Type

returns

sample(sample_shape=torch.Size([]))[source]

This sampler is implemented using MCMC run for a small number of steps after being initialized by a cheap approximate sampler. This sampler is approximate and cubic time. This is faster than the classic Aldous-Broder sampler [1,2], especially for graphs with large mixing time. Recent research [3,4] proposes samplers that run in sub-matrix-multiply time but are more complex to implement.

References

 Generating random spanning trees

Andrei Broder (1989)

 The Random Walk Construction of Uniform Spanning Trees and Uniform Labelled Trees,

David J. Aldous (1990)

 Sampling Random Spanning Trees Faster than Matrix Multiplication,

David Durfee, Rasmus Kyng, John Peebles, Anup B. Rao, Sushant Sachdeva (2017) https://arxiv.org/abs/1611.07451

 An almost-linear time algorithm for uniform random spanning tree generation,

Aaron Schild (2017) https://arxiv.org/abs/1711.06455

support = IntegerGreaterThan(lower_bound=0)
validate_edges(edges)[source]

Validates a batch of edges tensors, as returned by sample() or enumerate_support() or as input to log_prob().

Parameters

edges (torch.LongTensor) – A batch of edges.

Raises

ValueError

Returns

None

### Stable¶

class Stable(stability, skew, scale=1.0, loc=0.0, coords='S0', validate_args=None)[source]

Levy $$\alpha$$-stable distribution. See  for a review.

This uses Nolan’s parametrization  of the loc parameter, which is required for continuity and differentiability. This corresponds to the notation $$S^0_\alpha(\beta,\sigma,\mu_0)$$ of , where $$\alpha$$ = stability, $$\beta$$ = skew, $$\sigma$$ = scale, and $$\mu_0$$ = loc. To instead use the S parameterization as in scipy, pass coords="S", but BEWARE this is discontinuous at stability=1 and has poor geometry for inference.

This implements a reparametrized sampler rsample() , but does not implement log_prob() . Inference can be performed using either likelihood-free algorithms such as EnergyDistance, or reparameterization via the reparam() handler with one of the reparameterizers LatentStableReparam , SymmetricStableReparam , or StableReparam e.g.:

with poutine.reparam(config={"x": StableReparam()}):
pyro.sample("x", Stable(stability, skew, scale, loc))


or simply wrap in MinimalReparam or AutoReparam , e.g.:

@MinimalReparam()
def model():
...

 S. Borak, W. Hardle, R. Weron (2005).

Stable distributions. https://edoc.hu-berlin.de/bitstream/handle/18452/4526/8.pdf

 J.P. Nolan (1997).

Numerical calculation of stable densities and distribution functions.

 Rafal Weron (1996).

On the Chambers-Mallows-Stuck Method for Simulating Skewed Stable Random Variables.

 J.P. Nolan (2017).

Stable Distributions: Models for Heavy Tailed Data. http://fs2.american.edu/jpnolan/www/stable/chap1.pdf

Parameters
• stability (Tensor) – Levy stability parameter $$\alpha\in(0,2]$$ .

• skew (Tensor) – Skewness $$\beta\in[-1,1]$$ .

• scale (Tensor) – Scale $$\sigma > 0$$ . Defaults to 1.

• loc (Tensor) – Location $$\mu_0$$ when using Nolan’s S0 parametrization , or $$\mu$$ when using the S parameterization. Defaults to 0.

• coords (str) – Either “S0” (default) to use Nolan’s continuous S0 parametrization, or “S” to use the discontinuous parameterization.

arg_constraints = {'loc': Real(), 'scale': GreaterThan(lower_bound=0.0), 'skew': Interval(lower_bound=-1, upper_bound=1), 'stability': Interval(lower_bound=0, upper_bound=2)}
expand(batch_shape, _instance=None)[source]
has_rsample = True
log_prob(value)[source]
property mean
rsample(sample_shape=torch.Size([]))[source]
support = Real()
property variance

### TruncatedPolyaGamma¶

class TruncatedPolyaGamma(prototype, validate_args=None)[source]

This is a PolyaGamma(1, 0) distribution truncated to have finite support in the interval (0, 2.5). See  for details. As a consequence of the truncation the log_prob method is only accurate to about six decimal places. In addition the provided sampler is a rough approximation that is only meant to be used in contexts where sample accuracy is not important (e.g. in initialization). Broadly, this implementation is only intended for usage in cases where good approximations of the log_prob are sufficient, as is the case e.g. in HMC.

Parameters

prototype (tensor) – A prototype tensor of arbitrary shape used to determine the dtype and device returned by sample and log_prob.

References

 ‘Bayesian inference for logistic models using Polya-Gamma latent variables’

Nicholas G. Polson, James G. Scott, Jesse Windle.

arg_constraints = {}
expand(batch_shape, _instance=None)[source]
has_rsample = False
log_prob(value)[source]
num_gamma_variates = 8
num_log_prob_terms = 7
sample(sample_shape=())[source]
support = Interval(lower_bound=0.0, upper_bound=2.5)
truncation_point = 2.5

### Unit¶

class Unit(log_factor, *, has_rsample=None, validate_args=None)[source]

Trivial nonnormalized distribution representing the unit type.

The unit type has a single value with no data, i.e. value.numel() == 0.

This is used for pyro.factor() statements.

arg_constraints = {'log_factor': Real()}
expand(batch_shape, _instance=None)[source]
log_prob(value)[source]
rsample(sample_shape=torch.Size([]))[source]
sample(sample_shape=torch.Size([]))[source]
support = Real()

### VonMises3D¶

class VonMises3D(concentration, validate_args=None)[source]

Spherical von Mises distribution.

This implementation combines the direction parameter and concentration parameter into a single combined parameter that contains both direction and magnitude. The value arg is represented in cartesian coordinates: it must be a normalized 3-vector that lies on the 2-sphere.

See VonMises for a 2D polar coordinate cousin of this distribution. See projected_normal for a qualitatively similar distribution but implementing more functionality.

Currently only log_prob() is implemented.

Parameters

concentration (torch.Tensor) – A combined location-and-concentration vector. The direction of this vector is the location, and its magnitude is the concentration.

arg_constraints = {'concentration': Real()}
expand(batch_shape)[source]
log_prob(value)[source]
support = Sphere

### ZeroInflatedDistribution¶

class ZeroInflatedDistribution(base_dist, *, gate=None, gate_logits=None, validate_args=None)[source]

Generic Zero Inflated distribution.

This can be used directly or can be used as a base class as e.g. for ZeroInflatedPoisson and ZeroInflatedNegativeBinomial.

Parameters
• base_dist (TorchDistribution) – the base distribution.

• gate (torch.Tensor) – probability of extra zeros given via a Bernoulli distribution.

• gate_logits (torch.Tensor) – logits of extra zeros given via a Bernoulli distribution.

arg_constraints = {'gate': Interval(lower_bound=0.0, upper_bound=1.0), 'gate_logits': Real()}
expand(batch_shape, _instance=None)[source]
property gate
property gate_logits
log_prob(value)[source]
property mean
sample(sample_shape=torch.Size([]))[source]
property support
property variance

### ZeroInflatedNegativeBinomial¶

class ZeroInflatedNegativeBinomial(total_count, *, probs=None, logits=None, gate=None, gate_logits=None, validate_args=None)[source]

A Zero Inflated Negative Binomial distribution.

Parameters
• total_count (float or torch.Tensor) – non-negative number of negative Bernoulli trials.

• probs (torch.Tensor) – Event probabilities of success in the half open interval [0, 1).

• logits (torch.Tensor) – Event log-odds for probabilities of success.

• gate (torch.Tensor) – probability of extra zeros.

• gate_logits (torch.Tensor) – logits of extra zeros.

arg_constraints = {'gate': Interval(lower_bound=0.0, upper_bound=1.0), 'gate_logits': Real(), 'logits': Real(), 'probs': HalfOpenInterval(lower_bound=0.0, upper_bound=1.0), 'total_count': GreaterThanEq(lower_bound=0)}
property logits
property probs
support = IntegerGreaterThan(lower_bound=0)
property total_count

### ZeroInflatedPoisson¶

class ZeroInflatedPoisson(rate, *, gate=None, gate_logits=None, validate_args=None)[source]

A Zero Inflated Poisson distribution.

Parameters
arg_constraints = {'gate': Interval(lower_bound=0.0, upper_bound=1.0), 'gate_logits': Real(), 'rate': GreaterThan(lower_bound=0.0)}
property rate
support = IntegerGreaterThan(lower_bound=0)

## Transforms¶

### ConditionalTransform¶

class ConditionalTransform[source]

Bases: abc.ABC

abstract condition(context)[source]
Return type

torch.distributions.Transform

### CholeskyTransform¶

class CholeskyTransform(cache_size=0)[source]

Transform via the mapping $$y = safe_cholesky(x)$$, where x is a positive definite matrix.

bijective = True
codomain: torch.distributions.constraints.Constraint = LowerCholesky()
domain: torch.distributions.constraints.Constraint = PositiveDefinite()
log_abs_det_jacobian(x, y)[source]

### CorrMatrixCholeskyTransform¶

class CorrMatrixCholeskyTransform(cache_size=0)[source]

Transform via the mapping $$y = safe_cholesky(x)$$, where x is a correlation matrix.

bijective = True
codomain: torch.distributions.constraints.Constraint = CorrCholesky()
domain: torch.distributions.constraints.Constraint = CorrMatrix()
log_abs_det_jacobian(x, y)[source]

### DiscreteCosineTransform¶

class DiscreteCosineTransform(dim=- 1, smooth=0.0, cache_size=0)[source]

Discrete Cosine Transform of type-II.

