# Source code for pyro.distributions.hmm

```
# Copyright (c) 2017-2019 Uber Technologies, Inc.
# SPDX-License-Identifier: Apache-2.0
import torch
from torch.distributions import constraints
from pyro.distributions.torch import Categorical, Gamma, Independent, MultivariateNormal
from pyro.distributions.torch_distribution import TorchDistribution
from pyro.distributions.util import broadcast_shape
from pyro.ops.gamma_gaussian import (GammaGaussian, gamma_and_mvn_to_gamma_gaussian, gamma_gaussian_tensordot,
matrix_and_mvn_to_gamma_gaussian)
from pyro.ops.gaussian import Gaussian, gaussian_tensordot, matrix_and_mvn_to_gaussian, mvn_to_gaussian
@torch.jit.script
def _linear_integrate(init, trans, shift):
"""
Integrate the inhomogeneous linear shifterence equation::
x[0] = init
x[t] = x[t-1] @ trans[t] + shift[t]
:return: An integrated tensor ``x[:, :]``.
"""
# xs: List[Tensor]
xs = []
x = init.unsqueeze(-2)
shift = shift.unsqueeze(-3)
for t in range(trans.size(-3)):
x = x @ trans[..., t, :, :] + shift[..., t, :]
xs.append(x)
return torch.cat(xs, dim=-2)
def _logmatmulexp(x, y):
"""
Numerically stable version of ``(x.log() @ y.log()).exp()``.
"""
x_shift = x.max(-1, keepdim=True)[0]
y_shift = y.max(-2, keepdim=True)[0]
xy = torch.matmul((x - x_shift).exp(), (y - y_shift).exp()).log()
return xy + x_shift + y_shift
@torch.jit.script
def _sequential_logmatmulexp(logits):
"""
For a tensor ``x`` whose time dimension is -3, computes::
x[..., 0, :, :] @ x[..., 1, :, :] @ ... @ x[..., T-1, :, :]
but does so numerically stably in log space.
"""
batch_shape = logits.shape[:-3]
state_dim = logits.size(-1)
while logits.size(-3) > 1:
time = logits.size(-3)
even_time = time // 2 * 2
even_part = logits[..., :even_time, :, :]
x_y = even_part.reshape(batch_shape + (even_time // 2, 2, state_dim, state_dim))
x, y = x_y.unbind(-3)
contracted = _logmatmulexp(x, y)
if time > even_time:
contracted = torch.cat((contracted, logits[..., -1:, :, :]), dim=-3)
logits = contracted
return logits.squeeze(-3)
def _sequential_gaussian_tensordot(gaussian):
"""
Integrates a Gaussian ``x`` whose rightmost batch dimension is time, computes::
x[..., 0] @ x[..., 1] @ ... @ x[..., T-1]
"""
assert isinstance(gaussian, Gaussian)
assert gaussian.dim() % 2 == 0, "dim is not even"
batch_shape = gaussian.batch_shape[:-1]
state_dim = gaussian.dim() // 2
while gaussian.batch_shape[-1] > 1:
time = gaussian.batch_shape[-1]
even_time = time // 2 * 2
even_part = gaussian[..., :even_time]
x_y = even_part.reshape(batch_shape + (even_time // 2, 2))
x, y = x_y[..., 0], x_y[..., 1]
contracted = gaussian_tensordot(x, y, state_dim)
if time > even_time:
contracted = Gaussian.cat((contracted, gaussian[..., -1:]), dim=-1)
gaussian = contracted
return gaussian[..., 0]
def _is_subshape(x, y):
return broadcast_shape(x, y) == y
def _sequential_gaussian_filter_sample(init, trans, sample_shape):
"""
Draws a reparameterized sample from a Markov product of Gaussians via
parallel-scan forward-filter backward-sample.
"""
assert isinstance(init, Gaussian)
assert isinstance(trans, Gaussian)
assert trans.dim() == 2 * init.dim()
assert _is_subshape(trans.batch_shape[:-1], init.batch_shape)
state_dim = trans.dim() // 2
device = trans.precision.device
perm = torch.cat([torch.arange(1 * state_dim, 2 * state_dim, device=device),
torch.arange(0 * state_dim, 1 * state_dim, device=device),
torch.arange(2 * state_dim, 3 * state_dim, device=device)])
# Forward filter, similar to _sequential_gaussian_tensordot().
tape = []
shape = trans.batch_shape[:-1] # Note trans may be unbroadcasted.
