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)¶ Wraps
torch.distributions.bernoulli.Bernoulli
withTorchDistributionMixin
.
Beta¶

class
Beta
(concentration1, concentration0, validate_args=None)[source]¶ Wraps
torch.distributions.beta.Beta
withTorchDistributionMixin
.
Binomial¶

class
Binomial
(total_count=1, probs=None, logits=None, validate_args=None)[source]¶ Wraps
torch.distributions.binomial.Binomial
withTorchDistributionMixin
.
Categorical¶

class
Categorical
(probs=None, logits=None, validate_args=None)[source]¶ Wraps
torch.distributions.categorical.Categorical
withTorchDistributionMixin
.
Cauchy¶

class
Cauchy
(loc, scale, validate_args=None)¶ Wraps
torch.distributions.cauchy.Cauchy
withTorchDistributionMixin
.
Chi2¶

class
Chi2
(df, validate_args=None)¶ Wraps
torch.distributions.chi2.Chi2
withTorchDistributionMixin
.
ContinuousBernoulli¶

class
ContinuousBernoulli
(probs=None, logits=None, lims=(0.499, 0.501), validate_args=None)¶ Wraps
torch.distributions.continuous_bernoulli.ContinuousBernoulli
withTorchDistributionMixin
.
Dirichlet¶

class
Dirichlet
(concentration, validate_args=None)[source]¶ Wraps
torch.distributions.dirichlet.Dirichlet
withTorchDistributionMixin
.
Exponential¶

class
Exponential
(rate, validate_args=None)¶ Wraps
torch.distributions.exponential.Exponential
withTorchDistributionMixin
.
ExponentialFamily¶

class
ExponentialFamily
(batch_shape=torch.Size([]), event_shape=torch.Size([]), validate_args=None)¶ Wraps
torch.distributions.exp_family.ExponentialFamily
withTorchDistributionMixin
.
FisherSnedecor¶

class
FisherSnedecor
(df1, df2, validate_args=None)¶ Wraps
torch.distributions.fishersnedecor.FisherSnedecor
withTorchDistributionMixin
.
Gamma¶

class
Gamma
(concentration, rate, validate_args=None)[source]¶ Wraps
torch.distributions.gamma.Gamma
withTorchDistributionMixin
.
Geometric¶

class
Geometric
(probs=None, logits=None, validate_args=None)[source]¶ Wraps
torch.distributions.geometric.Geometric
withTorchDistributionMixin
.
Gumbel¶

class
Gumbel
(loc, scale, validate_args=None)¶ Wraps
torch.distributions.gumbel.Gumbel
withTorchDistributionMixin
.
HalfCauchy¶

class
HalfCauchy
(scale, validate_args=None)¶ Wraps
torch.distributions.half_cauchy.HalfCauchy
withTorchDistributionMixin
.
HalfNormal¶

class
HalfNormal
(scale, validate_args=None)¶ Wraps
torch.distributions.half_normal.HalfNormal
withTorchDistributionMixin
.
Independent¶

class
Independent
(base_distribution, reinterpreted_batch_ndims, validate_args=None)[source]¶ Wraps
torch.distributions.independent.Independent
withTorchDistributionMixin
.
Laplace¶

class
Laplace
(loc, scale, validate_args=None)¶ Wraps
torch.distributions.laplace.Laplace
withTorchDistributionMixin
.
LogNormal¶

class
LogNormal
(loc, scale, validate_args=None)[source]¶ Wraps
torch.distributions.log_normal.LogNormal
withTorchDistributionMixin
.
LogisticNormal¶

class
LogisticNormal
(loc, scale, validate_args=None)¶ Wraps
torch.distributions.logistic_normal.LogisticNormal
withTorchDistributionMixin
.
LowRankMultivariateNormal¶

class
LowRankMultivariateNormal
(loc, cov_factor, cov_diag, validate_args=None)¶ Wraps
torch.distributions.lowrank_multivariate_normal.LowRankMultivariateNormal
withTorchDistributionMixin
.
MixtureSameFamily¶

class
MixtureSameFamily
(mixture_distribution, component_distribution, validate_args=None)¶ Wraps
torch.distributions.mixture_same_family.MixtureSameFamily
withTorchDistributionMixin
.
Multinomial¶

class
Multinomial
(total_count=1, probs=None, logits=None, validate_args=None)¶ Wraps
torch.distributions.multinomial.Multinomial
withTorchDistributionMixin
.
MultivariateNormal¶

class
MultivariateNormal
(loc, covariance_matrix=None, precision_matrix=None, scale_tril=None, validate_args=None)[source]¶ Wraps
torch.distributions.multivariate_normal.MultivariateNormal
withTorchDistributionMixin
.
NegativeBinomial¶

class
NegativeBinomial
(total_count, probs=None, logits=None, validate_args=None)¶ Wraps
torch.distributions.negative_binomial.NegativeBinomial
withTorchDistributionMixin
.
Normal¶

class
Normal
(loc, scale, validate_args=None)[source]¶ Wraps
torch.distributions.normal.Normal
withTorchDistributionMixin
.
OneHotCategorical¶

class
OneHotCategorical
(probs=None, logits=None, validate_args=None)¶ Wraps
torch.distributions.one_hot_categorical.OneHotCategorical
withTorchDistributionMixin
.
Pareto¶

class
Pareto
(scale, alpha, validate_args=None)¶ Wraps
torch.distributions.pareto.Pareto
withTorchDistributionMixin
.
Poisson¶

class
Poisson
(rate, validate_args=None)¶ Wraps
torch.distributions.poisson.Poisson
withTorchDistributionMixin
.
RelaxedBernoulli¶

class
RelaxedBernoulli
(temperature, probs=None, logits=None, validate_args=None)¶ Wraps
torch.distributions.relaxed_bernoulli.RelaxedBernoulli
withTorchDistributionMixin
.
RelaxedOneHotCategorical¶

class
RelaxedOneHotCategorical
(temperature, probs=None, logits=None, validate_args=None)¶ Wraps
torch.distributions.relaxed_categorical.RelaxedOneHotCategorical
withTorchDistributionMixin
.
StudentT¶

class
StudentT
(df, loc=0.0, scale=1.0, validate_args=None)¶ Wraps
torch.distributions.studentT.StudentT
withTorchDistributionMixin
.
TransformedDistribution¶

class
TransformedDistribution
(base_distribution, transforms, validate_args=None)¶ Wraps
torch.distributions.transformed_distribution.TransformedDistribution
withTorchDistributionMixin
.
Uniform¶

class
Uniform
(low, high, validate_args=None)[source]¶ Wraps
torch.distributions.uniform.Uniform
withTorchDistributionMixin
.
VonMises¶

class
VonMises
(loc, concentration, validate_args=None)¶ Wraps
torch.distributions.von_mises.VonMises
withTorchDistributionMixin
.
Weibull¶

class
Weibull
(scale, concentration, validate_args=None)¶ Wraps
torch.distributions.weibull.Weibull
withTorchDistributionMixin
.
Pyro Distributions¶
Abstract Distribution¶

class
Distribution
[source]¶ Bases:
object
Base class for parameterized probability distributions.
Distributions in Pyro are stochastic function objects with
sample()
andlog_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