This uses dct() and idct() to compute orthonormal DCT and inverse DCT transforms. The jacobian is 1.

Parameters
• dim (int) – Dimension along which to transform. Must be negative. This is an absolute dim counting from the right.

• smooth (float) – Smoothing parameter. When 0, this transforms white noise to white noise; when 1 this transforms Brownian noise to to white noise; when -1 this transforms violet noise to white noise; etc. Any real number is allowed. https://en.wikipedia.org/wiki/Colors_of_noise.

bijective = True
property codomain
property domain
forward_shape(shape)[source]
inverse_shape(shape)[source]
log_abs_det_jacobian(x, y)[source]
with_cache(cache_size=1)[source]

### ELUTransform¶

class ELUTransform(cache_size=0)[source]

Bijective transform via the mapping $$y = \text{ELU}(x)$$.

bijective = True
codomain: torch.distributions.constraints.Constraint = GreaterThan(lower_bound=0.0)
domain: torch.distributions.constraints.Constraint = Real()
log_abs_det_jacobian(x, y)[source]
sign = 1

### HaarTransform¶

class HaarTransform(dim=- 1, flip=False, cache_size=0)[source]

Discrete Haar transform.

This uses haar_transform() and inverse_haar_transform() to compute (orthonormal) Haar and inverse Haar transforms. The jacobian is 1. For sequences with length T not a power of two, this implementation is equivalent to a block-structured Haar transform in which block sizes decrease by factors of one half from left to right.

Parameters
• dim (int) – Dimension along which to transform. Must be negative. This is an absolute dim counting from the right.

• flip (bool) – Whether to flip the time axis before applying the Haar transform. Defaults to false.

bijective = True
property codomain
property domain
forward_shape(shape)[source]
inverse_shape(shape)[source]
log_abs_det_jacobian(x, y)[source]
with_cache(cache_size=1)[source]

### LeakyReLUTransform¶

class LeakyReLUTransform(cache_size=0)[source]

Bijective transform via the mapping $$y = \text{LeakyReLU}(x)$$.

bijective = True
codomain: torch.distributions.constraints.Constraint = Real()
domain: torch.distributions.constraints.Constraint = Real()
log_abs_det_jacobian(x, y)[source]
sign = 1

### LowerCholeskyAffine¶

class LowerCholeskyAffine(loc, scale_tril, cache_size=0)[source]

A bijection of the form,

$$\mathbf{y} = \mathbf{L} \mathbf{x} + \mathbf{r}$$

where mathbf{L} is a lower triangular matrix and mathbf{r} is a vector.

Parameters
• loc (torch.tensor) – the fixed D-dimensional vector to shift the input by.

• scale_tril (torch.tensor) – the D x D lower triangular matrix used in the transformation.

bijective = True
codomain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 1)
domain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 1)
log_abs_det_jacobian(x, y)[source]

Calculates the elementwise determinant of the log Jacobian, i.e. log(abs(dy/dx)).

volume_preserving = False
with_cache(cache_size=1)[source]

### Normalize¶

class Normalize(p=2, cache_size=0)[source]

Safely project a vector onto the sphere wrt the p norm. This avoids the singularity at zero by mapping to the vector [1, 0, 0, ..., 0].

bijective = False
codomain: torch.distributions.constraints.Constraint = Sphere
domain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 1)
with_cache(cache_size=1)[source]

### OrderedTransform¶

class OrderedTransform(cache_size=0)[source]

Transforms a real vector into an ordered vector.

Specifically, enforces monotonically increasing order on the last dimension of a given tensor via the transformation $$y_0 = x_0$$, $$y_i = \sum_{1 \le j \le i} \exp(x_i)$$

bijective = True
codomain: torch.distributions.constraints.Constraint = OrderedVector()
domain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 1)
log_abs_det_jacobian(x, y)[source]

### Permute¶

class Permute(permutation, *, dim=- 1, cache_size=1)[source]

A bijection that reorders the input dimensions, that is, multiplies the input by a permutation matrix. This is useful in between AffineAutoregressive transforms to increase the flexibility of the resulting distribution and stabilize learning. Whilst not being an autoregressive transform, the log absolute determinate of the Jacobian is easily calculable as 0. Note that reordering the input dimension between two layers of AffineAutoregressive is not equivalent to reordering the dimension inside the MADE networks that those IAFs use; using a Permute transform results in a distribution with more flexibility.

Example usage:

>>> from pyro.nn import AutoRegressiveNN
>>> from pyro.distributions.transforms import AffineAutoregressive, Permute
>>> base_dist = dist.Normal(torch.zeros(10), torch.ones(10))
>>> iaf1 = AffineAutoregressive(AutoRegressiveNN(10, ))
>>> ff = Permute(torch.randperm(10, dtype=torch.long))
>>> iaf2 = AffineAutoregressive(AutoRegressiveNN(10, ))
>>> flow_dist = dist.TransformedDistribution(base_dist, [iaf1, ff, iaf2])
>>> flow_dist.sample()

Parameters
• permutation (torch.LongTensor) – a permutation ordering that is applied to the inputs.

• dim (int) – the tensor dimension to permute. This value must be negative and defines the event dim as abs(dim).

bijective = True
property codomain
property domain
property inv_permutation
log_abs_det_jacobian(x, y)[source]

Calculates the elementwise determinant of the log Jacobian, i.e. log(abs([dy_0/dx_0, …, dy_{N-1}/dx_{N-1}])). Note that this type of transform is not autoregressive, so the log Jacobian is not the sum of the previous expression. However, it turns out it’s always 0 (since the determinant is -1 or +1), and so returning a vector of zeros works.

volume_preserving = True
with_cache(cache_size=1)[source]

### PositivePowerTransform¶

class PositivePowerTransform(exponent, *, cache_size=0, validate_args=None)[source]

Transform via the mapping $$y=\operatorname{sign}(x)|x|^{\text{exponent}}$$.

Whereas PowerTransform allows arbitrary exponent and restricts domain and codomain to postive values, this class restricts exponent > 0 and allows real domain and codomain.

Warning

The Jacobian is typically zero or infinite at the origin.

bijective = True
codomain: torch.distributions.constraints.Constraint = Real()
domain: torch.distributions.constraints.Constraint = Real()
forward_shape(shape)[source]
inverse_shape(shape)[source]
log_abs_det_jacobian(x, y)[source]
sign = 1
with_cache(cache_size=1)[source]

### SoftplusLowerCholeskyTransform¶

class SoftplusLowerCholeskyTransform(cache_size=0)[source]

Transform from unconstrained matrices to lower-triangular matrices with nonnegative diagonal entries. This is useful for parameterizing positive definite matrices in terms of their Cholesky factorization.

codomain: torch.distributions.constraints.Constraint = LowerCholesky()
domain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 2)

### SoftplusTransform¶

class SoftplusTransform(cache_size=0)[source]

Transform via the mapping $$\text{Softplus}(x) = \log(1 + \exp(x))$$.

bijective = True
codomain: torch.distributions.constraints.Constraint = GreaterThan(lower_bound=0.0)
domain: torch.distributions.constraints.Constraint = Real()
log_abs_det_jacobian(x, y)[source]
sign = 1

### UnitLowerCholeskyTransform¶

class UnitLowerCholeskyTransform(cache_size=0)[source]

Transform from unconstrained matrices to lower-triangular matrices with all ones diagonals.

codomain: torch.distributions.constraints.Constraint = UnitLowerCholesky()
domain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 2)

## TransformModules¶

### AffineAutoregressive¶

class AffineAutoregressive(autoregressive_nn, log_scale_min_clip=- 5.0, log_scale_max_clip=3.0, sigmoid_bias=2.0, stable=False)[source]

An implementation of the bijective transform of Inverse Autoregressive Flow (IAF), using by default Eq (10) from Kingma Et Al., 2016,

$$\mathbf{y} = \mu_t + \sigma_t\odot\mathbf{x}$$

where $$\mathbf{x}$$ are the inputs, $$\mathbf{y}$$ are the outputs, $$\mu_t,\sigma_t$$ are calculated from an autoregressive network on $$\mathbf{x}$$, and $$\sigma_t>0$$.

If the stable keyword argument is set to True then the transformation used is,

$$\mathbf{y} = \sigma_t\odot\mathbf{x} + (1-\sigma_t)\odot\mu_t$$

where $$\sigma_t$$ is restricted to $$(0,1)$$. This variant of IAF is claimed by the authors to be more numerically stable than one using Eq (10), although in practice it leads to a restriction on the distributions that can be represented, presumably since the input is restricted to rescaling by a number on $$(0,1)$$.

Together with TransformedDistribution this provides a way to create richer variational approximations.

Example usage:

>>> from pyro.nn import AutoRegressiveNN
>>> base_dist = dist.Normal(torch.zeros(10), torch.ones(10))
>>> transform = AffineAutoregressive(AutoRegressiveNN(10, ))
>>> pyro.module("my_transform", transform)
>>> flow_dist = dist.TransformedDistribution(base_dist, [transform])
>>> flow_dist.sample()


The inverse of the Bijector is required when, e.g., scoring the log density of a sample with TransformedDistribution. This implementation caches the inverse of the Bijector when its forward operation is called, e.g., when sampling from TransformedDistribution. However, if the cached value isn’t available, either because it was overwritten during sampling a new value or an arbitrary value is being scored, it will calculate it manually. Note that this is an operation that scales as O(D) where D is the input dimension, and so should be avoided for large dimensional uses. So in general, it is cheap to sample from IAF and score a value that was sampled by IAF, but expensive to score an arbitrary value.