gaussian = trans
while gaussian.batch_shape[-1] > 1:
time = gaussian.batch_shape[-1]
even_time = time // 2 * 2
even_part = gaussian[..., :even_time]
x_y = even_part.reshape(shape + (even_time // 2, 2))
x, y = x_y[..., 0], x_y[..., 1]
x = x.event_pad(right=state_dim)
y = y.event_pad(left=state_dim)
joint = (x + y).event_permute(perm)
tape.append(joint)
contracted = joint.marginalize(left=state_dim)
if time > even_time:
contracted = Gaussian.cat((contracted, gaussian[..., -1:]), dim=-1)
gaussian = contracted
gaussian = gaussian[..., 0] + init.event_pad(right=state_dim)
# Backward sample.
shape = sample_shape + init.batch_shape
result = gaussian.rsample(sample_shape).reshape(shape + (2, state_dim))
for joint in reversed(tape):
# The following comments demonstrate two example computations, one
# EVEN, one ODD. Ignoring sample_shape and batch_shape, let each zn be
# a single sampled event of shape (state_dim,).
if joint.batch_shape[-1] == result.size(-2) - 1: # EVEN case.
# Suppose e.g. result = [z0, z2, z4]
cond = result.repeat_interleave(2, dim=-2) # [z0, z0, z2, z2, z4, z4]
cond = cond[..., 1:-1, :] # [z0, z2, z2, z4]
cond = cond.reshape(shape + (-1, 2 * state_dim)) # [z0z2, z2z4]
sample = joint.condition(cond).rsample() # [z1, z3]
sample = torch.nn.functional.pad(sample, (0, 0, 0, 1)) # [z1, z3, 0]
result = torch.stack([
result, # [z0, z2, z4]
sample, # [z1, z3, 0]
], dim=-2) # [[z0, z1], [z2, z3], [z4, 0]]
result = result.reshape(shape + (-1, state_dim)) # [z0, z1, z2, z3, z4, 0]
result = result[..., :-1, :] # [z0, z1, z2, z3, z4]
else: # ODD case.
assert joint.batch_shape[-1] == result.size(-2) - 2
# Suppose e.g. result = [z0, z2, z3]
cond = result[..., :-1, :].repeat_interleave(2, dim=-2) # [z0, z0, z2, z2]
cond = cond[..., 1:-1, :] # [z0, z2]
cond = cond.reshape(shape + (-1, 2 * state_dim)) # [z0z2]
sample = joint.condition(cond).rsample() # [z1]
sample = torch.cat([sample, result[..., -1:, :]], dim=-2) # [z1, z3]
result = torch.stack([
result[..., :-1, :], # [z0, z2]
sample, # [z1, z3]
], dim=-2) # [[z0, z1], [z2, z3]]
result = result.reshape(shape + (-1, state_dim)) # [z0, z1, z2, z3]
return result[..., 1:, :] # [z1, z2, z3, ...]
def _sequential_gamma_gaussian_tensordot(gamma_gaussian):
"""
Integrates a GammaGaussian ``x`` whose rightmost batch dimension is time, computes::
x[..., 0] @ x[..., 1] @ ... @ x[..., T-1]
"""
assert isinstance(gamma_gaussian, GammaGaussian)
assert gamma_gaussian.dim() % 2 == 0, "dim is not even"
batch_shape = gamma_gaussian.batch_shape[:-1]
state_dim = gamma_gaussian.dim() // 2
while gamma_gaussian.batch_shape[-1] > 1:
time = gamma_gaussian.batch_shape[-1]
even_time = time // 2 * 2
even_part = gamma_gaussian[..., :even_time]
x_y = even_part.reshape(batch_shape + (even_time // 2, 2))
x, y = x_y[..., 0], x_y[..., 1]
contracted = gamma_gaussian_tensordot(x, y, state_dim)
if time > even_time:
contracted = GammaGaussian.cat((contracted, gamma_gaussian[..., -1:]), dim=-1)
gamma_gaussian = contracted
return gamma_gaussian[..., 0]
class HiddenMarkovModel(TorchDistribution):
"""
Abstract base class for Hidden Markov Models.
The purpose of this class is to handle duration logic for homogeneous HMMs.
:param int duration: 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.
"""
def __init__(self, duration, batch_shape, event_shape, validate_args=None):
if duration is None:
if event_shape[0] != 1:
# Infer duration from event_shape.
duration = event_shape[0]
elif duration != event_shape[0]:
if event_shape[0] != 1:
raise ValueError("duration, event_shape mismatch: {} vs {}"
.format(duration, event_shape))
# Infer event_shape from duration.
event_shape = torch.Size((duration,) + event_shape[1:])
self._duration = duration
super().__init__(batch_shape, event_shape, validate_args)
@property
def duration(self):
"""
Returns the size of the time axis, or None if unknown.