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

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 onedimensional Tensor
with same batch size as value and params. The shape of the result should beself.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 nonreparameterized 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 likeSVI
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 lockstep, 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
onFunsor
distributions, but we return a lazy sum(updated, log_normalizer)
because PyTorch distributions must be normalized. Thusconjugate_update()
should commute withdist_to_funsor()
andtensor_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(datalatent)
but normalized overlatent
rather thandata
. Herelatent
is a candidate sample fromself
anddata
is a ground observation of unrelated type.Returns: a pair (updated,log_normalizer)
whereupdated
is an updated distribution of typetype(self)
, andlog_normalizer
is aTensor
representing the normalization factor.

has_rsample_
(value)[source]¶ Force reparameterized or detached sampling on a single distribution instance. This sets the
.has_rsample
attribute inplace.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

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
[source]¶ Bases:
pyro.distributions.distribution.Distribution
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 fromTorchDistributionMixin
.
__call__
(sample_shape=torch.Size([]))[source]¶ Samples a random value.
This is reparameterized whenever possible, calling
rsample()
for reparameterized distributions andsample()
for nonreparameterized 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

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

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

to_event
(reinterpreted_batch_ndims=None)[source]¶ Reinterprets the
n
rightmost dimensions of this distributionsbatch_shape
as event dims, adding them to the left side ofevent_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([2]), 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

mask
(mask)[source]¶ Masks a distribution by a boolean or booleanvalued 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]¶ Bases:
torch.distributions.distribution.Distribution
,pyro.distributions.torch_distribution.TorchDistributionMixin
Base class for PyTorchcompatible 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 typeTensor
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 nonidentical (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 shapesample_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()
(orrsample()
if.has_rsample == True
) andlog_prob()
, and must implement the propertiesbatch_shape
, andevent_shape
. Discrete classes may also implement theenumerate_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
AVFMultivariateNormal¶

class
AVFMultivariateNormal
(loc, scale_tril, control_var)[source]¶ Bases:
pyro.distributions.torch.MultivariateNormal
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) – Ddimensional 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() opt_cv.zero_grad()

arg_constraints
= {'control_var': Real(), 'loc': Real(), 'scale_tril': LowerTriangular()}¶
BetaBinomial¶

class
BetaBinomial
(concentration1, concentration0, total_count=1, validate_args=None)[source]¶ Bases:
pyro.distributions.torch_distribution.TorchDistribution
Compound distribution comprising of a betabinomial pair. The probability of success (
probs
for theBinomial
distribution) is unknown and randomly drawn from aBeta
distribution prior to a certain number of Bernoulli trials given bytotal_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)}¶

concentration0
¶

concentration1
¶

has_enumerate_support
= True¶

mean
¶

support
¶

variance
¶
CoalescentTimes¶

class
CoalescentTimes
(leaf_times, *, validate_args=None)[source]¶ Bases:
pyro.distributions.torch_distribution.TorchDistribution
Distribution over coalescent times given irregular sampled
leaf_times
.Sample values will be sorted sets of binary coalescent times. Each sample
value
will have cardinalityvalue.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
 [1] J.F.C. Kingman (1982)
 “On the Genealogy of Large Populations” Journal of Applied Probability
 [2] 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. 
arg_constraints
= {'leaf_times': Real()}¶

support
¶
CoalescentTimesWithRate¶

class
CoalescentTimesWithRate
(leaf_times, rate_grid, *, validate_args=None)[source]¶ Bases:
pyro.distributions.torch_distribution.TorchDistribution
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]
, …,[T1,inf)
, whereT = 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 cardinalityvalue.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
 [1] J.F.C. Kingman (1982)
 “On the Genealogy of Large Populations” Journal of Applied Probability
 [2] J.F.C. Kingman (1982)
 “The Coalescent” Stochastic Processes and their Applications
 [3] 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 piecwise constant in time.

arg_constraints
= {'leaf_times': Real(), 'rate_grid': GreaterThan(lower_bound=0.0)}¶

duration
¶

log_prob
(value)[source]¶ Computes likelihood as in equations 78 of [3].
This has time complexity
O(T + S N log(N))
whereT
is the number of time steps,N
is the number of leaves, andS = sample_shape.numel()
is the number of samples ofvalue
.This is differentiable wrt
rate_grid
but neitherleaf_times
norvalue = coal_times
.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

support
¶
ConditionalDistribution¶
ConditionalTransformedDistribution¶
Delta¶

class
Delta
(v, log_density=0.0, event_dim=0, validate_args=None)[source]¶ Bases:
pyro.distributions.torch_distribution.TorchDistribution
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': Real()}¶

has_rsample
= True¶

mean
¶

support
= Real()¶

variance
¶
DirichletMultinomial¶

class
DirichletMultinomial
(concentration, total_count=1, is_sparse=False, validate_args=None)[source]¶ Bases:
pyro.distributions.torch_distribution.TorchDistribution
Compound distribution comprising of a dirichletmultinomial pair. The probability of classes (
probs
for theMultinomial
distribution) is unknown and randomly drawn from aDirichlet
distribution prior to a certain number of Categorical trials given bytotal_count
.Parameters:  or torch.Tensor concentration (float) – concentration parameter (alpha) for the Dirichlet distribution.
 or torch.Tensor total_count (int) – 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': GreaterThan(lower_bound=0.0), 'total_count': IntegerGreaterThan(lower_bound=0)}¶

concentration
¶

mean
¶

support
¶

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 [1] to parallelize over time, achieving O(log(time)) parallel complexity.
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
andobservation_dist
. However, because time is included in this distribution’s event_shape, the homogeneous+homogeneous case will have a broadcastable event_shape withnum_steps = 1
, allowinglog_prob()
to work with arbitrary length data:# homogeneous + homogeneous case: event_shape = (1,) + observation_dist.event_shape
References:
 [1] Simo Sarkka, Angel F. GarciaFernandez (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 tobatch_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 tobatch_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 sizestate_dim
and be broadcastable tobatch_shape + (num_steps, state_dim)
. The.event_shape
may be arbitrary.  duration (int) – Optional size of the time axis
event_shape[0]
. 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()}¶

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 asinitial_logits
in a sequential Pyro model for prediction.Return type: Categorical

support
¶
EmpiricalDistribution¶

class
Empirical
(samples, log_weights, validate_args=None)[source]¶ Bases:
pyro.distributions.torch_distribution.TorchDistribution
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([2]) >>> emp_dist.event_shape torch.Size([10])
>>> 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([2]) >>> # 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()

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

log_weights
¶

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

sample
(sample_shape=torch.Size([]))[source]¶ See
pyro.distributions.torch_distribution.TorchDistribution.sample()

sample_size
¶ Number of samples that constitute the empirical distribution.
Return int: number of samples collected.

support
= Real()¶

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

class
ExtendedBetaBinomial
(concentration1, concentration0, total_count=1, validate_args=None)[source]¶ Bases:
pyro.distributions.conjugate.BetaBinomial
EXPERIMENTAL
BetaBinomial
distribution extended to have logical support the entire integers and to allow arbitrary integertotal_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}¶

support
= Integer¶

ExtendedBinomial¶

class
ExtendedBinomial
(total_count=1, probs=None, logits=None, validate_args=None)[source]¶ Bases:
pyro.distributions.torch.Binomial
EXPERIMENTAL
Binomial
distribution extended to have logical support the entire integers and to allow arbitrary integertotal_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}¶

support
= Integer¶

FoldedDistribution¶

class
FoldedDistribution
(base_dist, validate_args=None)[source]¶ Bases:
pyro.distributions.torch.TransformedDistribution
Equivalent to
TransformedDistribution(base_dist, AbsTransform())
, but additionally supportslog_prob()
.Parameters: base_dist (Distribution) – The distribution to reflect. 
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 [2] and [3]. This adapts [1] 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
andobservation_dist
. However, because time is included in this distribution’s event_shape, the homogeneous+homogeneous case will have a broadcastable event_shape withnum_steps = 1
, allowinglog_prob()
to work with arbitrary length data:event_shape = (1, obs_dim) # homogeneous + homogeneous case
References:
 [1] Simo Sarkka, Angel F. GarciaFernandez (2019)
 “Temporal Parallelization of Bayesian Filters and Smoothers” https://arxiv.org/pdf/1905.13002.pdf
 [2] F. J. Giron and J. C. Rojano (1994)
 “Bayesian Kalman filtering with elliptically contoured errors”
 [3] Filip Tronarp, Toni Karvonen, and Simo Sarkka (2019)
 “Student’s tfilters for noise scale estimation” https://users.aalto.fi/~ssarkka/pub/SPL2019.pdf
Variables: 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[0]
. This is required when sampling from homogeneous HMMs whose parameters are not expanded along the time axis.