Parameters
• autoregressive_nn (callable) – an autoregressive neural network whose forward call returns a real-valued mean and logit-scale as a tuple

• log_scale_min_clip (float) – The minimum value for clipping the log(scale) from the autoregressive NN

• log_scale_max_clip (float) – The maximum value for clipping the log(scale) from the autoregressive NN

• sigmoid_bias (float) – A term to add the logit of the input when using the stable tranform.

• stable (bool) – When true, uses the alternative “stable” version of the transform (see above).

References:

 Diederik P. Kingma, Tim Salimans, Rafal Jozefowicz, Xi Chen, Ilya Sutskever, Max Welling. Improving Variational Inference with Inverse Autoregressive Flow. [arXiv:1606.04934]

 Danilo Jimenez Rezende, Shakir Mohamed. Variational Inference with Normalizing Flows. [arXiv:1505.05770]

 Mathieu Germain, Karol Gregor, Iain Murray, Hugo Larochelle. MADE: Masked Autoencoder for Distribution Estimation. [arXiv:1502.03509]

autoregressive = True
bijective = True
codomain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 1)
domain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 1)
log_abs_det_jacobian(x, y)[source]

Calculates the elementwise determinant of the log Jacobian

sign = 1

### AffineCoupling¶

class AffineCoupling(split_dim, hypernet, *, dim=- 1, log_scale_min_clip=- 5.0, log_scale_max_clip=3.0)[source]

An implementation of the affine coupling layer of RealNVP (Dinh et al., 2017) that uses the bijective transform,

$$\mathbf{y}_{1:d} = \mathbf{x}_{1:d}$$ $$\mathbf{y}_{(d+1):D} = \mu + \sigma\odot\mathbf{x}_{(d+1):D}$$

where $$\mathbf{x}$$ are the inputs, $$\mathbf{y}$$ are the outputs, e.g. $$\mathbf{x}_{1:d}$$ represents the first $$d$$ elements of the inputs, and $$\mu,\sigma$$ are shift and translation parameters calculated as the output of a function inputting only $$\mathbf{x}_{1:d}$$.

That is, the first $$d$$ components remain unchanged, and the subsequent $$D-d$$ are shifted and translated by a function of the previous components.

Together with TransformedDistribution this provides a way to create richer variational approximations.

Example usage:

>>> from pyro.nn import DenseNN
>>> input_dim = 10
>>> split_dim = 6
>>> base_dist = dist.Normal(torch.zeros(input_dim), torch.ones(input_dim))
>>> param_dims = [input_dim-split_dim, input_dim-split_dim]
>>> hypernet = DenseNN(split_dim, [10*input_dim], param_dims)
>>> transform = AffineCoupling(split_dim, hypernet)
>>> pyro.module("my_transform", transform)
>>> flow_dist = dist.TransformedDistribution(base_dist, [transform])
>>> flow_dist.sample()


The inverse of the Bijector is required when, e.g., scoring the log density of a sample with TransformedDistribution. This implementation caches the inverse of the Bijector when its forward operation is called, e.g., when sampling from TransformedDistribution. However, if the cached value isn’t available, either because it was overwritten during sampling a new value or an arbitary value is being scored, it will calculate it manually.

This is an operation that scales as O(1), i.e. constant in the input dimension. So in general, it is cheap to sample and score (an arbitrary value) from AffineCoupling.

Parameters
• split_dim (int) – Zero-indexed dimension $$d$$ upon which to perform input/ output split for transformation.

• hypernet (callable) – a neural network whose forward call returns a real-valued mean and logit-scale as a tuple. The input should have final dimension split_dim and the output final dimension input_dim-split_dim for each member of the tuple.

• dim (int) – the tensor dimension on which to split. This value must be negative and defines the event dim as abs(dim).

• log_scale_min_clip (float) – The minimum value for clipping the log(scale) from the autoregressive NN

• log_scale_max_clip (float) – The maximum value for clipping the log(scale) from the autoregressive NN

References:

 Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. Density estimation using Real NVP. ICLR 2017.

bijective = True
property codomain
property domain
log_abs_det_jacobian(x, y)[source]

Calculates the elementwise determinant of the log jacobian

### BatchNorm¶

class BatchNorm(input_dim, momentum=0.1, epsilon=1e-05)[source]

A type of batch normalization that can be used to stabilize training in normalizing flows. The inverse operation is defined as

$$x = (y - \hat{\mu}) \oslash \sqrt{\hat{\sigma^2}} \otimes \gamma + \beta$$

that is, the standard batch norm equation, where $$x$$ is the input, $$y$$ is the output, $$\gamma,\beta$$ are learnable parameters, and $$\hat{\mu}$$/$$\hat{\sigma^2}$$ are smoothed running averages of the sample mean and variance, respectively. The constraint $$\gamma>0$$ is enforced to ease calculation of the log-det-Jacobian term.

This is an element-wise transform, and when applied to a vector, learns two parameters ($$\gamma,\beta$$) for each dimension of the input.

When the module is set to training mode, the moving averages of the sample mean and variance are updated every time the inverse operator is called, e.g., when a normalizing flow scores a minibatch with the log_prob method.

Also, when the module is set to training mode, the sample mean and variance on the current minibatch are used in place of the smoothed averages, $$\hat{\mu}$$ and $$\hat{\sigma^2}$$, for the inverse operator. For this reason it is not the case that $$x=g(g^{-1}(x))$$ during training, i.e., that the inverse operation is the inverse of the forward one.

Example usage:

>>> from pyro.nn import AutoRegressiveNN
>>> from pyro.distributions.transforms import AffineAutoregressive
>>> base_dist = dist.Normal(torch.zeros(10), torch.ones(10))
>>> iafs = [AffineAutoregressive(AutoRegressiveNN(10, )) for _ in range(2)]
>>> bn = BatchNorm(10)
>>> flow_dist = dist.TransformedDistribution(base_dist, [iafs, bn, iafs])
>>> flow_dist.sample()

Parameters
• input_dim (int) – the dimension of the input

• momentum (float) – momentum parameter for updating moving averages

• epsilon (float) – small number to add to variances to ensure numerical stability

References:

 Sergey Ioffe and Christian Szegedy. Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift. In International Conference on Machine Learning, 2015. https://arxiv.org/abs/1502.03167

 Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. Density Estimation using Real NVP. In International Conference on Learning Representations, 2017. https://arxiv.org/abs/1605.08803

 George Papamakarios, Theo Pavlakou, and Iain Murray. Masked Autoregressive Flow for Density Estimation. In Neural Information Processing Systems, 2017. https://arxiv.org/abs/1705.07057

bijective = True
codomain: torch.distributions.constraints.Constraint = Real()
property constrained_gamma
domain: torch.distributions.constraints.Constraint = Real()
log_abs_det_jacobian(x, y)[source]

Calculates the elementwise determinant of the log Jacobian, dx/dy

### BlockAutoregressive¶

class BlockAutoregressive(input_dim, hidden_factors=[8, 8], activation='tanh', residual=None)[source]

An implementation of Block Neural Autoregressive Flow (block-NAF) (De Cao et al., 2019) bijective transform. Block-NAF uses a similar transformation to deep dense NAF, building the autoregressive NN into the structure of the transform, in a sense.

Together with TransformedDistribution this provides a way to create richer variational approximations.

Example usage:

>>> base_dist = dist.Normal(torch.zeros(10), torch.ones(10))
>>> naf = BlockAutoregressive(input_dim=10)
>>> pyro.module("my_naf", naf)
>>> naf_dist = dist.TransformedDistribution(base_dist, [naf])
>>> naf_dist.sample()


The inverse operation is not implemented. This would require numerical inversion, e.g., using a root finding method - a possibility for a future implementation.

Parameters
• input_dim (int) – The dimensionality of the input and output variables.

• hidden_factors (list) – Hidden layer i has hidden_factors[i] hidden units per input dimension. This corresponds to both $$a$$ and $$b$$ in De Cao et al. (2019). The elements of hidden_factors must be integers.

• activation (string) – Activation function to use. One of ‘ELU’, ‘LeakyReLU’, ‘sigmoid’, or ‘tanh’.

• residual (string) – Type of residual connections to use. Choices are “None”, “normal” for $$\mathbf{y}+f(\mathbf{y})$$, and “gated” for $$\alpha\mathbf{y} + (1 - \alpha\mathbf{y})$$ for learnable parameter $$\alpha$$.

References:

 Nicola De Cao, Ivan Titov, Wilker Aziz. Block Neural Autoregressive Flow. [arXiv:1904.04676]

autoregressive = True
bijective = True
codomain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 1)
domain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 1)
log_abs_det_jacobian(x, y)[source]

Calculates the elementwise determinant of the log jacobian

### ConditionalAffineAutoregressive¶

class ConditionalAffineAutoregressive(autoregressive_nn, **kwargs)[source]

An implementation of the bijective transform of Inverse Autoregressive Flow (IAF) that conditions on an additional context variable and uses, by default, Eq (10) from Kingma Et Al., 2016,

$$\mathbf{y} = \mu_t + \sigma_t\odot\mathbf{x}$$

where $$\mathbf{x}$$ are the inputs, $$\mathbf{y}$$ are the outputs, $$\mu_t,\sigma_t$$ are calculated from an autoregressive network on $$\mathbf{x}$$ and context $$\mathbf{z}\in\mathbb{R}^M$$, and $$\sigma_t>0$$.

If the stable keyword argument is set to True then the transformation used is,

$$\mathbf{y} = \sigma_t\odot\mathbf{x} + (1-\sigma_t)\odot\mu_t$$

where $$\sigma_t$$ is restricted to $$(0,1)$$. This variant of IAF is claimed by the authors to be more numerically stable than one using Eq (10), although in practice it leads to a restriction on the distributions that can be represented, presumably since the input is restricted to rescaling by a number on $$(0,1)$$.