"""
return self._duration
def _validate_sample(self, value):
if value.dim() < self.event_dim:
raise ValueError("value has too few dimensions: {}".format(value.shape))
if self.duration is not None:
super()._validate_sample(value)
return
# Temporarily infer duration from value.shape.
duration = value.size(-self.event_dim)
old = self._event_shape
new = torch.Size((duration,)) + self._event_shape[1:]
try:
self._event_shape = new
super()._validate_sample(value)
finally:
self._event_shape = old
[docs]class DiscreteHMM(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`` and ``observation_dist``. However,
because time is included in this distribution's event_shape, the
homogeneous+homogeneous case will have a broadcastable event_shape with
``num_steps = 1``, allowing :meth:`log_prob` to work with arbitrary length
data::
# homogeneous + homogeneous case:
event_shape = (1,) + observation_dist.event_shape
**References:**
[1] Simo Sarkka, Angel F. Garcia-Fernandez (2019)
"Temporal Parallelization of Bayesian Filters and Smoothers"
https://arxiv.org/pdf/1905.13002.pdf
:param ~torch.Tensor initial_logits: A logits tensor for an initial
categorical distribution over latent states. Should have rightmost size
``state_dim`` and be broadcastable to ``batch_shape + (state_dim,)``.
:param ~torch.Tensor transition_logits: A logits tensor for transition
conditional distributions between latent states. Should have rightmost
shape ``(state_dim, state_dim)`` (old, new), and be broadcastable to
``batch_shape + (num_steps, state_dim, state_dim)``.
:param ~torch.distributions.Distribution observation_dist: A conditional
distribution of observed data conditioned on latent state. The
``.batch_shape`` should have rightmost size ``state_dim`` and be
broadcastable to ``batch_shape + (num_steps, state_dim)``. The
``.event_shape`` may be arbitrary.
:param int duration: 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": constraints.real,
"transition_logits": constraints.real}
def __init__(self, initial_logits, transition_logits, observation_dist,
validate_args=None, duration=None):
if initial_logits.dim() < 1:
raise ValueError("expected initial_logits to have at least one dim, "
"actual shape = {}".format(initial_logits.shape))
if transition_logits.dim() < 2:
raise ValueError("expected transition_logits to have at least two dims, "
"actual shape = {}".format(transition_logits.shape))
if len(observation_dist.batch_shape) < 1:
raise ValueError("expected observation_dist to have at least one batch dim, "
"actual .batch_shape = {}".format(observation_dist.batch_shape))
shape = broadcast_shape(initial_logits.shape[:-1] + (1,),
transition_logits.shape[:-2],
observation_dist.batch_shape[:-1])
batch_shape, time_shape = shape[:-1], shape[-1:]
event_shape = time_shape + observation_dist.event_shape
self.initial_logits = initial_logits - initial_logits.logsumexp(-1, True)
self.transition_logits = transition_logits - transition_logits.logsumexp(-1, True)
self.observation_dist = observation_dist
super().__init__(duration, batch_shape, event_shape, validate_args=validate_args)
@property
def support(self):
return self.observation_dist.support
[docs] def expand(self, batch_shape, _instance=None):
new = self._get_checked_instance(DiscreteHMM, _instance)
batch_shape = torch.Size(broadcast_shape(self.batch_shape, batch_shape))
# We only need to expand one of the inputs, since batch_shape is determined
# by broadcasting all three. To save computation in _sequential_logmatmulexp(),
# we expand only initial_logits, which is applied only after the logmatmulexp.
# This is similar to the ._unbroadcasted_* pattern used elsewhere in distributions.
new.initial_logits = self.initial_logits.expand(batch_shape + (-1,))
new.transition_logits = self.transition_logits
new.observation_dist = self.observation_dist
super(DiscreteHMM, new).__init__(self.duration, batch_shape, self.event_shape, validate_args=False)
new._validate_args = self.__dict__.get('_validate_args')
return new
[docs] def log_prob(self, value):
if self._validate_args:
self._validate_sample(value)
# Combine observation and transition factors.
value = value.unsqueeze(-1 - self.observation_dist.event_dim)
observation_logits = self.observation_dist.log_prob(value)
result = self.transition_logits + observation_logits.unsqueeze(-2)
# Eliminate time dimension.
result = _sequential_logmatmulexp(result)
# Combine initial factor.
result = self.initial_logits + result.logsumexp(-1)
# Marginalize out final state.
result = result.logsumexp(-1)
return result
[docs] def filter(self, value):
"""
Compute posterior over final state given a sequence of observations.
:param ~torch.Tensor value: A sequence of observations.
:return: A posterior distribution
over latent states at the final time step. ``result.logits`` can
then be used as ``initial_logits`` in a sequential Pyro model for
prediction.
:rtype: ~pyro.distributions.Categorical
"""
if self._validate_args:
self._validate_sample(value)
# Combine observation and transition factors.
value = value.unsqueeze(-1 - self.observation_dist.event_dim)
observation_logits = self.observation_dist.log_prob(value)
logp = self.transition_logits + observation_logits.unsqueeze(-2)
# Eliminate time dimension.
logp = _sequential_logmatmulexp(logp)
# Combine initial factor.
logp = (self.initial_logits.unsqueeze(-1) + logp).logsumexp(-2)
# Convert to a distribution.
return Categorical(logits=logp, validate_args=self._validate_args)
[docs]class GaussianHMM(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 :meth:`log_prob` is differentiable.