arg_constraints
= {}¶

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

support
= Real()¶
GammaPoisson¶

class
GammaPoisson
(concentration, rate, validate_args=None)[source]¶ Bases:
pyro.distributions.torch_distribution.TorchDistribution
Compound distribution comprising of a gammapoisson pair, also referred to as a gammapoisson mixture. The
rate
parameter for thePoisson
distribution is unknown and randomly drawn from aGamma
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: 
arg_constraints
= {'concentration': GreaterThan(lower_bound=0.0), 'rate': GreaterThan(lower_bound=0.0)}¶

concentration
¶

mean
¶

rate
¶

support
= IntegerGreaterThan(lower_bound=0)¶

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 [1] 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
andobservation_dist
. However, because time is included in this distribution’s event_shape, the homogeneous+homogeneous case will have a broadcastable event_shape withnum_steps = 1
, allowinglog_prob()
to work with arbitrary length data:event_shape = (1, obs_dim) # homogeneous + homogeneous case
References:
 [1] Simo Sarkka, Angel F. GarciaFernandez (2019)
 “Temporal Parallelization of Bayesian Filters and Smoothers” https://arxiv.org/pdf/1905.13002.pdf
Variables: 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[0]
. 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(dataself.probs)
but normalized overself.probs
rather thandata
.Returns: a pair (updated,log_normalizer)
whereupdated
is an updatedGaussianHMM
, andlog_normalizer
is aTensor
representing the normalization factor.

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 asinitial_dist
in a sequential Pyro model for prediction.Return type: MultivariateNormal

has_rsample
= True¶

prefix_condition
(data)[source]¶ EXPERIMENTAL Given self has
event_shape == (t+f, d)
and datax
of shapebatch_shape + (t, d)
, compute a conditional distribution of event_shape(f, d)
. Typicallyt
is the number of training time steps,f
is the number of forecast time steps, andd
is the data dimension.Parameters: data (Tensor) – data of dimension at least 2.

rsample_posterior
(value, sample_shape=torch.Size([]))[source]¶ EXPERIMENTAL Sample from the latent state conditioned on observation.

support
= Real()¶
GaussianMRF¶

class
GaussianMRF
(initial_dist, transition_dist, observation_dist, validate_args=None)[source]¶ Bases:
pyro.distributions.torch_distribution.TorchDistribution
Temporal Markov Random Field with Gaussian factors for initial, transition, and observation distributions. This adapts [1] 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
andobservation_dist
. However, because time is included in this distribution’s event_shape, the homogeneous+homogeneous case will have a broadcastable event_shape withnum_steps = 1
, allowinglog_prob()
to work with arbitrary length data:event_shape = (1, obs_dim) # homogeneous + homogeneous case
References:
 [1] Simo Sarkka, Angel F. GarciaFernandez (2019)
 “Temporal Parallelization of Bayesian Filters and Smoothers” https://arxiv.org/pdf/1905.13002.pdf
Variables: 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
= {}¶
GaussianScaleMixture¶

class
GaussianScaleMixture
(coord_scale, component_logits, component_scale)[source]¶ Bases:
pyro.distributions.torch_distribution.TorchDistribution
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 Ddimensional Normal distribution with zero mean and a Ddimensional 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 Kdimensional 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 [1] for details on the implementations of the pathwise derivative. Please consider citing this reference if you use the pathwise derivative in your research.
[1] 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) – Ddimensional vector of scales
 component_logits (torch.tensor) – Kdimensional vector of logits
 component_scale (torch.tensor) – Kdimensional 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¶
IndependentHMM¶

class
IndependentHMM
(base_dist)[source]¶ Bases:
pyro.distributions.torch_distribution.TorchDistribution
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
= {}¶

duration
¶

has_rsample
¶

support
¶

InverseGamma¶

class
InverseGamma
(concentration, rate, validate_args=None)[source]¶ Bases:
pyro.distributions.torch.TransformedDistribution
Creates an inversegamma 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
= {'concentration': GreaterThan(lower_bound=0.0), 'rate': GreaterThan(lower_bound=0.0)}¶

concentration
¶

has_rsample
= True¶

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
orIndependent
ofNormal
,StudentT
, orStable
. 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 intoobs_base_dist
and an optionalobs_transform
(defaulting to the identity).This implements a reparameterized
rsample()
method but does not implement alog_prob()
method. Derived classes may implementlog_prob()
.Inference without
log_prob()
can be performed using either reparameterization withLinearHMMReparam
or likelihoodfree algorithms such asEnergyDistance
. 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
andobservation_dist
. However at least one of the distributions or matrices must be expanded to contain the time dimension.Variables: 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[0]
. This is required when sampling from homogeneous HMMs whose parameters are not expanded along the time axis.

arg_constraints
= {}¶

has_rsample
= True¶

support
¶
 initial_dist – A distribution over initial states. This should have
batch_shape broadcastable to
LKJCorrCholesky¶

class
LKJCorrCholesky
(d, eta, validate_args=None)[source]¶ Bases:
pyro.distributions.torch_distribution.TorchDistribution
Generates cholesky factors of correlation matrices using an LKJ prior.
The expected use is to combine it with a vector of variances and pass it to the scale_tril parameter of a multivariate distribution such as MultivariateNormal.
E.g., if theta is a (positive) vector of covariances with the same dimensionality as this distribution, and Omega is sampled from this distribution, scale_tril=torch.mm(torch.diag(sqrt(theta)), Omega)
Note that the event_shape of this distribution is [d, d]
Note
When using this distribution with HMC/NUTS, it is important to use a step_size such as 1e4. If not, you are likely to experience LAPACK errors regarding positivedefiniteness.
For example usage, refer to pyro/examples/lkj.py.
Parameters:  d (int) – Dimensionality of the matrix
 eta (torch.Tensor) – A single positive number parameterizing the distribution.

arg_constraints
= {'eta': GreaterThan(lower_bound=0.0)}¶

has_rsample
= False¶

support
= CorrCholesky()¶
MaskedDistribution¶

class
MaskedDistribution
(base_dist, mask)[source]¶ Bases:
pyro.distributions.torch_distribution.TorchDistribution
Masks a distribution by a boolean tensor that is broadcastable to the distribution’s
batch_shape
.In the special case
mask is False
, computation oflog_prob()
,score_parts()
, andkl_divergence()
is skipped, and constant zero values are returned instead.Parameters: mask (torch.Tensor or bool) – A boolean or booleanvalued tensor. 
arg_constraints
= {}¶

has_enumerate_support
¶

has_rsample
¶

mean
¶

support
¶

variance
¶

MaskedMixture¶

class
MaskedMixture
(mask, component0, component1, validate_args=None)[source]¶ Bases:
pyro.distributions.torch_distribution.TorchDistribution
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)): pyro.sample("obs", MaskedMixture(mask, dist1, dist2), obs=data)
Parameters:  mask (torch.Tensor) – A byte tensor toggling between
component0
andcomponent1
.  component0 (pyro.distributions.TorchDistribution) – a distribution
for batch elements
mask == 0
.  component1 (pyro.distributions.TorchDistribution) – a distribution
for batch elements
mask == 1
.