Together with ConditionalTransformedDistribution this provides a way to create richer variational approximations.

Example usage:

>>> from pyro.nn import ConditionalAutoRegressiveNN
>>> input_dim = 10
>>> context_dim = 4
>>> batch_size = 3
>>> hidden_dims = [10*input_dim, 10*input_dim]
>>> base_dist = dist.Normal(torch.zeros(input_dim), torch.ones(input_dim))
>>> hypernet = ConditionalAutoRegressiveNN(input_dim, context_dim, hidden_dims)
>>> transform = ConditionalAffineAutoregressive(hypernet)
>>> pyro.module("my_transform", transform)
>>> z = torch.rand(batch_size, context_dim)
>>> flow_dist = dist.ConditionalTransformedDistribution(base_dist,
... [transform]).condition(z)
>>> flow_dist.sample(sample_shape=torch.Size([batch_size]))


The inverse of the Bijector is required when, e.g., scoring the log density of a sample with TransformedDistribution. This implementation caches the inverse of the Bijector when its forward operation is called, e.g., when sampling from TransformedDistribution. However, if the cached value isn’t available, either because it was overwritten during sampling a new value or an arbitrary value is being scored, it will calculate it manually. Note that this is an operation that scales as O(D) where D is the input dimension, and so should be avoided for large dimensional uses. So in general, it is cheap to sample from IAF and score a value that was sampled by IAF, but expensive to score an arbitrary value.

Parameters
• autoregressive_nn (nn.Module) – an autoregressive neural network whose forward call returns a real-valued mean and logit-scale as a tuple

• log_scale_min_clip (float) – The minimum value for clipping the log(scale) from the autoregressive NN

• log_scale_max_clip (float) – The maximum value for clipping the log(scale) from the autoregressive NN

• sigmoid_bias (float) – A term to add the logit of the input when using the stable tranform.

• stable (bool) – When true, uses the alternative “stable” version of the transform (see above).

References:

 Diederik P. Kingma, Tim Salimans, Rafal Jozefowicz, Xi Chen, Ilya Sutskever, Max Welling. Improving Variational Inference with Inverse Autoregressive Flow. [arXiv:1606.04934]

 Danilo Jimenez Rezende, Shakir Mohamed. Variational Inference with Normalizing Flows. [arXiv:1505.05770]

 Mathieu Germain, Karol Gregor, Iain Murray, Hugo Larochelle. MADE: Masked Autoencoder for Distribution Estimation. [arXiv:1502.03509]

bijective = True
codomain = IndependentConstraint(Real(), 1)
condition(context)[source]

Conditions on a context variable, returning a non-conditional transform of of type AffineAutoregressive.

domain = IndependentConstraint(Real(), 1)
training: bool

### ConditionalAffineCoupling¶

class ConditionalAffineCoupling(split_dim, hypernet, **kwargs)[source]

An implementation of the affine coupling layer of RealNVP (Dinh et al., 2017) that conditions on an additional context variable and uses the bijective transform,

$$\mathbf{y}_{1:d} = \mathbf{x}_{1:d}$$ $$\mathbf{y}_{(d+1):D} = \mu + \sigma\odot\mathbf{x}_{(d+1):D}$$

where $$\mathbf{x}$$ are the inputs, $$\mathbf{y}$$ are the outputs, e.g. $$\mathbf{x}_{1:d}$$ represents the first $$d$$ elements of the inputs, and $$\mu,\sigma$$ are shift and translation parameters calculated as the output of a function input $$\mathbf{x}_{1:d}$$ and a context variable $$\mathbf{z}\in\mathbb{R}^M$$.

That is, the first $$d$$ components remain unchanged, and the subsequent $$D-d$$ are shifted and translated by a function of the previous components.

Together with ConditionalTransformedDistribution this provides a way to create richer variational approximations.

Example usage:

>>> from pyro.nn import ConditionalDenseNN
>>> input_dim = 10
>>> split_dim = 6
>>> context_dim = 4
>>> batch_size = 3
>>> base_dist = dist.Normal(torch.zeros(input_dim), torch.ones(input_dim))
>>> param_dims = [input_dim-split_dim, input_dim-split_dim]
>>> hypernet = ConditionalDenseNN(split_dim, context_dim, [10*input_dim],
... param_dims)
>>> transform = ConditionalAffineCoupling(split_dim, hypernet)
>>> pyro.module("my_transform", transform)
>>> z = torch.rand(batch_size, context_dim)
>>> flow_dist = dist.ConditionalTransformedDistribution(base_dist,
... [transform]).condition(z)
>>> flow_dist.sample(sample_shape=torch.Size([batch_size]))


The inverse of the Bijector is required when, e.g., scoring the log density of a sample with ConditionalTransformedDistribution. This implementation caches the inverse of the Bijector when its forward operation is called, e.g., when sampling from ConditionalTransformedDistribution. However, if the cached value isn’t available, either because it was overwritten during sampling a new value or an arbitary value is being scored, it will calculate it manually.

This is an operation that scales as O(1), i.e. constant in the input dimension. So in general, it is cheap to sample and score (an arbitrary value) from ConditionalAffineCoupling.

Parameters
• split_dim (int) – Zero-indexed dimension $$d$$ upon which to perform input/ output split for transformation.

• hypernet (callable) – A neural network whose forward call returns a real-valued mean and logit-scale as a tuple. The input should have final dimension split_dim and the output final dimension input_dim-split_dim for each member of the tuple. The network also inputs a context variable as a keyword argument in order to condition the output upon it.

• log_scale_min_clip (float) – The minimum value for clipping the log(scale) from the NN

• log_scale_max_clip (float) – The maximum value for clipping the log(scale) from the NN

References:

Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. Density estimation using Real NVP. ICLR 2017.

bijective = True
codomain = IndependentConstraint(Real(), 1)
condition(context)[source]

See pyro.distributions.conditional.ConditionalTransformModule.condition()

domain = IndependentConstraint(Real(), 1)
training: bool

### ConditionalGeneralizedChannelPermute¶

class ConditionalGeneralizedChannelPermute(nn, channels=3, permutation=None)[source]

A bijection that generalizes a permutation on the channels of a batch of 2D image in $$[\ldots,C,H,W]$$ format conditioning on an additional context variable. Specifically this transform performs the operation,

$$\mathbf{y} = \text{torch.nn.functional.conv2d}(\mathbf{x}, W)$$

where $$\mathbf{x}$$ are the inputs, $$\mathbf{y}$$ are the outputs, and $$W\sim C\times C\times 1\times 1$$ is the filter matrix for a 1x1 convolution with $$C$$ input and output channels.

Ignoring the final two dimensions, $$W$$ is restricted to be the matrix product,

$$W = PLU$$

where $$P\sim C\times C$$ is a permutation matrix on the channel dimensions, and $$LU\sim C\times C$$ is an invertible product of a lower triangular and an upper triangular matrix that is the output of an NN with input $$z\in\mathbb{R}^{M}$$ representing the context variable to condition on.

The input $$\mathbf{x}$$ and output $$\mathbf{y}$$ both have shape […,C,H,W], where C is the number of channels set at initialization.

This operation was introduced in  for Glow normalizing flow, and is also known as 1x1 invertible convolution. It appears in other notable work such as [2,3], and corresponds to the class tfp.bijectors.MatvecLU of TensorFlow Probability.

Example usage:

>>> from pyro.nn.dense_nn import DenseNN
>>> context_dim = 5
>>> batch_size = 3
>>> channels = 3
>>> base_dist = dist.Normal(torch.zeros(channels, 32, 32),
... torch.ones(channels, 32, 32))
>>> hidden_dims = [context_dim*10, context_dim*10]
>>> nn = DenseNN(context_dim, hidden_dims, param_dims=[channels*channels])
>>> transform = ConditionalGeneralizedChannelPermute(nn, channels=channels)
>>> z = torch.rand(batch_size, context_dim)
>>> flow_dist = dist.ConditionalTransformedDistribution(base_dist,
... [transform]).condition(z)
>>> flow_dist.sample(sample_shape=torch.Size([batch_size]))

Parameters
• nn – a function inputting the context variable and outputting real-valued parameters of dimension $$C^2$$.

• channels (int) – Number of channel dimensions in the input.

 Diederik P. Kingma, Prafulla Dhariwal. Glow: Generative Flow with Invertible 1x1 Convolutions. [arXiv:1807.03039]

 Ryan Prenger, Rafael Valle, Bryan Catanzaro. WaveGlow: A Flow-based Generative Network for Speech Synthesis. [arXiv:1811.00002]

 Conor Durkan, Artur Bekasov, Iain Murray, George Papamakarios. Neural Spline Flows. [arXiv:1906.04032]

bijective = True
codomain = IndependentConstraint(Real(), 3)
condition(context)[source]

See pyro.distributions.conditional.ConditionalTransformModule.condition()

domain = IndependentConstraint(Real(), 3)
training: bool

### ConditionalHouseholder¶

class ConditionalHouseholder(input_dim, nn, count_transforms=1)[source]

Represents multiple applications of the Householder bijective transformation conditioning on an additional context. A single Householder transformation takes the form,

$$\mathbf{y} = (I - 2*\frac{\mathbf{u}\mathbf{u}^T}{||\mathbf{u}||^2})\mathbf{x}$$

where $$\mathbf{x}$$ are the inputs with dimension $$D$$, $$\mathbf{y}$$ are the outputs, and $$\mathbf{u}\in\mathbb{R}^D$$ is the output of a function, e.g. a NN, with input $$z\in\mathbb{R}^{M}$$ representing the context variable to condition on.