This corresponds to the generative model::
z = initial_distribution.sample()
x = []
for t in range(num_events):
z = z @ transition_matrix + transition_dist.sample()
x.append(z @ observation_matrix + observation_dist.sample())
The event_shape of this distribution includes time on the left::
event_shape = (num_steps,) + observation_dist.event_shape
This distribution supports any combination of homogeneous/heterogeneous
time dependency of ``transition_dist`` and ``observation_dist``. However,
because time is included in this distribution's event_shape, the
homogeneous+homogeneous case will have a broadcastable event_shape with
``num_steps = 1``, allowing :meth:`log_prob` to work with arbitrary length
data::
event_shape = (1, obs_dim) # homogeneous + homogeneous case
**References:**
[1] Simo Sarkka, Angel F. Garcia-Fernandez (2019)
"Temporal Parallelization of Bayesian Filters and Smoothers"
https://arxiv.org/pdf/1905.13002.pdf
:ivar int hidden_dim: The dimension of the hidden state.
:ivar int obs_dim: The dimension of the observed state.
:param ~torch.distributions.MultivariateNormal initial_dist: A distribution
over initial states. This should have batch_shape broadcastable to
``self.batch_shape``. This should have event_shape ``(hidden_dim,)``.
:param ~torch.Tensor transition_matrix: 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)``.
:param ~torch.distributions.MultivariateNormal transition_dist: A process
noise distribution. This should have batch_shape broadcastable to
``self.batch_shape + (num_steps,)``. This should have event_shape
``(hidden_dim,)``.
:param ~torch.Tensor observation_matrix: A linear transformation from hidden
to observed state. This should have shape broadcastable to
``self.batch_shape + (num_steps, hidden_dim, obs_dim)``.
:param observation_dist: An observation noise distribution. This should
have batch_shape broadcastable to ``self.batch_shape + (num_steps,)``.
This should have event_shape ``(obs_dim,)``.
:type observation_dist: ~torch.distributions.MultivariateNormal or
~torch.distributions.Independent of ~torch.distributions.Normal
:param int duration: 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.
"""
has_rsample = True
arg_constraints = {}
support = constraints.real
def __init__(self, initial_dist, transition_matrix, transition_dist,
observation_matrix, observation_dist, validate_args=None, duration=None):
assert (isinstance(initial_dist, torch.distributions.MultivariateNormal) or
(isinstance(initial_dist, torch.distributions.Independent) and
isinstance(initial_dist.base_dist, torch.distributions.Normal)))
assert isinstance(transition_matrix, torch.Tensor)
assert (isinstance(transition_dist, torch.distributions.MultivariateNormal) or
(isinstance(transition_dist, torch.distributions.Independent) and
isinstance(transition_dist.base_dist, torch.distributions.Normal)))
assert isinstance(observation_matrix, torch.Tensor)
assert (isinstance(observation_dist, torch.distributions.MultivariateNormal) or
(isinstance(observation_dist, torch.distributions.Independent) and
isinstance(observation_dist.base_dist, torch.distributions.Normal)))
hidden_dim, obs_dim = observation_matrix.shape[-2:]
assert initial_dist.event_shape == (hidden_dim,)
assert transition_matrix.shape[-2:] == (hidden_dim, hidden_dim)
assert transition_dist.event_shape == (hidden_dim,)
assert observation_dist.event_shape == (obs_dim,)
shape = broadcast_shape(initial_dist.batch_shape + (1,),
transition_matrix.shape[:-2],
transition_dist.batch_shape,
observation_matrix.shape[:-2],
observation_dist.batch_shape)
batch_shape, time_shape = shape[:-1], shape[-1:]
event_shape = time_shape + (obs_dim,)
super().__init__(duration, batch_shape, event_shape, validate_args=validate_args)
self.hidden_dim = hidden_dim
self.obs_dim = obs_dim
self._init = mvn_to_gaussian(initial_dist).expand(self.batch_shape)
self._trans = matrix_and_mvn_to_gaussian(transition_matrix, transition_dist)
self._obs = matrix_and_mvn_to_gaussian(observation_matrix, observation_dist)
[docs] def expand(self, batch_shape, _instance=None):
new = self._get_checked_instance(GaussianHMM, _instance)