arg_constraints
= {}¶

has_rsample
¶

support
¶
 mask (torch.Tensor) – A byte tensor toggling between
MixtureOfDiagNormals¶

class
MixtureOfDiagNormals
(locs, coord_scale, component_logits)[source]¶ Bases:
pyro.distributions.torch_distribution.TorchDistribution
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 Ddimensional Normal distribution with a Ddimensional mean parameter and a Ddimensional 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 Kdimensional vector of softmax logits, component_logits. This distribution implements pathwise derivatives for samples from the distribution.
See reference [1] 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.
[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) – K x D scale matrix
 component_logits (torch.Tensor) – Kdimensional vector of softmax logits

arg_constraints
= {'component_logits': Real(), 'coord_scale': GreaterThan(lower_bound=0.0), 'locs': Real()}¶

has_rsample
= True¶
MultivariateStudentT¶

class
MultivariateStudentT
(df, loc, scale_tril, validate_args=None)[source]¶ Bases:
pyro.distributions.torch_distribution.TorchDistribution
Creates a multivariate Student’s tdistribution parameterized by degree of freedom
df
, meanloc
and scalescale_tril
.Parameters: 
arg_constraints
= {'df': GreaterThan(lower_bound=0.0), 'loc': RealVector(), 'scale_tril': LowerCholesky()}¶

has_rsample
= True¶

mean
¶

support
= RealVector()¶

variance
¶

OMTMultivariateNormal¶

class
OMTMultivariateNormal
(loc, scale_tril)[source]¶ Bases:
pyro.distributions.torch.MultivariateNormal
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:  loc (torch.Tensor) – Mean.
 scale_tril (torch.Tensor) – Cholesky of Covariance matrix.

arg_constraints
= {'loc': Real(), 'scale_tril': LowerTriangular()}¶
RelaxedBernoulliStraightThrough¶

class
RelaxedBernoulliStraightThrough
(temperature, probs=None, logits=None, validate_args=None)[source]¶ Bases:
pyro.distributions.torch.RelaxedBernoulli
An implementation of
RelaxedBernoulli
with a straightthrough 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:
 [1] The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables,
 Chris J. Maddison, Andriy Mnih, Yee Whye Teh
 [2] Categorical Reparameterization with GumbelSoftmax,
 Eric Jang, Shixiang Gu, Ben Poole
 The samples returned by the
RelaxedOneHotCategoricalStraightThrough¶

class
RelaxedOneHotCategoricalStraightThrough
(temperature, probs=None, logits=None, validate_args=None)[source]¶ Bases:
pyro.distributions.torch.RelaxedOneHotCategorical
An implementation of
RelaxedOneHotCategorical
with a straightthrough 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:
 [1] The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables,
 Chris J. Maddison, Andriy Mnih, Yee Whye Teh
 [2] Categorical Reparameterization with GumbelSoftmax,
 Eric Jang, Shixiang Gu, Ben Poole
 The samples returned by the
Rejector¶

class
Rejector
(propose, log_prob_accept, log_scale, *, batch_shape=None, event_shape=None)[source]¶ Bases:
pyro.distributions.torch_distribution.TorchDistribution
Rejection sampled distribution given an acceptance rate function.
Parameters:  propose (Distribution) – A proposal distribution that samples batched
proposals via
propose()
.rsample()
supports asample_shape
arg only ifpropose()
supports asample_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¶
 propose (Distribution) – A proposal distribution that samples batched
proposals via
SpanningTree¶

class
SpanningTree
(edge_logits, sampler_options=None, validate_args=None)[source]¶ Bases:
pyro.distributions.torch_distribution.TorchDistribution
Distribution over spanning trees on a fixed number
V
of vertices.A tree is represented as
torch.LongTensor
edges
of shape(V1,2)
satisfying the following properties: The edges constitute a tree, i.e. are connected and cycle free.
 Each edge
(v1,v2) = edges[e]
is sorted, i.e.v1 < v2
.  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 theV*(V1)//2
edges in the complete graph onV
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 sizeindependent 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*(V1)//2
containing logits (aka negative energies) of all edges in the complete graph onV
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()}¶

enumerate_support
(expand=True)[source]¶ This is implemented for trees with up to 6 vertices (and 5 edges).

has_enumerate_support
= True¶

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 AldousBroder sampler [1,2], especially for graphs with large mixing time. Recent research [3,4] proposes samplers that run in submatrixmultiply time but are more complex to implement.
References
 [1] Generating random spanning trees
 Andrei Broder (1989)
 [2] The Random Walk Construction of Uniform Spanning Trees and Uniform Labelled Trees,
 David J. Aldous (1990)
 [3] 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
 [4] An almostlinear 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 bysample()
orenumerate_support()
or as input tolog_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]¶ Bases:
pyro.distributions.torch_distribution.TorchDistribution
Levy \(\alpha\)stable distribution. See [1] for a review.
This uses Nolan’s parametrization [2] of the
loc
parameter, which is required for continuity and differentiability. This corresponds to the notation \(S^0_\alpha(\beta,\sigma,\mu_0)\) of [1], where \(\alpha\) = stability, \(\beta\) = skew, \(\sigma\) = scale, and \(\mu_0\) = loc. To instead use the S parameterization as in scipy, passcoords="S"
, but BEWARE this is discontinuous atstability=1
and has poor geometry for inference.This implements a reparametrized sampler
rsample()
, but does not implementlog_prob()
. Inference can be performed using either likelihoodfree algorithms such asEnergyDistance
, or reparameterization via thereparam()
handler with one of the reparameterizersLatentStableReparam
,SymmetricStableReparam
, orStableReparam
e.g.:with poutine.reparam(config={"x": StableReparam()}): pyro.sample("x", Stable(stability, skew, scale, loc))
 [1] S. Borak, W. Hardle, R. Weron (2005).
 Stable distributions. https://edoc.huberlin.de/bitstream/handle/18452/4526/8.pdf
 [2] J.P. Nolan (1997).
 Numerical calculation of stable densities and distribution functions.
 [3] Rafal Weron (1996).
 On the ChambersMallowsStuck Method for Simulating Skewed Stable Random Variables.
 [4] 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 [2], 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)}¶

has_rsample
= True¶

mean
¶

support
= Real()¶

variance
¶
TruncatedPolyaGamma¶

class
TruncatedPolyaGamma
(prototype, validate_args=None)[source]¶ Bases:
pyro.distributions.torch_distribution.TorchDistribution
This is a PolyaGamma(1, 0) distribution truncated to have finite support in the interval (0, 2.5). See [1] 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
 [1] ‘Bayesian inference for logistic models using PolyaGamma latent variables’
 Nicholas G. Polson, James G. Scott, Jesse Windle.