The transformation represents the reflection of $$\mathbf{x}$$ through the plane passing through the origin with normal $$\mathbf{u}$$.

$$D$$ applications of this transformation are able to transform standard i.i.d. standard Gaussian noise into a Gaussian variable with an arbitrary covariance matrix. With $$K<D$$ transformations, one is able to approximate a full-rank Gaussian distribution using a linear transformation of rank $$K$$.

Together with ConditionalTransformedDistribution this provides a way to create richer variational approximations.

Example usage:

>>> from pyro.nn.dense_nn import DenseNN
>>> input_dim = 10
>>> context_dim = 5
>>> batch_size = 3
>>> base_dist = dist.Normal(torch.zeros(input_dim), torch.ones(input_dim))
>>> param_dims = [input_dim]
>>> hypernet = DenseNN(context_dim, [50, 50], param_dims)
>>> transform = ConditionalHouseholder(input_dim, hypernet)
>>> z = torch.rand(batch_size, context_dim)
>>> flow_dist = dist.ConditionalTransformedDistribution(base_dist,
... [transform]).condition(z)
>>> flow_dist.sample(sample_shape=torch.Size([batch_size]))

Parameters
• input_dim (int) – the dimension of the input (and output) variable.

• nn (callable) – a function inputting the context variable and outputting a triplet of real-valued parameters of dimensions $$(1, D, D)$$.

• count_transforms (int) – number of applications of Householder transformation to apply.

References:

 Jakub M. Tomczak, Max Welling. Improving Variational Auto-Encoders using Householder Flow. [arXiv:1611.09630]

bijective = True
codomain = IndependentConstraint(Real(), 1)
condition(context)[source]

See pyro.distributions.conditional.ConditionalTransformModule.condition()

domain = IndependentConstraint(Real(), 1)
training: bool

### ConditionalMatrixExponential¶

class ConditionalMatrixExponential(input_dim, nn, iterations=8, normalization='none', bound=None)[source]

A dense matrix exponential bijective transform (Hoogeboom et al., 2020) that conditions on an additional context variable with equation,

$$\mathbf{y} = \exp(M)\mathbf{x}$$

where $$\mathbf{x}$$ are the inputs, $$\mathbf{y}$$ are the outputs, $$\exp(\cdot)$$ represents the matrix exponential, and $$M\in\mathbb{R}^D\times\mathbb{R}^D$$ is the output of a neural network conditioning on a context variable $$\mathbf{z}$$ for input dimension $$D$$. In general, $$M$$ is not required to be invertible.

Due to the favourable mathematical properties of the matrix exponential, the transform has an exact inverse and a log-determinate-Jacobian that scales in time-complexity as $$O(D)$$. Both the forward and reverse operations are approximated with a truncated power series. For numerical stability, the norm of $$M$$ can be restricted with the normalization keyword argument.

Example usage:

>>> from pyro.nn.dense_nn import DenseNN
>>> input_dim = 10
>>> context_dim = 5
>>> batch_size = 3
>>> base_dist = dist.Normal(torch.zeros(input_dim), torch.ones(input_dim))
>>> param_dims = [input_dim*input_dim]
>>> hypernet = DenseNN(context_dim, [50, 50], param_dims)
>>> transform = ConditionalMatrixExponential(input_dim, hypernet)
>>> z = torch.rand(batch_size, context_dim)
>>> flow_dist = dist.ConditionalTransformedDistribution(base_dist,
... [transform]).condition(z)
>>> flow_dist.sample(sample_shape=torch.Size([batch_size]))

Parameters
• input_dim (int) – the dimension of the input (and output) variable.

• iterations (int) – the number of terms to use in the truncated power series that approximates matrix exponentiation.

• normalization (string) – One of [‘none’, ‘weight’, ‘spectral’] normalization that selects what type of normalization to apply to the weight matrix. weight corresponds to weight normalization (Salimans and Kingma, 2016) and spectral to spectral normalization (Miyato et al, 2018).

• bound (float) – a bound on either the weight or spectral norm, when either of those two types of regularization are chosen by the normalization argument. A lower value for this results in fewer required terms of the truncated power series to closely approximate the exact value of the matrix exponential.

References:

 Emiel Hoogeboom, Victor Garcia Satorras, Jakub M. Tomczak, Max Welling. The

Convolution Exponential and Generalized Sylvester Flows. [arXiv:2006.01910]

 Tim Salimans, Diederik P. Kingma. Weight Normalization: A Simple

Reparameterization to Accelerate Training of Deep Neural Networks. [arXiv:1602.07868]

 Takeru Miyato, Toshiki Kataoka, Masanori Koyama, Yuichi Yoshida. Spectral

Normalization for Generative Adversarial Networks. ICLR 2018.

bijective = True
codomain = IndependentConstraint(Real(), 1)
condition(context)[source]

See pyro.distributions.conditional.ConditionalTransformModule.condition()

domain = IndependentConstraint(Real(), 1)
training: bool

### ConditionalNeuralAutoregressive¶

class ConditionalNeuralAutoregressive(autoregressive_nn, **kwargs)[source]

An implementation of the deep Neural Autoregressive Flow (NAF) bijective transform of the “IAF flavour” conditioning on an additiona context variable that can be used for sampling and scoring samples drawn from it (but not arbitrary ones).

Example usage:

>>> from pyro.nn import ConditionalAutoRegressiveNN
>>> input_dim = 10
>>> context_dim = 5
>>> batch_size = 3
>>> base_dist = dist.Normal(torch.zeros(input_dim), torch.ones(input_dim))
>>> arn = ConditionalAutoRegressiveNN(input_dim, context_dim, ,
... param_dims=*3)
>>> transform = ConditionalNeuralAutoregressive(arn, hidden_units=16)
>>> pyro.module("my_transform", transform)
>>> z = torch.rand(batch_size, context_dim)
>>> flow_dist = dist.ConditionalTransformedDistribution(base_dist,
... [transform]).condition(z)
>>> flow_dist.sample(sample_shape=torch.Size([batch_size]))


The inverse operation is not implemented. This would require numerical inversion, e.g., using a root finding method - a possibility for a future implementation.

Parameters
• autoregressive_nn (nn.Module) – an autoregressive neural network whose forward call returns a tuple of three real-valued tensors, whose last dimension is the input dimension, and whose penultimate dimension is equal to hidden_units.

• hidden_units (int) – the number of hidden units to use in the NAF transformation (see Eq (8) in reference)

• activation (string) – Activation function to use. One of ‘ELU’, ‘LeakyReLU’, ‘sigmoid’, or ‘tanh’.

Reference:

 Chin-Wei Huang, David Krueger, Alexandre Lacoste, Aaron Courville. Neural Autoregressive Flows. [arXiv:1804.00779]

bijective = True
codomain = IndependentConstraint(Real(), 1)
condition(context)[source]

Conditions on a context variable, returning a non-conditional transform of of type NeuralAutoregressive.

domain = IndependentConstraint(Real(), 1)
training: bool

### ConditionalPlanar¶

class ConditionalPlanar(nn)[source]

A conditional ‘planar’ bijective transform using the equation,

$$\mathbf{y} = \mathbf{x} + \mathbf{u}\tanh(\mathbf{w}^T\mathbf{z}+b)$$

where $$\mathbf{x}$$ are the inputs with dimension $$D$$, $$\mathbf{y}$$ are the outputs, and the pseudo-parameters $$b\in\mathbb{R}$$, $$\mathbf{u}\in\mathbb{R}^D$$, and $$\mathbf{w}\in\mathbb{R}^D$$ are the output of a function, e.g. a NN, with input $$z\in\mathbb{R}^{M}$$ representing the context variable to condition on. For this to be an invertible transformation, the condition $$\mathbf{w}^T\mathbf{u}>-1$$ is enforced.

Together with ConditionalTransformedDistribution this provides a way to create richer variational approximations.

Example usage:

>>> from pyro.nn.dense_nn import DenseNN
>>> input_dim = 10
>>> context_dim = 5
>>> batch_size = 3
>>> base_dist = dist.Normal(torch.zeros(input_dim), torch.ones(input_dim))
>>> param_dims = [1, input_dim, input_dim]
>>> hypernet = DenseNN(context_dim, [50, 50], param_dims)
>>> transform = ConditionalPlanar(hypernet)
>>> z = torch.rand(batch_size, context_dim)
>>> flow_dist = dist.ConditionalTransformedDistribution(base_dist,
... [transform]).condition(z)
>>> flow_dist.sample(sample_shape=torch.Size([batch_size]))


The inverse of this transform does not possess an analytical solution and is left unimplemented. However, the inverse is cached when the forward operation is called during sampling, and so samples drawn using the planar transform can be scored.

Parameters

nn (callable) – a function inputting the context variable and outputting a triplet of real-valued parameters of dimensions $$(1, D, D)$$.

References:  Variational Inference with Normalizing Flows [arXiv:1505.05770] Danilo Jimenez Rezende, Shakir Mohamed

bijective = True
codomain = IndependentConstraint(Real(), 1)
condition(context)[source]

See pyro.distributions.conditional.ConditionalTransformModule.condition()

domain = IndependentConstraint(Real(), 1)
training: bool

A conditional ‘radial’ bijective transform context using the equation,

$$\mathbf{y} = \mathbf{x} + \beta h(\alpha,r)(\mathbf{x} - \mathbf{x}_0)$$

where $$\mathbf{x}$$ are the inputs, $$\mathbf{y}$$ are the outputs, and $$\alpha\in\mathbb{R}^+$$, $$\beta\in\mathbb{R}$$, and $$\mathbf{x}_0\in\mathbb{R}^D$$, are the output of a function, e.g. a NN, with input $$z\in\mathbb{R}^{M}$$ representing the context variable to condition on. The input dimension is $$D$$, $$r=||\mathbf{x}-\mathbf{x}_0||_2$$, and $$h(\alpha,r)=1/(\alpha+r)$$. For this to be an invertible transformation, the condition $$\beta>-\alpha$$ is enforced.