new.hidden_dim = self.hidden_dim
new.obs_dim = self.obs_dim
new._obs = self._obs
new._trans = self._trans
# To save computation in _sequential_gaussian_tensordot(), we expand
# only _init, which is applied only after
# _sequential_gaussian_tensordot().
batch_shape = torch.Size(broadcast_shape(self.batch_shape, batch_shape))
new._init = self._init.expand(batch_shape)
super(GaussianHMM, new).__init__(self.duration, batch_shape, self.event_shape, validate_args=False)
new._validate_args = self.__dict__.get('_validate_args')
return new
[docs] def log_prob(self, value):
if self._validate_args:
self._validate_sample(value)
# Combine observation and transition factors.
result = self._trans + self._obs.condition(value).event_pad(left=self.hidden_dim)
# Eliminate time dimension.
result = _sequential_gaussian_tensordot(result.expand(result.batch_shape))
# Combine initial factor.
result = gaussian_tensordot(self._init, result, dims=self.hidden_dim)
# Marginalize out final state.
result = result.event_logsumexp()
return result
[docs] def rsample(self, sample_shape=torch.Size()):
assert self.duration is not None
sample_shape = torch.Size(sample_shape)
obs_dim = self.obs_dim
hidden_dim = self.hidden_dim
trans = self._trans + self._obs.marginalize(right=self.obs_dim).event_pad(left=self.hidden_dim)
trans = trans.expand(trans.batch_shape[:-1] + (self.duration,))
z = _sequential_gaussian_filter_sample(self._init, trans, sample_shape)
perm = torch.cat([torch.arange(hidden_dim, hidden_dim + obs_dim, device=z.device),
torch.arange(hidden_dim, device=z.device)])
x = self._obs.event_permute(perm).condition(z).rsample()
return x
[docs] def rsample_posterior(self, value, sample_shape=torch.Size()):
"""
EXPERIMENTAL Sample from the latent state conditioned on observation.
"""
trans = self._trans + self._obs.condition(value).event_pad(left=self.hidden_dim)
trans = trans.expand(trans.batch_shape)
z = _sequential_gaussian_filter_sample(self._init, trans, sample_shape)
return z
[docs] def filter(self, value):
"""
Compute posterior over final state given a sequence of observations.
:param ~torch.Tensor value: A sequence of observations.
:return: A posterior
distribution over latent states at the final time step. ``result``
can then be used as ``initial_dist`` in a sequential Pyro model for
prediction.
:rtype: ~pyro.distributions.MultivariateNormal
"""
if self._validate_args:
self._validate_sample(value)
# Combine observation and transition factors.
logp = self._trans + self._obs.condition(value).event_pad(left=self.hidden_dim)
# Eliminate time dimension.
logp = _sequential_gaussian_tensordot(logp.expand(logp.batch_shape))
# Combine initial factor.
logp = gaussian_tensordot(self._init, logp, dims=self.hidden_dim)
# Convert to a distribution
precision = logp.precision
loc = logp.info_vec.unsqueeze(-1).cholesky_solve(precision.cholesky()).squeeze(-1)
return MultivariateNormal(loc, precision_matrix=precision,
validate_args=self._validate_args)
[docs] def conjugate_update(self, other):
"""
EXPERIMENTAL Creates an updated :class:`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
:param other: A distribution representing ``p(data|self.probs)`` but
normalized over ``self.probs`` rather than ``data``.
:type other: ~torch.distributions.Independent of
~torch.distributions.MultivariateNormal or ~torch.distributions.Normal
:return: a pair ``(updated,log_normalizer)`` where ``updated`` is an
updated :class:`GaussianHMM` , and ``log_normalizer`` is a
:class:`~torch.Tensor` representing the normalization factor.
"""
assert (isinstance(other, torch.distributions.Independent) and
(isinstance(other.base_dist, torch.distributions.Normal) or
isinstance(other.base_dist, torch.distributions.MultivariateNormal)))
duration = other.event_shape[0] if self.duration is None else self.duration
event_shape = torch.Size((duration, self.obs_dim))
assert other.event_shape == event_shape
new = self._get_checked_instance(GaussianHMM)
new.hidden_dim = self.hidden_dim
new.obs_dim = self.obs_dim
new._init = self._init
new._trans = self._trans
new._obs = self._obs + mvn_to_gaussian(other.to_event(-1)).event_pad(left=self.hidden_dim)
# Normalize.
# TODO cache this computation for the forward pass of .rsample().
logp = new._trans + new._obs.marginalize(right=new.obs_dim).event_pad(left=new.hidden_dim)
logp = _sequential_gaussian_tensordot(logp.expand(logp.batch_shape))
logp = gaussian_tensordot(new._init, logp, dims=new.hidden_dim)
log_normalizer = logp.event_logsumexp()
new._init = new._init - log_normalizer
batch_shape = log_normalizer.shape
super(GaussianHMM, new).__init__(duration, batch_shape, event_shape, validate_args=False)
new._validate_args = self.__dict__.get('_validate_args')
return new, log_normalizer
[docs] def prefix_condition(self, data):
"""
EXPERIMENTAL Given self has ``event_shape == (t+f, d)`` and data ``x``
of shape ``batch_shape + (t, d)``, compute a conditional distribution
of event_shape ``(f, d)``. Typically ``t`` is the number of training
time steps, ``f`` is the number of forecast time steps, and ``d`` is
the data dimension.