arg_constraints
= {}¶

has_rsample
= False¶

num_gamma_variates
= 8¶

num_log_prob_terms
= 7¶

support
= Interval(lower_bound=0.0, upper_bound=2.5)¶

truncation_point
= 2.5¶
Unit¶

class
Unit
(log_factor, validate_args=None)[source]¶ Bases:
pyro.distributions.torch_distribution.TorchDistribution
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()}¶

support
= Real()¶

VonMises3D¶

class
VonMises3D
(concentration, validate_args=None)[source]¶ Bases:
pyro.distributions.torch_distribution.TorchDistribution
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 3vector that lies on the 2sphere.See
VonMises
for a 2D polar coordinate cousin of this distribution.Currently only
log_prob()
is implemented.Parameters: concentration (torch.Tensor) – A combined locationandconcentration vector. The direction of this vector is the location, and its magnitude is the concentration. 
arg_constraints
= {'concentration': Real()}¶

support
= Real()¶

ZeroInflatedPoisson¶

class
ZeroInflatedPoisson
(gate, rate, validate_args=None)[source]¶ Bases:
pyro.distributions.zero_inflated.ZeroInflatedDistribution
A Zero Inflated Poisson distribution.
Parameters:  gate (torch.Tensor) – probability of extra zeros.
 rate (torch.Tensor) – rate of poisson distribution.

arg_constraints
= {'gate': Interval(lower_bound=0.0, upper_bound=1.0), 'rate': GreaterThan(lower_bound=0.0)}¶

rate
¶

support
= IntegerGreaterThan(lower_bound=0)¶
ZeroInflatedNegativeBinomial¶

class
ZeroInflatedNegativeBinomial
(gate, total_count, probs=None, logits=None, validate_args=None)[source]¶ Bases:
pyro.distributions.zero_inflated.ZeroInflatedDistribution
A Zero Inflated Negative Binomial distribution.
Parameters:  gate (torch.Tensor) – probability of extra zeros.
 total_count (float or torch.Tensor) – nonnegative number of negative Bernoulli trials.
 probs (torch.Tensor) – Event probabilities of success in the half open interval [0, 1).
 logits (torch.Tensor) – Event logodds for probabilities of success.

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

logits
¶

probs
¶

support
= IntegerGreaterThan(lower_bound=0)¶

total_count
¶
ZeroInflatedDistribution¶

class
ZeroInflatedDistribution
(gate, base_dist, validate_args=None)[source]¶ Bases:
pyro.distributions.torch_distribution.TorchDistribution
Base class for a Zero Inflated distribution.
Parameters:  gate (torch.Tensor) – probability of extra zeros given via a Bernoulli distribution.
 base_dist (TorchDistribution) – the base distribution.

arg_constraints
= {'gate': Interval(lower_bound=0.0, upper_bound=1.0)}¶
Transforms¶
ConditionalTransform¶
CorrLCholeskyTransform¶

class
CorrLCholeskyTransform
(cache_size=0)[source]¶ Bases:
torch.distributions.transforms.Transform
Transforms a vector into the cholesky factor of a correlation matrix.
The input should have shape [batch_shape] + [d * (d1)/2]. The output will have shape [batch_shape] + [d, d].
References:
[1] Cholesky Factors of Correlation Matrices. Stan Reference Manual v2.18, Section 10.12.

bijective
= True¶

codomain
= CorrCholesky()¶

domain
= Real()¶

event_dim
= 1¶

sign
= 1¶

ELUTransform¶
HaarTransform¶

class
HaarTransform
(dim=1, flip=False, cache_size=0)[source]¶ Bases:
torch.distributions.transforms.Transform
Discrete Haar transform.
This uses
haar_transform()
andinverse_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 blockstructured Haar transform in which block sizes decrease by factors of one half from left to right.Parameters: 
bijective
= True¶

codomain
= RealVector()¶

domain
= RealVector()¶

LeakyReLUTransform¶
LowerCholeskyAffine¶

class
LowerCholeskyAffine
(loc, scale_tril, cache_size=0)[source]¶ Bases:
torch.distributions.transforms.Transform
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 Ddimensional vector to shift the input by.
 scale_tril (torch.tensor) – the D x D lower triangular matrix used in the transformation.

bijective
= True¶

codomain
= RealVector()¶

event_dim
= 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¶
Permute¶

class
Permute
(permutation, *, dim=1, cache_size=1)[source]¶ Bases:
torch.distributions.transforms.Transform
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 ofAffineAutoregressive
is not equivalent to reordering the dimension inside the MADE networks that those IAFs use; using aPermute
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, [40])) >>> ff = Permute(torch.randperm(10, dtype=torch.long)) >>> iaf2 = AffineAutoregressive(AutoRegressiveNN(10, [40])) >>> flow_dist = dist.TransformedDistribution(base_dist, [iaf1, ff, iaf2]) >>> flow_dist.sample() # doctest: +SKIP
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¶

codomain
= Real()¶

log_abs_det_jacobian
(x, y)[source]¶ Calculates the elementwise determinant of the log Jacobian, i.e. log(abs([dy_0/dx_0, …, dy_{N1}/dx_{N1}])). 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¶
DiscreteCosineTransform¶

class
DiscreteCosineTransform
(dim=1, smooth=0.0, cache_size=0)[source]¶ Bases:
torch.distributions.transforms.Transform
Discrete Cosine Transform of typeII.
This uses
dct()
andidct()
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¶

codomain
= RealVector()¶

domain
= RealVector()¶
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]¶ Bases:
pyro.distributions.torch_transform.TransformModule
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, [40])) >>> pyro.module("my_transform", transform) # doctest: +SKIP >>> flow_dist = dist.TransformedDistribution(base_dist, [transform]) >>> flow_dist.sample() # doctest: +SKIP
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 fromTransformedDistribution
. 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 realvalued mean and logitscale 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:
[1] Diederik P. Kingma, Tim Salimans, Rafal Jozefowicz, Xi Chen, Ilya Sutskever, Max Welling. Improving Variational Inference with Inverse Autoregressive Flow. [arXiv:1606.04934]
[2] Danilo Jimenez Rezende, Shakir Mohamed. Variational Inference with Normalizing Flows. [arXiv:1505.05770]
[3] Mathieu Germain, Karol Gregor, Iain Murray, Hugo Larochelle. MADE: Masked Autoencoder for Distribution Estimation. [arXiv:1502.03509]

autoregressive
= True¶

bijective
= True¶

codomain
= Real()¶

domain
= Real()¶

event_dim
= 1¶

sign
= 1¶
AffineCoupling¶

class
AffineCoupling
(split_dim, hypernet, *, dim=1, log_scale_min_clip=5.0, log_scale_max_clip=3.0)[source]¶ Bases:
pyro.distributions.torch_transform.TransformModule
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 \(Dd\) 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_dimsplit_dim, input_dimsplit_dim] >>> hypernet = DenseNN(split_dim, [10*input_dim], param_dims) >>> transform = AffineCoupling(split_dim, hypernet) >>> pyro.module("my_transform", transform) # doctest: +SKIP >>> flow_dist = dist.TransformedDistribution(base_dist, [transform]) >>> flow_dist.sample() # doctest: +SKIP
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 fromTransformedDistribution
. 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) – Zeroindexed dimension \(d\) upon which to perform input/ output split for transformation.
 hypernet (callable) – an autoregressive neural network whose forward call returns a realvalued mean and logitscale as a tuple. The input should have final dimension split_dim and the output final dimension input_dimsplit_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:
[1] Laurent Dinh, Jascha SohlDickstein, and Samy Bengio. Density estimation using Real NVP. ICLR 2017.