Example usage:

>>> from pyro.nn.dense_nn import DenseNN
>>> input_dim = 10
>>> context_dim = 5
>>> batch_size = 3
>>> base_dist = dist.Normal(torch.zeros(input_dim), torch.ones(input_dim))
>>> param_dims = [input_dim, 1, 1]
>>> hypernet = DenseNN(context_dim, [50, 50], param_dims)
>>> z = torch.rand(batch_size, context_dim)
>>> flow_dist = dist.ConditionalTransformedDistribution(base_dist,
... [transform]).condition(z)
>>> flow_dist.sample(sample_shape=torch.Size([batch_size]))


The inverse of this transform does not possess an analytical solution and is left unimplemented. However, the inverse is cached when the forward operation is called during sampling, and so samples drawn using the radial transform can be scored.

Parameters

input_dim (int) – the dimension of the input (and output) variable.

References:

 Danilo Jimenez Rezende, Shakir Mohamed. Variational Inference with Normalizing Flows. [arXiv:1505.05770]

bijective = True
codomain = IndependentConstraint(Real(), 1)
condition(context)[source]

See pyro.distributions.conditional.ConditionalTransformModule.condition()

domain = IndependentConstraint(Real(), 1)
training: bool

### ConditionalSpline¶

class ConditionalSpline(nn, input_dim, count_bins, bound=3.0, order='linear')[source]

An implementation of the element-wise rational spline bijections of linear and quadratic order (Durkan et al., 2019; Dolatabadi et al., 2020) conditioning on an additional context variable.

Rational splines are functions that are comprised of segments that are the ratio of two polynomials. For instance, for the $$d$$-th dimension and the $$k$$-th segment on the spline, the function will take the form,

$$y_d = \frac{\alpha^{(k)}(x_d)}{\beta^{(k)}(x_d)},$$

where $$\alpha^{(k)}$$ and $$\beta^{(k)}$$ are two polynomials of order $$d$$ whose parameters are the output of a function, e.g. a NN, with input $$z\\in\\mathbb{R}^{M}$$ representing the context variable to condition on.. For $$d=1$$, we say that the spline is linear, and for $$d=2$$, quadratic. The spline is constructed on the specified bounding box, $$[-K,K]\times[-K,K]$$, with the identity function used elsewhere.

Rational splines offer an excellent combination of functional flexibility whilst maintaining a numerically stable inverse that is of the same computational and space complexities as the forward operation. This element-wise transform permits the accurate represention of complex univariate distributions.

Example usage:

>>> from pyro.nn.dense_nn import DenseNN
>>> input_dim = 10
>>> context_dim = 5
>>> batch_size = 3
>>> count_bins = 8
>>> base_dist = dist.Normal(torch.zeros(input_dim), torch.ones(input_dim))
>>> param_dims = [input_dim * count_bins, input_dim * count_bins,
... input_dim * (count_bins - 1), input_dim * count_bins]
>>> hypernet = DenseNN(context_dim, [50, 50], param_dims)
>>> transform = ConditionalSpline(hypernet, input_dim, count_bins)
>>> z = torch.rand(batch_size, context_dim)
>>> flow_dist = dist.ConditionalTransformedDistribution(base_dist,
... [transform]).condition(z)
>>> flow_dist.sample(sample_shape=torch.Size([batch_size]))

Parameters
• input_dim (int) – Dimension of the input vector. This is required so we know how many parameters to store.

• count_bins (int) – The number of segments comprising the spline.

• bound (float) – The quantity $$K$$ determining the bounding box, $$[-K,K]\times[-K,K]$$, of the spline.

• order (string) – One of [‘linear’, ‘quadratic’] specifying the order of the spline.

References:

Conor Durkan, Artur Bekasov, Iain Murray, George Papamakarios. Neural Spline Flows. NeurIPS 2019.

Hadi M. Dolatabadi, Sarah Erfani, Christopher Leckie. Invertible Generative Modeling using Linear Rational Splines. AISTATS 2020.

bijective = True
codomain = Real()
condition(context)[source]

See pyro.distributions.conditional.ConditionalTransformModule.condition()

domain = Real()
training: bool

### ConditionalSplineAutoregressive¶

class ConditionalSplineAutoregressive(input_dim, autoregressive_nn, **kwargs)[source]

An implementation of the autoregressive layer with rational spline bijections of linear and quadratic order (Durkan et al., 2019; Dolatabadi et al., 2020) that conditions on an additional context variable. Rational splines are functions that are comprised of segments that are the ratio of two polynomials (see Spline).

The autoregressive layer uses the transformation,

$$y_d = g_{\theta_d}(x_d)\ \ \ d=1,2,\ldots,D$$

where $$\mathbf{x}=(x_1,x_2,\ldots,x_D)$$ are the inputs, $$\mathbf{y}=(y_1,y_2,\ldots,y_D)$$ are the outputs, $$g_{\theta_d}$$ is an elementwise rational monotonic spline with parameters $$\theta_d$$, and $$\theta=(\theta_1,\theta_2,\ldots,\theta_D)$$ is the output of a conditional autoregressive NN inputting $$\mathbf{x}$$ and conditioning on the context variable $$\mathbf{z}$$.

Example usage:

>>> from pyro.nn import ConditionalAutoRegressiveNN
>>> input_dim = 10
>>> count_bins = 8
>>> context_dim = 5
>>> batch_size = 3
>>> base_dist = dist.Normal(torch.zeros(input_dim), torch.ones(input_dim))
>>> hidden_dims = [input_dim * 10, input_dim * 10]
>>> param_dims = [count_bins, count_bins, count_bins - 1, count_bins]
>>> hypernet = ConditionalAutoRegressiveNN(input_dim, context_dim, hidden_dims,
... param_dims=param_dims)
>>> transform = ConditionalSplineAutoregressive(input_dim, hypernet,
... count_bins=count_bins)
>>> pyro.module("my_transform", transform)
>>> z = torch.rand(batch_size, context_dim)
>>> flow_dist = dist.ConditionalTransformedDistribution(base_dist,
... [transform]).condition(z)
>>> flow_dist.sample(sample_shape=torch.Size([batch_size]))

Parameters
• input_dim (int) – Dimension of the input vector. Despite operating element-wise, this is required so we know how many parameters to store.

• autoregressive_nn (callable) – an autoregressive neural network whose forward call returns tuple of the spline parameters

• count_bins (int) – The number of segments comprising the spline.

• bound (float) – The quantity $$K$$ determining the bounding box, $$[-K,K]\times[-K,K]$$, of the spline.

• order (string) – One of [‘linear’, ‘quadratic’] specifying the order of the spline.

References:

Conor Durkan, Artur Bekasov, Iain Murray, George Papamakarios. Neural Spline Flows. NeurIPS 2019.

Hadi M. Dolatabadi, Sarah Erfani, Christopher Leckie. Invertible Generative Modeling using Linear Rational Splines. AISTATS 2020.

bijective = True
codomain = IndependentConstraint(Real(), 1)
condition(context)[source]

Conditions on a context variable, returning a non-conditional transform of of type SplineAutoregressive.

domain = IndependentConstraint(Real(), 1)
training: bool

### ConditionalTransformModule¶

class ConditionalTransformModule(*args, **kwargs)[source]

Conditional transforms with learnable parameters such as normalizing flows should inherit from this class rather than ConditionalTransform so they are also a subclass of Module and inherit all the useful methods of that class.

property inv: pyro.distributions.conditional.ConditionalTransformModule
training: bool

### GeneralizedChannelPermute¶

class GeneralizedChannelPermute(channels=3, permutation=None)[source]

Bases: pyro.distributions.transforms.generalized_channel_permute.ConditionedGeneralizedChannelPermute, pyro.distributions.torch_transform.TransformModule

A bijection that generalizes a permutation on the channels of a batch of 2D image in $$[\ldots,C,H,W]$$ format. Specifically this transform performs the operation,

$$\mathbf{y} = \text{torch.nn.functional.conv2d}(\mathbf{x}, W)$$

where $$\mathbf{x}$$ are the inputs, $$\mathbf{y}$$ are the outputs, and $$W\sim C\times C\times 1\times 1$$ is the filter matrix for a 1x1 convolution with $$C$$ input and output channels.

Ignoring the final two dimensions, $$W$$ is restricted to be the matrix product,

$$W = PLU$$

where $$P\sim C\times C$$ is a permutation matrix on the channel dimensions, $$L\sim C\times C$$ is a lower triangular matrix with ones on the diagonal, and $$U\sim C\times C$$ is an upper triangular matrix. $$W$$ is initialized to a random orthogonal matrix. Then, $$P$$ is fixed and the learnable parameters set to $$L,U$$.

The input $$\mathbf{x}$$ and output $$\mathbf{y}$$ both have shape […,C,H,W], where C is the number of channels set at initialization.

This operation was introduced in  for Glow normalizing flow, and is also known as 1x1 invertible convolution. It appears in other notable work such as [2,3], and corresponds to the class tfp.bijectors.MatvecLU of TensorFlow Probability.