:param data: data of dimension at least 2.
:type data: ~torch.Tensor
"""
assert data.dim() >= 2
assert data.size(-1) == self.event_shape[-1]
assert data.size(-2) < self.duration
t = data.size(-2)
f = self.duration - t
left = self._get_checked_instance(GaussianHMM)
left.hidden_dim = self.hidden_dim
left.obs_dim = self.obs_dim
left._init = self._init
right = self._get_checked_instance(GaussianHMM)
right.hidden_dim = self.hidden_dim
right.obs_dim = self.obs_dim
if self._obs.batch_shape == () or self._obs.batch_shape[-1] == 1: # homogeneous
left._obs = self._obs
right._obs = self._obs
else: # heterogeneous
left._obs = self._obs[..., :t]
right._obs = self._obs[..., t:]
if self._trans.batch_shape == () or self._trans.batch_shape[-1] == 1: # homogeneous
left._trans = self._trans
right._trans = self._trans
else: # heterogeneous
left._trans = self._trans[..., :t]
right._trans = self._trans[..., t:]
super(GaussianHMM, left).__init__(t, self.batch_shape, (t, self.obs_dim),
validate_args=self._validate_args)
initial_dist = left.filter(data)
right._init = mvn_to_gaussian(initial_dist)
batch_shape = broadcast_shape(right._init.batch_shape, self.batch_shape)
super(GaussianHMM, right).__init__(f, batch_shape, (f, self.obs_dim),
validate_args=self._validate_args)
return right
[docs]class GammaGaussianHMM(HiddenMarkovModel):
"""
Hidden Markov Model with the joint distribution of initial state, hidden
state, and observed state is a :class:`~pyro.distributions.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`` and ``observation_dist``. However,
because time is included in this distribution's event_shape, the
homogeneous+homogeneous case will have a broadcastable event_shape with
``num_steps = 1``, allowing :meth:`log_prob` to work with arbitrary length
data::
event_shape = (1, obs_dim) # homogeneous + homogeneous case
**References:**
[1] Simo Sarkka, Angel F. Garcia-Fernandez (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 t-filters for noise scale estimation"
https://users.aalto.fi/~ssarkka/pub/SPL2019.pdf
:ivar int hidden_dim: The dimension of the hidden state.
:ivar int obs_dim: The dimension of the observed state.
:param Gamma scale_dist: Prior of the mixing distribution.
:param MultivariateNormal initial_dist: 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,)``.
:param ~torch.Tensor transition_matrix: 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)``.
:param MultivariateNormal transition_dist: 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,)``.
:param ~torch.Tensor observation_matrix: A linear transformation from hidden
to observed state. This should have shape broadcastable to
``self.batch_shape + (num_steps, hidden_dim, obs_dim)``.
:param MultivariateNormal observation_dist: 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,)``.
:param int duration: 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 = {}
support = constraints.real
def __init__(self, scale_dist, initial_dist, transition_matrix, transition_dist,
observation_matrix, observation_dist, validate_args=None, duration=None):
assert isinstance(scale_dist, Gamma)
assert isinstance(initial_dist, MultivariateNormal)
assert isinstance(transition_matrix, torch.Tensor)
assert isinstance(transition_dist, MultivariateNormal)
assert isinstance(observation_matrix, torch.Tensor)
assert isinstance(observation_dist, MultivariateNormal)
hidden_dim, obs_dim = observation_matrix.shape[-2:]
assert initial_dist.event_shape == (hidden_dim,)
assert transition_matrix.shape[-2:] == (hidden_dim, hidden_dim)
assert transition_dist.event_shape == (hidden_dim,)
assert observation_dist.event_shape == (obs_dim,)
shape = broadcast_shape(scale_dist.batch_shape + (1,),
initial_dist.batch_shape + (1,),
transition_matrix.shape[:-2],
transition_dist.batch_shape,
observation_matrix.shape[:-2],
observation_dist.batch_shape)
batch_shape, time_shape = shape[:-1], shape[-1:]
event_shape = time_shape + (obs_dim,)
super().__init__(duration, batch_shape, event_shape, validate_args=validate_args)
self.hidden_dim = hidden_dim
self.obs_dim = obs_dim
self._init = gamma_and_mvn_to_gamma_gaussian(scale_dist, initial_dist)
self._trans = matrix_and_mvn_to_gamma_gaussian(transition_matrix, transition_dist)
self._obs = matrix_and_mvn_to_gamma_gaussian(observation_matrix, observation_dist)
[docs] def expand(self, batch_shape, _instance=None):
new = self._get_checked_instance(GammaGaussianHMM, _instance)
batch_shape = torch.Size(broadcast_shape(self.batch_shape, batch_shape))