bijective
= True¶

codomain
= Real()¶

domain
= Real()¶
BatchNorm¶

class
BatchNorm
(input_dim, momentum=0.1, epsilon=1e05)[source]¶ Bases:
pyro.distributions.torch_transform.TransformModule
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 logdetJacobian term.
This is an elementwise 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, [40])) for _ in range(2)] >>> bn = BatchNorm(10) >>> flow_dist = dist.TransformedDistribution(base_dist, [iafs[0], bn, iafs[1]]) >>> flow_dist.sample() # doctest: +SKIP
Parameters: References:
[1] 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
[2] Laurent Dinh, Jascha SohlDickstein, and Samy Bengio. Density Estimation using Real NVP. In International Conference on Learning Representations, 2017. https://arxiv.org/abs/1605.08803
[3] 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
= Real()¶

constrained_gamma
¶

domain
= Real()¶

event_dim
= 0¶

BlockAutoregressive¶

class
BlockAutoregressive
(input_dim, hidden_factors=[8, 8], activation='tanh', residual=None)[source]¶ Bases:
pyro.distributions.torch_transform.TransformModule
An implementation of Block Neural Autoregressive Flow (blockNAF) (De Cao et al., 2019) bijective transform. BlockNAF 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) # doctest: +SKIP >>> naf_dist = dist.TransformedDistribution(base_dist, [naf]) >>> naf_dist.sample() # doctest: +SKIP
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:
[1] Nicola De Cao, Ivan Titov, Wilker Aziz. Block Neural Autoregressive Flow. [arXiv:1904.04676]

autoregressive
= True¶

bijective
= True¶

codomain
= Real()¶

domain
= Real()¶

event_dim
= 1¶
ConditionalAffineAutoregressive¶

class
ConditionalAffineAutoregressive
(autoregressive_nn, **kwargs)[source]¶ Bases:
pyro.distributions.conditional.ConditionalTransformModule
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) # doctest: +SKIP >>> 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])) # doctest: +SKIP
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 fromTransformedDistribution
. 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 realvalued mean and logitscale 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:
[1] Diederik P. Kingma, Tim Salimans, Rafal Jozefowicz, Xi Chen, Ilya Sutskever, Max Welling. Improving Variational Inference with Inverse Autoregressive Flow. [arXiv:1606.04934]
[2] Danilo Jimenez Rezende, Shakir Mohamed. Variational Inference with Normalizing Flows. [arXiv:1505.05770]
[3] Mathieu Germain, Karol Gregor, Iain Murray, Hugo Larochelle. MADE: Masked Autoencoder for Distribution Estimation. [arXiv:1502.03509]

bijective
= True¶

codomain
= Real()¶

condition
(context)[source]¶ Conditions on a context variable, returning a nonconditional transform of of type
AffineAutoregressive
.

domain
= Real()¶

event_dim
= 1¶
ConditionalAffineCoupling¶

class
ConditionalAffineCoupling
(split_dim, hypernet, **kwargs)[source]¶ Bases:
pyro.distributions.conditional.ConditionalTransformModule
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 \(Dd\) 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_dimsplit_dim, input_dimsplit_dim] >>> hypernet = ConditionalDenseNN(split_dim, context_dim, [10*input_dim], ... param_dims) >>> transform = ConditionalAffineCoupling(split_dim, hypernet) >>> pyro.module("my_transform", transform) # doctest: +SKIP >>> 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])) # doctest: +SKIP
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 fromConditionalTransformedDistribution
. 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) – Zeroindexed dimension \(d\) upon which to perform input/ output split for transformation.
 hypernet (callable) – A neural network whose forward call returns a realvalued mean and logitscale as a tuple. The input should have final dimension split_dim and the output final dimension input_dimsplit_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 SohlDickstein, and Samy Bengio. Density estimation using Real NVP. ICLR 2017.

bijective
= True¶

codomain
= Real()¶

condition
(context)[source]¶ See
pyro.distributions.conditional.ConditionalTransformModule.condition()

domain
= Real()¶

event_dim
= 1¶
ConditionalGeneralizedChannelPermute¶

class
ConditionalGeneralizedChannelPermute
(nn, channels=3, permutation=None)[source]¶ Bases:
pyro.distributions.conditional.ConditionalTransformModule
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 [1] 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])) # doctest: +SKIP
Parameters:  nn – a function inputting the context variable and outputting realvalued parameters of dimension \(C^2\).
 channels (int) – Number of channel dimensions in the input.
[1] Diederik P. Kingma, Prafulla Dhariwal. Glow: Generative Flow with Invertible 1x1 Convolutions. [arXiv:1807.03039]
[2] Ryan Prenger, Rafael Valle, Bryan Catanzaro. WaveGlow: A Flowbased Generative Network for Speech Synthesis. [arXiv:1811.00002]
[3] Conor Durkan, Artur Bekasov, Iain Murray, George Papamakarios. Neural Spline Flows. [arXiv:1906.04032]

bijective
= True¶

codomain
= Real()¶

condition
(context)[source]¶ See
pyro.distributions.conditional.ConditionalTransformModule.condition()

domain
= Real()¶

event_dim
= 3¶
ConditionalHouseholder¶

class
ConditionalHouseholder
(input_dim, nn, count_transforms=1)[source]¶ Bases:
pyro.distributions.conditional.ConditionalTransformModule
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 fullrank 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])) # doctest: +SKIP
Parameters: References:
[1] Jakub M. Tomczak, Max Welling. Improving Variational AutoEncoders using Householder Flow. [arXiv:1611.09630]

bijective
= True¶

codomain
= Real()¶

condition
(context)[source]¶ See
pyro.distributions.conditional.ConditionalTransformModule.condition()

domain
= Real()¶

event_dim
= 1¶

ConditionalNeuralAutoregressive¶

class
ConditionalNeuralAutoregressive
(autoregressive_nn, **kwargs)[source]¶ Bases:
pyro.distributions.conditional.ConditionalTransformModule
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, [40], ... param_dims=[16]*3) >>> transform = ConditionalNeuralAutoregressive(arn, hidden_units=16) >>> pyro.module("my_transform", transform) # doctest: +SKIP >>> 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])) # doctest: +SKIP
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 realvalued 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:
[1] ChinWei Huang, David Krueger, Alexandre Lacoste, Aaron Courville. Neural Autoregressive Flows. [arXiv:1804.00779]

bijective
= True¶

codomain
= Real()¶

condition
(context)[source]¶ Conditions on a context variable, returning a nonconditional transform of of type
NeuralAutoregressive
.

domain
= Real()¶

event_dim
= 1¶
ConditionalPlanar¶

class
ConditionalPlanar
(nn)[source]¶ Bases:
pyro.distributions.conditional.ConditionalTransformModule
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 pseudoparameters \(b\in\mathbb{R}\), \(\mathbf{u}\in\mathbb{R}^D\), 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])) # doctest: +SKIP
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 realvalued parameters of dimensions \((1, D, D)\). References: [1] Variational Inference with Normalizing Flows [arXiv:1505.05770] Danilo Jimenez Rezende, Shakir Mohamed

bijective
= True¶

codomain
= Real()¶

condition
(context)[source]¶ See
pyro.distributions.conditional.ConditionalTransformModule.condition()

domain
= Real()¶

event_dim
= 1¶

ConditionalRadial¶

class
ConditionalRadial
(nn)[source]¶ Bases:
pyro.distributions.conditional.ConditionalTransformModule
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.
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 = Radial(10) >>> pyro.module("my_transform", transform) # doctest: +SKIP >>> flow_dist = dist.TransformedDistribution(base_dist, [transform]) >>> flow_dist.sample() # doctest: +SKIP
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:
[1] Danilo Jimenez Rezende, Shakir Mohamed. Variational Inference with Normalizing Flows. [arXiv:1505.05770]