Example usage:

>>> channels = 3
>>> base_dist = dist.Normal(torch.zeros(channels, 32, 32),
... torch.ones(channels, 32, 32))
>>> inv_conv = GeneralizedChannelPermute(channels=channels)
>>> flow_dist = dist.TransformedDistribution(base_dist, [inv_conv])
>>> flow_dist.sample()

Parameters

channels (int) – Number of channel dimensions in the input.

 Diederik P. Kingma, Prafulla Dhariwal. Glow: Generative Flow with Invertible 1x1 Convolutions. [arXiv:1807.03039]

 Ryan Prenger, Rafael Valle, Bryan Catanzaro. WaveGlow: A Flow-based Generative Network for Speech Synthesis. [arXiv:1811.00002]

 Conor Durkan, Artur Bekasov, Iain Murray, George Papamakarios. Neural Spline Flows. [arXiv:1906.04032]

bijective = True
codomain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 3)
domain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 3)

### Householder¶

class Householder(input_dim, count_transforms=1)[source]

Bases: pyro.distributions.transforms.householder.ConditionedHouseholder, pyro.distributions.torch_transform.TransformModule

Represents multiple applications of the Householder bijective transformation. A single Householder transformation takes the form,

$$\mathbf{y} = (I - 2*\frac{\mathbf{u}\mathbf{u}^T}{||\mathbf{u}||^2})\mathbf{x}$$

where $$\mathbf{x}$$ are the inputs, $$\mathbf{y}$$ are the outputs, and the learnable parameters are $$\mathbf{u}\in\mathbb{R}^D$$ for input dimension $$D$$.

The transformation represents the reflection of $$\mathbf{x}$$ through the plane passing through the origin with normal $$\mathbf{u}$$.

$$D$$ applications of this transformation are able to transform standard i.i.d. standard Gaussian noise into a Gaussian variable with an arbitrary covariance matrix. With $$K<D$$ transformations, one is able to approximate a full-rank Gaussian distribution using a linear transformation of rank $$K$$.

Together with TransformedDistribution this provides a way to create richer variational approximations.

Example usage:

>>> base_dist = dist.Normal(torch.zeros(10), torch.ones(10))
>>> transform = Householder(10, count_transforms=5)
>>> pyro.module("my_transform", p)
>>> flow_dist = dist.TransformedDistribution(base_dist, [transform])
>>> flow_dist.sample()

Parameters
• input_dim (int) – the dimension of the input (and output) variable.

• count_transforms (int) – number of applications of Householder transformation to apply.

References:

 Jakub M. Tomczak, Max Welling. Improving Variational Auto-Encoders using Householder Flow. [arXiv:1611.09630]

bijective = True
codomain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 1)
domain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 1)
reset_parameters()[source]
volume_preserving = True

### MatrixExponential¶

class MatrixExponential(input_dim, iterations=8, normalization='none', bound=None)[source]

Bases: pyro.distributions.transforms.matrix_exponential.ConditionedMatrixExponential, pyro.distributions.torch_transform.TransformModule

A dense matrix exponential bijective transform (Hoogeboom et al., 2020) with equation,

$$\mathbf{y} = \exp(M)\mathbf{x}$$

where $$\mathbf{x}$$ are the inputs, $$\mathbf{y}$$ are the outputs, $$\exp(\cdot)$$ represents the matrix exponential, and the learnable parameters are $$M\in\mathbb{R}^D\times\mathbb{R}^D$$ for input dimension $$D$$. In general, $$M$$ is not required to be invertible.

Due to the favourable mathematical properties of the matrix exponential, the transform has an exact inverse and a log-determinate-Jacobian that scales in time-complexity as $$O(D)$$. Both the forward and reverse operations are approximated with a truncated power series. For numerical stability, the norm of $$M$$ can be restricted with the normalization keyword argument.

Example usage:

>>> base_dist = dist.Normal(torch.zeros(10), torch.ones(10))
>>> transform = MatrixExponential(10)
>>> pyro.module("my_transform", transform)
>>> flow_dist = dist.TransformedDistribution(base_dist, [transform])
>>> flow_dist.sample()

Parameters
• input_dim (int) – the dimension of the input (and output) variable.

• iterations (int) – the number of terms to use in the truncated power series that approximates matrix exponentiation.

• normalization (string) – One of [‘none’, ‘weight’, ‘spectral’] normalization that selects what type of normalization to apply to the weight matrix. weight corresponds to weight normalization (Salimans and Kingma, 2016) and spectral to spectral normalization (Miyato et al, 2018).

• bound (float) – a bound on either the weight or spectral norm, when either of those two types of regularization are chosen by the normalization argument. A lower value for this results in fewer required terms of the truncated power series to closely approximate the exact value of the matrix exponential.

References:

 Emiel Hoogeboom, Victor Garcia Satorras, Jakub M. Tomczak, Max Welling. The

Convolution Exponential and Generalized Sylvester Flows. [arXiv:2006.01910]

 Tim Salimans, Diederik P. Kingma. Weight Normalization: A Simple

Reparameterization to Accelerate Training of Deep Neural Networks. [arXiv:1602.07868]

 Takeru Miyato, Toshiki Kataoka, Masanori Koyama, Yuichi Yoshida. Spectral

Normalization for Generative Adversarial Networks. ICLR 2018.

bijective = True
codomain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 1)
domain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 1)
reset_parameters()[source]

### NeuralAutoregressive¶

class NeuralAutoregressive(autoregressive_nn, hidden_units=16, activation='sigmoid')[source]

An implementation of the deep Neural Autoregressive Flow (NAF) bijective transform of the “IAF flavour” that can be used for sampling and scoring samples drawn from it (but not arbitrary ones).

Example usage:

>>> from pyro.nn import AutoRegressiveNN
>>> base_dist = dist.Normal(torch.zeros(10), torch.ones(10))
>>> arn = AutoRegressiveNN(10, , param_dims=*3)
>>> transform = NeuralAutoregressive(arn, hidden_units=16)
>>> pyro.module("my_transform", transform)
>>> flow_dist = dist.TransformedDistribution(base_dist, [transform])
>>> flow_dist.sample()


The inverse operation is not implemented. This would require numerical inversion, e.g., using a root finding method - a possibility for a future implementation.

Parameters
• autoregressive_nn (nn.Module) – an autoregressive neural network whose forward call returns a tuple of three real-valued tensors, whose last dimension is the input dimension, and whose penultimate dimension is equal to hidden_units.

• hidden_units (int) – the number of hidden units to use in the NAF transformation (see Eq (8) in reference)

• activation (string) – Activation function to use. One of ‘ELU’, ‘LeakyReLU’, ‘sigmoid’, or ‘tanh’.

Reference:

 Chin-Wei Huang, David Krueger, Alexandre Lacoste, Aaron Courville. Neural Autoregressive Flows. [arXiv:1804.00779]

autoregressive = True
bijective = True
codomain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 1)
domain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 1)
eps = 1e-08
log_abs_det_jacobian(x, y)[source]

Calculates the elementwise determinant of the log Jacobian

### Planar¶

class Planar(input_dim)[source]

Bases: pyro.distributions.transforms.planar.ConditionedPlanar, pyro.distributions.torch_transform.TransformModule

A ‘planar’ bijective transform with equation,

$$\mathbf{y} = \mathbf{x} + \mathbf{u}\tanh(\mathbf{w}^T\mathbf{z}+b)$$

where $$\mathbf{x}$$ are the inputs, $$\mathbf{y}$$ are the outputs, and the learnable parameters are $$b\in\mathbb{R}$$, $$\mathbf{u}\in\mathbb{R}^D$$, $$\mathbf{w}\in\mathbb{R}^D$$ for input dimension $$D$$. For this to be an invertible transformation, the condition $$\mathbf{w}^T\mathbf{u}>-1$$ is enforced.

Together with TransformedDistribution this provides a way to create richer variational approximations.

Example usage:

>>> base_dist = dist.Normal(torch.zeros(10), torch.ones(10))
>>> transform = Planar(10)
>>> pyro.module("my_transform", transform)
>>> flow_dist = dist.TransformedDistribution(base_dist, [transform])
>>> flow_dist.sample()


The inverse of this transform does not possess an analytical solution and is left unimplemented. However, the inverse is cached when the forward operation is called during sampling, and so samples drawn using the planar transform can be scored.

Parameters

input_dim (int) – the dimension of the input (and output) variable.

References:

 Danilo Jimenez Rezende, Shakir Mohamed. Variational Inference with Normalizing Flows. [arXiv:1505.05770]

bijective = True
codomain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 1)
domain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 1)
reset_parameters()[source]

### Polynomial¶

class Polynomial(autoregressive_nn, input_dim, count_degree, count_sum)[source]

An autoregressive bijective transform as described in Jaini et al. (2019) applying following equation element-wise,

$$y_n = c_n + \int^{x_n}_0\sum^K_{k=1}\left(\sum^R_{r=0}a^{(n)}_{r,k}u^r\right)du$$

where $$x_n$$ is the $$n$$ is the $$n$$, $$\left\{a^{(n)}_{r,k}\in\mathbb{R}\right\}$$ are learnable parameters that are the output of an autoregressive NN inputting $$x_{\prec n}={x_1,x_2,\ldots,x_{n-1}}$$.

Together with TransformedDistribution this provides a way to create richer variational approximations.

Example usage:

>>> from pyro.nn import AutoRegressiveNN
>>> input_dim = 10
>>> count_degree = 4
>>> count_sum = 3
>>> base_dist = dist.Normal(torch.zeros(input_dim), torch.ones(input_dim))
>>> param_dims = [(count_degree + 1)*count_sum]
>>> arn = AutoRegressiveNN(input_dim, [input_dim*10], param_dims)
>>> transform = Polynomial(arn, input_dim=input_dim, count_degree=count_degree,
... count_sum=count_sum)
>>> pyro.module("my_transform", transform)
>>> flow_dist = dist.TransformedDistribution(base_dist, [transform])
>>> flow_dist.sample()


The inverse of this transform does not possess an analytical solution and is left unimplemented. However, the inverse is cached when the forward operation is called during sampling, and so samples drawn using a polynomial transform can be scored.