new.hidden_dim = self.hidden_dim
new.obs_dim = self.obs_dim
# We only need to expand one of the inputs, since batch_shape is determined
# by broadcasting all three. To save computation in _sequential_gaussian_tensordot(),
# we expand only _init, which is applied only after _sequential_gaussian_tensordot().
new._init = self._init.expand(batch_shape)
new._trans = self._trans
new._obs = self._obs
super(GammaGaussianHMM, new).__init__(self.duration, batch_shape, self.event_shape, validate_args=False)
new._validate_args = self.__dict__.get('_validate_args')
return new
[docs] def log_prob(self, value):
if self._validate_args:
self._validate_sample(value)
# Combine observation and transition factors.
result = self._trans + self._obs.condition(value).event_pad(left=self.hidden_dim)
# Eliminate time dimension.
result = _sequential_gamma_gaussian_tensordot(result.expand(result.batch_shape))
# Combine initial factor.
result = gamma_gaussian_tensordot(self._init, result, dims=self.hidden_dim)
# Marginalize out final state.
result = result.event_logsumexp()
# Marginalize out multiplier.
result = result.logsumexp()
return result
[docs] def filter(self, value):
"""
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).
:param ~torch.Tensor value: A sequence of observations.
:return: A pair of posterior distributions over the mixing and the latent
state at the final time step.
:rtype: a tuple of ~pyro.distributions.Gamma and ~pyro.distributions.MultivariateNormal
"""
if self._validate_args:
self._validate_sample(value)
# Combine observation and transition factors.
logp = self._trans + self._obs.condition(value).event_pad(left=self.hidden_dim)
# Eliminate time dimension.
logp = _sequential_gamma_gaussian_tensordot(logp.expand(logp.batch_shape))
# Combine initial factor.
logp = gamma_gaussian_tensordot(self._init, logp, dims=self.hidden_dim)
# Posterior of the scale
gamma_dist = logp.event_logsumexp()
scale_post = Gamma(gamma_dist.concentration, gamma_dist.rate,
validate_args=self._validate_args)
# Conditional of last state on unit scale
scale_tril = logp.precision.cholesky()
loc = logp.info_vec.unsqueeze(-1).cholesky_solve(scale_tril).squeeze(-1)
mvn = MultivariateNormal(loc, scale_tril=scale_tril,
validate_args=self._validate_args)
return scale_post, mvn
[docs]class LinearHMM(HiddenMarkovModel):
r"""
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.
`:class:`~pyro.distributions.MultivariateNormal` or
`:class:`~pyro.distributions.Independent` of
`:class:`~pyro.distributions.Normal`,
`:class:`~pyro.distributions.StudentT`, or `~pyro.distributions.Stable` .
Additionally the observation distribution may be constrained, e.g.
`~pyro.distributions.LogNormal`
This corresponds to the generative model::
z = initial_distribution.sample()
x = []
for t in range(num_events):
z = z @ transition_matrix + transition_dist.sample()
y = z @ observation_matrix + obs_base_dist.sample()
x.append(obs_transform(y))
where ``observation_dist`` is split into ``obs_base_dist`` and an optional
``obs_transform`` (defaulting to the identity).
This implements a reparameterized :meth:`rsample` method but does not
implement a :meth:`log_prob` method. Derived classes may implement
:meth:`log_prob` .
Inference without :meth:`log_prob` can be performed using either
reparameterization with :class:`~pyro.infer.reparam.hmm.LinearHMMReparam`
or likelihood-free algorithms such as
:class:`~pyro.infer.energy_distance.EnergyDistance` . Note that while
stable processes generally require a common shared stability parameter
:math:`\alpha` , this distribution and the above inference algorithms allow
heterogeneous stability parameters.
The event_shape of this distribution includes time on the left::
event_shape = (num_steps,) + observation_dist.event_shape
This distribution supports any combination of homogeneous/heterogeneous
time dependency of ``transition_dist`` and ``observation_dist``. However at
least one of the distributions or matrices must be expanded to contain the
time dimension.
:ivar int hidden_dim: The dimension of the hidden state.
:ivar int obs_dim: The dimension of the observed state.
:param initial_dist: A distribution over initial states. This should have
batch_shape broadcastable to ``self.batch_shape``. This should have
event_shape ``(hidden_dim,)``.
:param ~torch.Tensor transition_matrix: 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)``.
:param 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,)``.
:param ~torch.Tensor observation_matrix: A linear transformation from hidden
to observed state. This should have shape broadcastable to
``self.batch_shape + (num_steps, hidden_dim, obs_dim)``.
:param 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,)``.