bijective
= True¶

codomain
= Real()¶

condition
(context)[source]¶ See
pyro.distributions.conditional.ConditionalTransformModule.condition()

domain
= Real()¶

event_dim
= 1¶

ConditionalTransformModule¶

class
ConditionalTransformModule
(*args, **kwargs)[source]¶ Bases:
pyro.distributions.conditional.ConditionalTransform
,torch.nn.modules.module.Module
Conditional transforms with learnable parameters such as normalizing flows should inherit from this class rather than
ConditionalTransform
so they are also a subclass ofModule
and inherit all the useful methods of that class.
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 [1] 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() # doctest: +SKIP
Parameters: channels (int) – Number of channel dimensions in the input. [1] Diederik P. Kingma, Prafulla Dhariwal. Glow: Generative Flow with Invertible 1x1 Convolutions. [arXiv:1807.03039]
[2] Ryan Prenger, Rafael Valle, Bryan Catanzaro. WaveGlow: A Flowbased Generative Network for Speech Synthesis. [arXiv:1811.00002]
[3] Conor Durkan, Artur Bekasov, Iain Murray, George Papamakarios. Neural Spline Flows. [arXiv:1906.04032]

bijective
= True¶

codomain
= Real()¶

domain
= Real()¶

event_dim
= 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 fullrank 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) # doctest: +SKIP >>> flow_dist = dist.TransformedDistribution(base_dist, [transform]) >>> flow_dist.sample() # doctest: +SKIP
Parameters: References:
[1] Jakub M. Tomczak, Max Welling. Improving Variational AutoEncoders using Householder Flow. [arXiv:1611.09630]

bijective
= True¶

codomain
= Real()¶

domain
= Real()¶

event_dim
= 1¶

volume_preserving
= True¶

NeuralAutoregressive¶

class
NeuralAutoregressive
(autoregressive_nn, hidden_units=16, activation='sigmoid')[source]¶ Bases:
pyro.distributions.torch_transform.TransformModule
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, [40], param_dims=[16]*3) >>> transform = NeuralAutoregressive(arn, hidden_units=16) >>> pyro.module("my_transform", transform) # doctest: +SKIP >>> flow_dist = dist.TransformedDistribution(base_dist, [transform]) >>> flow_dist.sample() # doctest: +SKIP
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 realvalued 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:
[1] ChinWei Huang, David Krueger, Alexandre Lacoste, Aaron Courville. Neural Autoregressive Flows. [arXiv:1804.00779]

autoregressive
= True¶

bijective
= True¶

codomain
= Real()¶

domain
= Real()¶

eps
= 1e08¶

event_dim
= 1¶
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) # doctest: +SKIP >>> flow_dist = dist.TransformedDistribution(base_dist, [transform]) >>> flow_dist.sample() # doctest: +SKIP
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:
[1] Danilo Jimenez Rezende, Shakir Mohamed. Variational Inference with Normalizing Flows. [arXiv:1505.05770]

bijective
= True¶

codomain
= Real()¶

domain
= Real()¶

event_dim
= 1¶

Polynomial¶

class
Polynomial
(autoregressive_nn, input_dim, count_degree, count_sum)[source]¶ Bases:
pyro.distributions.torch_transform.TransformModule
An autoregressive bijective transform as described in Jaini et al. (2019) applying following equation elementwise,
\(y_n = c_n + \int^{x_n}_0\sum^K_{k=1}\left(\sum^R_{r=0}a^{(n)}_{r,k}u^r\right)du\)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 \(x_{\prec n}={x_1,x_2,\ldots,x_{n1}}\).
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) # doctest: +SKIP >>> flow_dist = dist.TransformedDistribution(base_dist, [transform]) >>> flow_dist.sample() # doctest: +SKIP
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 realvalued numbers of size (batch_size, (count_degree+1)*count_sum, input_dim)
 count_degree (int) – The degree of the polynomial to use for each elementwise transformation.
 count_sum (int) – The number of polynomials to sum in each elementwise transformation.
References:
[1] Priyank Jaini, Kira A. Shelby, Yaoliang Yu. Sumofsquares polynomial flow. [arXiv:1905.02325]

autoregressive
= True¶

bijective
= True¶

codomain
= Real()¶

domain
= Real()¶

event_dim
= 1¶
Radial¶

class
Radial
(input_dim)[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.
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 = Radial(10) >>> pyro.module("my_transform", transform) # doctest: +SKIP >>> flow_dist = dist.TransformedDistribution(base_dist, [transform]) >>> flow_dist.sample() # doctest: +SKIP
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:
[1] Danilo Jimenez Rezende, Shakir Mohamed. Variational Inference with Normalizing Flows. [arXiv:1505.05770]

bijective
= True¶

codomain
= Real()¶

domain
= Real()¶

event_dim
= 1¶

Spline¶

class
Spline
(input_dim, count_bins=8, bound=3.0, order='linear')[source]¶ Bases:
pyro.distributions.torch_transform.TransformModule
An implementation of the elementwise 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 elementwise 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) # doctest: +SKIP >>> flow_dist = dist.TransformedDistribution(base_dist, [transform]) >>> flow_dist.sample() # doctest: +SKIP
Parameters:  input_dim (int) – Dimension of the input vector. Despite operating elementwise, 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()¶

domain
= Real()¶

event_dim
= 0¶
Sylvester¶

class
Sylvester
(input_dim, count_transforms=1)[source]¶ Bases:
pyro.distributions.transforms.householder.Householder
An implementation of the Sylvester bijective transform of the Householder variety (Van den Berg Et Al., 2018),
\(\mathbf{y} = \mathbf{x} + QR\tanh(SQ^T\mathbf{x}+\mathbf{b})\)where \(\mathbf{x}\) are the inputs, \(\mathbf{y}\) are the outputs, \(R,S\sim D\times D\) are upper triangular matrices for input dimension \(D\), \(Q\sim D\times D\) is an orthogonal matrix, and \(\mathbf{b}\sim D\) is learnable bias term.
The Sylvester transform is a generalization of
Planar
. In the Householder type of the Sylvester transform, the orthogonality of \(Q\) is enforced by representing it as the product of Householder transformations.Together with
TransformedDistribution
it provides a way to create richer variational approximations.Example usage:
>>> base_dist = dist.Normal(torch.zeros(10), torch.ones(10)) >>> transform = Sylvester(10, count_transforms=4) >>> pyro.module("my_transform", transform) # doctest: +SKIP >>> flow_dist = dist.TransformedDistribution(base_dist, [transform]) >>> flow_dist.sample() # doctest: +SKIP tensor([0.4071, 0.5030, 0.7924, 0.2366, 0.2387, 0.1417, 0.0868, 0.1389, 0.4629, 0.0986])
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 Sylvester transform can be scored.
References:
[1] Rianne van den Berg, Leonard Hasenclever, Jakub M. Tomczak, Max Welling. Sylvester Normalizing Flows for Variational Inference. UAI 2018.