Parameters
• autoregressive_nn (nn.Module) – an autoregressive neural network whose forward call returns a tensor of real-valued numbers of size (batch_size, (count_degree+1)*count_sum, input_dim)

• count_degree (int) – The degree of the polynomial to use for each element-wise transformation.

• count_sum (int) – The number of polynomials to sum in each element-wise transformation.

References:

 Priyank Jaini, Kira A. Shelby, Yaoliang Yu. Sum-of-squares polynomial flow. [arXiv:1905.02325]

autoregressive = True
bijective = True
codomain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 1)
domain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 1)
log_abs_det_jacobian(x, y)[source]

Calculates the elementwise determinant of the log Jacobian

reset_parameters()[source]

Bases: pyro.distributions.transforms.radial.ConditionedRadial, pyro.distributions.torch_transform.TransformModule

A ‘radial’ bijective transform using the equation,

$$\mathbf{y} = \mathbf{x} + \beta h(\alpha,r)(\mathbf{x} - \mathbf{x}_0)$$

where $$\mathbf{x}$$ are the inputs, $$\mathbf{y}$$ are the outputs, and the learnable parameters are $$\alpha\in\mathbb{R}^+$$, $$\beta\in\mathbb{R}$$, $$\mathbf{x}_0\in\mathbb{R}^D$$, for input dimension $$D$$, $$r=||\mathbf{x}-\mathbf{x}_0||_2$$, $$h(\alpha,r)=1/(\alpha+r)$$. For this to be an invertible transformation, the condition $$\beta>-\alpha$$ is enforced.

Example usage:

>>> base_dist = dist.Normal(torch.zeros(10), torch.ones(10))
>>> pyro.module("my_transform", transform)
>>> flow_dist = dist.TransformedDistribution(base_dist, [transform])
>>> flow_dist.sample()


The inverse of this transform does not possess an analytical solution and is left unimplemented. However, the inverse is cached when the forward operation is called during sampling, and so samples drawn using the radial transform can be scored.

Parameters

input_dim (int) – the dimension of the input (and output) variable.

References:

 Danilo Jimenez Rezende, Shakir Mohamed. Variational Inference with Normalizing Flows. [arXiv:1505.05770]

bijective = True
codomain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 1)
domain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 1)
reset_parameters()[source]

### Spline¶

class Spline(input_dim, count_bins=8, bound=3.0, order='linear')[source]

Bases: pyro.distributions.transforms.spline.ConditionedSpline, pyro.distributions.torch_transform.TransformModule

An implementation of the element-wise rational spline bijections of linear and quadratic order (Durkan et al., 2019; Dolatabadi et al., 2020). Rational splines are functions that are comprised of segments that are the ratio of two polynomials. For instance, for the $$d$$-th dimension and the $$k$$-th segment on the spline, the function will take the form,

$$y_d = \frac{\alpha^{(k)}(x_d)}{\beta^{(k)}(x_d)},$$

where $$\alpha^{(k)}$$ and $$\beta^{(k)}$$ are two polynomials of order $$d$$. For $$d=1$$, we say that the spline is linear, and for $$d=2$$, quadratic. The spline is constructed on the specified bounding box, $$[-K,K]\times[-K,K]$$, with the identity function used elsewhere.

Rational splines offer an excellent combination of functional flexibility whilst maintaining a numerically stable inverse that is of the same computational and space complexities as the forward operation. This element-wise transform permits the accurate represention of complex univariate distributions.

Example usage:

>>> base_dist = dist.Normal(torch.zeros(10), torch.ones(10))
>>> transform = Spline(10, count_bins=4, bound=3.)
>>> pyro.module("my_transform", transform)
>>> flow_dist = dist.TransformedDistribution(base_dist, [transform])
>>> flow_dist.sample()

Parameters
• input_dim (int) – Dimension of the input vector. This is required so we know how many parameters to store.

• count_bins (int) – The number of segments comprising the spline.

• bound (float) – The quantity $$K$$ determining the bounding box, $$[-K,K]\times[-K,K]$$, of the spline.

• order (string) – One of [‘linear’, ‘quadratic’] specifying the order of the spline.

References:

Conor Durkan, Artur Bekasov, Iain Murray, George Papamakarios. Neural Spline Flows. NeurIPS 2019.

Hadi M. Dolatabadi, Sarah Erfani, Christopher Leckie. Invertible Generative Modeling using Linear Rational Splines. AISTATS 2020.

bijective = True
codomain: torch.distributions.constraints.Constraint = Real()
domain: torch.distributions.constraints.Constraint = Real()

### SplineAutoregressive¶

class SplineAutoregressive(input_dim, autoregressive_nn, count_bins=8, bound=3.0, order='linear')[source]

An implementation of the autoregressive layer with rational spline bijections of linear and quadratic order (Durkan et al., 2019; Dolatabadi et al., 2020). Rational splines are functions that are comprised of segments that are the ratio of two polynomials (see Spline).

The autoregressive layer uses the transformation,

$$y_d = g_{\theta_d}(x_d)\ \ \ d=1,2,\ldots,D$$

where $$\mathbf{x}=(x_1,x_2,\ldots,x_D)$$ are the inputs, $$\mathbf{y}=(y_1,y_2,\ldots,y_D)$$ are the outputs, $$g_{\theta_d}$$ is an elementwise rational monotonic spline with parameters $$\theta_d$$, and $$\theta=(\theta_1,\theta_2,\ldots,\theta_D)$$ is the output of an autoregressive NN inputting $$\mathbf{x}$$.

Example usage:

>>> from pyro.nn import AutoRegressiveNN
>>> input_dim = 10
>>> count_bins = 8
>>> base_dist = dist.Normal(torch.zeros(input_dim), torch.ones(input_dim))
>>> hidden_dims = [input_dim * 10, input_dim * 10]
>>> param_dims = [count_bins, count_bins, count_bins - 1, count_bins]
>>> hypernet = AutoRegressiveNN(input_dim, hidden_dims, param_dims=param_dims)
>>> transform = SplineAutoregressive(input_dim, hypernet, count_bins=count_bins)
>>> pyro.module("my_transform", transform)
>>> flow_dist = dist.TransformedDistribution(base_dist, [transform])
>>> flow_dist.sample()

Parameters
• input_dim (int) – Dimension of the input vector. Despite operating element-wise, this is required so we know how many parameters to store.

• autoregressive_nn (callable) – an autoregressive neural network whose forward call returns tuple of the spline parameters

• count_bins (int) – The number of segments comprising the spline.

• bound (float) – The quantity $$K$$ determining the bounding box, $$[-K,K]\times[-K,K]$$, of the spline.

• order (string) – One of [‘linear’, ‘quadratic’] specifying the order of the spline.

References:

Conor Durkan, Artur Bekasov, Iain Murray, George Papamakarios. Neural Spline Flows. NeurIPS 2019.

Hadi M. Dolatabadi, Sarah Erfani, Christopher Leckie. Invertible Generative Modeling using Linear Rational Splines. AISTATS 2020.

autoregressive = True
bijective = True
codomain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 1)
domain: torch.distributions.constraints.Constraint = IndependentConstraint(Real(), 1)
log_abs_det_jacobian(x, y)[source]

Calculates the elementwise determinant of the log Jacobian

### SplineCoupling¶

class SplineCoupling(input_dim, split_dim, hypernet, count_bins=8, bound=3.0, order='linear', identity=False)[source]

An implementation of the coupling layer with rational spline bijections of linear and quadratic order (Durkan et al., 2019; Dolatabadi et al., 2020). Rational splines are functions that are comprised of segments that are the ratio of two polynomials (see Spline).

The spline coupling layer uses the transformation,

$$\mathbf{y}_{1:d} = g_\theta(\mathbf{x}_{1:d})$$ $$\mathbf{y}_{(d+1):D} = h_\phi(\mathbf{x}_{(d+1):D};\mathbf{x}_{1:d})$$

where $$\mathbf{x}$$ are the inputs, $$\mathbf{y}$$ are the outputs, e.g. $$\mathbf{x}_{1:d}$$ represents the first $$d$$ elements of the inputs, $$g_\theta$$ is either the identity function or an elementwise rational monotonic spline with parameters $$\theta$$, and $$h_\phi$$ is a conditional elementwise spline spline, conditioning on the first $$d$$ elements.

Example usage:

>>> from pyro.nn import DenseNN
>>> input_dim = 10
>>> split_dim = 6
>>> count_bins = 8
>>> base_dist = dist.Normal(torch.zeros(input_dim), torch.ones(input_dim))
>>> param_dims = [(input_dim - split_dim) * count_bins,
... (input_dim - split_dim) * count_bins,
... (input_dim - split_dim) * (count_bins - 1),
... (input_dim - split_dim) * count_bins]
>>> hypernet = DenseNN(split_dim, [10*input_dim], param_dims)
>>> transform = SplineCoupling(input_dim, split_dim, hypernet)
>>> pyro.module("my_transform", transform)
>>> flow_dist = dist.TransformedDistribution(base_dist, [transform])
>>> flow_dist.sample()

Parameters
• input_dim (int) – Dimension of the input vector. Despite operating element-wise, this is required so we know how many parameters to store.

• split_dim – Zero-indexed dimension $$d$$ upon which