bijective
= True¶

codomain
= Real()¶

domain
= Real()¶

event_dim
= 1¶

TransformModule¶

class
TransformModule
(*args, **kwargs)[source]¶ Bases:
torch.distributions.transforms.Transform
,torch.nn.modules.module.Module
Transforms with learnable parameters such as normalizing flows should inherit from this class rather than Transform so they are also a subclass of nn.Module and inherit all the useful methods of that class.
ComposeTransformModule¶

class
ComposeTransformModule
(parts)[source]¶ Bases:
torch.distributions.transforms.ComposeTransform
,torch.nn.modules.container.ModuleList
This allows us to use a list of TransformModule in the same way as
ComposeTransform
. This is needed so that transform parameters are automatically registered by Pyro’s param store when used inPyroModule
instances.
Transform Factories¶
Each Transform
and TransformModule
includes a corresponding helper function in lower case that inputs, at minimum, the input dimensions of the transform, and possibly additional arguments to customize the transform in an intuitive way. The purpose of these helper functions is to hide from the user whether or not the transform requires the construction of a hypernet, and if so, the input and output dimensions of that hypernet.
iterated¶

iterated
(repeats, base_fn, *args, **kwargs)[source]¶ Helper function to compose a sequence of bijective transforms with potentially learnable parameters using
ComposeTransformModule
.Parameters:  repeats – number of repeated transforms.
 base_fn – function to construct the bijective transform.
 args – arguments taken by base_fn.
 kwargs – keyword arguments taken by base_fn.
Returns: instance of
TransformModule
.
affine_autoregressive¶

affine_autoregressive
(input_dim, hidden_dims=None, **kwargs)[source]¶ A helper function to create an
AffineAutoregressive
object that takes care of constructing an autoregressive network with the correct input/output dimensions.Parameters:  input_dim (int) – Dimension of input variable
 hidden_dims (list[int]) – The desired hidden dimensions of the autoregressive network. Defaults to using [3*input_dim + 1]
 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).
affine_coupling¶

affine_coupling
(input_dim, hidden_dims=None, split_dim=None, dim=1, **kwargs)[source]¶ A helper function to create an
AffineCoupling
object that takes care of constructing a dense network with the correct input/output dimensions.Parameters:  input_dim (int) – Dimension(s) of input variable to permute. Note that when dim < 1 this must be a tuple corresponding to the event shape.
 hidden_dims (list[int]) – The desired hidden dimensions of the dense network. Defaults to using [10*input_dim]
 split_dim (int) – The dimension to split the input on for the coupling transform. Defaults to using input_dim // 2
 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
batchnorm¶
block_autoregressive¶

block_autoregressive
(input_dim, **kwargs)[source]¶ A helper function to create a
BlockAutoregressive
object for consistency with other helpers.Parameters:  input_dim (int) – Dimension of input variable
 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 \(\alpha\).
conditional_affine_autoregressive¶

conditional_affine_autoregressive
(input_dim, context_dim, hidden_dims=None, **kwargs)[source]¶ A helper function to create an
ConditionalAffineAutoregressive
object that takes care of constructing a dense network with the correct input/output dimensions.Parameters:  input_dim (int) – Dimension of input variable
 context_dim (int) – Dimension of context variable
 hidden_dims (list[int]) – The desired hidden dimensions of the dense network. Defaults to using [10*input_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
 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).
conditional_affine_coupling¶

conditional_affine_coupling
(input_dim, context_dim, hidden_dims=None, split_dim=None, dim=1, **kwargs)[source]¶ A helper function to create an
ConditionalAffineCoupling
object that takes care of constructing a dense network with the correct input/output dimensions.Parameters:  input_dim (int) – Dimension of input variable
 context_dim (int) – Dimension of context variable
 hidden_dims (list[int]) – The desired hidden dimensions of the dense network. Defaults to using [10*input_dim]
 split_dim (int) – The dimension to split the input on for the coupling transform. Defaults to using input_dim // 2
 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
conditional_generalized_channel_permute¶

conditional_generalized_channel_permute
(context_dim, channels=3, hidden_dims=None)[source]¶ A helper function to create a
ConditionalGeneralizedChannelPermute
object for consistency with other helpers.Parameters: channels (int) – Number of channel dimensions in the input.
conditional_householder¶

conditional_householder
(input_dim, context_dim, hidden_dims=None, count_transforms=1)[source]¶ A helper function to create a
ConditionalHouseholder
object that takes care of constructing a dense network with the correct input/output dimensions.Parameters:
conditional_neural_autoregressive¶

conditional_neural_autoregressive
(input_dim, context_dim, hidden_dims=None, activation='sigmoid', width=16)[source]¶ A helper function to create a
ConditionalNeuralAutoregressive
object that takes care of constructing an autoregressive network with the correct input/output dimensions.Parameters:  input_dim (int) – Dimension of input variable
 context_dim (int) – Dimension of context variable
 hidden_dims (list[int]) – The desired hidden dimensions of the autoregressive network. Defaults to using [3*input_dim + 1]
 activation (string) – Activation function to use. One of ‘ELU’, ‘LeakyReLU’, ‘sigmoid’, or ‘tanh’.
 width (int) – The width of the “multilayer perceptron” in the transform (see paper). Defaults to 16
conditional_planar¶

conditional_planar
(input_dim, context_dim, hidden_dims=None)[source]¶ A helper function to create a
ConditionalPlanar
object that takes care of constructing a dense network with the correct input/output dimensions.Parameters:
conditional_radial¶

conditional_radial
(input_dim, context_dim, hidden_dims=None)[source]¶ A helper function to create a
ConditionalRadial
object that takes care of constructing a dense network with the correct input/output dimensions.Parameters:
elu¶
generalized_channel_permute¶

generalized_channel_permute
(**kwargs)[source]¶ A helper function to create a
GeneralizedChannelPermute
object for consistency with other helpers.Parameters: channels (int) – Number of channel dimensions in the input.
householder¶

householder
(input_dim, count_transforms=None)[source]¶ A helper function to create a
Householder
object for consistency with other helpers.Parameters:
leaky_relu¶

leaky_relu
()[source]¶ A helper function to create a
LeakyReLUTransform
object for consistency with other helpers.
neural_autoregressive¶

neural_autoregressive
(input_dim, hidden_dims=None, activation='sigmoid', width=16)[source]¶ A helper function to create a
NeuralAutoregressive
object that takes care of constructing an autoregressive network with the correct input/output dimensions.Parameters:  input_dim (int) – Dimension of input variable
 hidden_dims (list[int]) – The desired hidden dimensions of the autoregressive network. Defaults to using [3*input_dim + 1]
 activation (string) – Activation function to use. One of ‘ELU’, ‘LeakyReLU’, ‘sigmoid’, or ‘tanh’.
 width (int) – The width of the “multilayer perceptron” in the transform (see paper). Defaults to 16
permute¶

permute
(input_dim, permutation=None, dim=1)[source]¶ A helper function to create a
Permute
object for consistency with other helpers.Parameters:  input_dim (int) – Dimension(s) of input variable to permute. Note that when dim < 1 this must be a tuple corresponding to the event shape.
 permutation (torch.LongTensor) – Torch tensor of integer indices representing permutation. Defaults to a random permutation.
 dim (int) – the tensor dimension to permute. This value must be negative and defines the event dim as abs(dim).
planar¶
polynomial¶

polynomial
(input_dim, hidden_dims=None)[source]¶ A helper function to create a
Polynomial
object that takes care of constructing an autoregressive network with the correct input/output dimensions.Parameters:  input_dim (int) – Dimension of input variable
 hidden_dims – The desired hidden dimensions of of the autoregressive network. Defaults to using [input_dim * 10]
radial¶
spline¶
sylvester¶

sylvester
(input_dim, count_transforms=None)[source]¶ A helper function to create a
Sylvester
object for consistency with other helpers.Parameters:  input_dim (int) – Dimension of input variable
 count_transforms – Number of Sylvester operations to apply. Defaults to input_dim // 2 + 1. :type count_transforms: int