:param int duration: 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 = {}
support = constraints.real
has_rsample = True
def __init__(self, initial_dist, transition_matrix, transition_dist,
observation_matrix, observation_dist,
validate_args=None, duration=None):
assert initial_dist.has_rsample
assert initial_dist.event_dim == 1
assert (isinstance(transition_matrix, torch.Tensor) and
transition_matrix.dim() >= 2)
assert transition_dist.has_rsample
assert transition_dist.event_dim == 1
assert (isinstance(observation_matrix, torch.Tensor) and
observation_matrix.dim() >= 2)
assert observation_dist.has_rsample
assert observation_dist.event_dim == 1
hidden_dim, obs_dim = observation_matrix.shape[-2:]
assert initial_dist.event_shape == (hidden_dim,)
assert transition_matrix.shape[-2:] == (hidden_dim, hidden_dim)
assert transition_dist.event_shape == (hidden_dim,)
assert observation_dist.event_shape == (obs_dim,)
shape = broadcast_shape(initial_dist.batch_shape + (1,),
transition_matrix.shape[:-2],
transition_dist.batch_shape,
observation_matrix.shape[:-2],
observation_dist.batch_shape)
batch_shape, time_shape = shape[:-1], shape[-1:]
event_shape = time_shape + (obs_dim,)
super().__init__(duration, batch_shape, event_shape, validate_args=validate_args)
# Expand eagerly.
if initial_dist.batch_shape != batch_shape:
initial_dist = initial_dist.expand(batch_shape)
if transition_matrix.shape[:-2] != batch_shape + time_shape:
transition_matrix = transition_matrix.expand(
batch_shape + time_shape + (hidden_dim, hidden_dim))
if transition_dist.batch_shape != batch_shape + time_shape:
transition_dist = transition_dist.expand(batch_shape + time_shape)
if observation_matrix.shape[:-2] != batch_shape + time_shape:
observation_matrix = observation_matrix.expand(
batch_shape + time_shape + (hidden_dim, obs_dim))
if observation_dist.batch_shape != batch_shape + time_shape:
observation_dist = observation_dist.expand(batch_shape + time_shape)
# Extract observation transforms.
transforms = []
while True:
if isinstance(observation_dist, torch.distributions.Independent):
observation_dist = observation_dist.base_dist
elif isinstance(observation_dist, torch.distributions.TransformedDistribution):
transforms = observation_dist.transforms + transforms
observation_dist = observation_dist.base_dist
else:
break
if not observation_dist.event_shape:
observation_dist = Independent(observation_dist, 1)
self.hidden_dim = hidden_dim
self.obs_dim = obs_dim
self.initial_dist = initial_dist
self.transition_matrix = transition_matrix
self.transition_dist = transition_dist
self.observation_matrix = observation_matrix
self.observation_dist = observation_dist
self.transforms = transforms
@property
def support(self):
return self.observation_dist.support
[docs] def expand(self, batch_shape, _instance=None):
new = self._get_checked_instance(LinearHMM, _instance)
batch_shape = torch.Size(batch_shape)
time_shape = self.transition_dist.batch_shape[-1:]
new.hidden_dim = self.hidden_dim
new.obs_dim = self.obs_dim
new.initial_dist = self.initial_dist.expand(batch_shape)
new.transition_matrix = self.transition_matrix.expand(
batch_shape + time_shape + (self.hidden_dim, self.hidden_dim))
new.transition_dist = self.transition_dist.expand(batch_shape + time_shape)
new.observation_matrix = self.observation_matrix.expand(
batch_shape + time_shape + (self.hidden_dim, self.obs_dim))
new.observation_dist = self.observation_dist.expand(batch_shape + time_shape)
new.transforms = self.transforms
super(LinearHMM, new).__init__(self.duration, batch_shape, self.event_shape, validate_args=False)
new._validate_args = self.__dict__.get('_validate_args')
return new
[docs] def log_prob(self, value):
raise NotImplementedError("LinearHMM.log_prob() is not implemented")
[docs] def rsample(self, sample_shape=torch.Size()):
assert self.duration is not None
init = self.initial_dist.rsample(sample_shape)
trans = self.transition_dist.expand(self.batch_shape + (self.duration,)).rsample(sample_shape)
obs = self.observation_dist.expand(self.batch_shape + (self.duration,)).rsample(sample_shape)
trans_matrix = self.transition_matrix.expand(self.batch_shape + (self.duration, -1, -1))
z = _linear_integrate(init, trans_matrix, trans)
x = (z.unsqueeze(-2) @ self.observation_matrix).squeeze(-2) + obs
for t in self.transforms:
x = t(x)
return x
[docs]class GaussianMRF(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 :meth:`log_prob` is differentiable.
The event_shape of this distribution includes time on the left::
event_shape = (num_steps,) + observation_dist.event_shape
This distribution supports any combination of homogeneous/heterogeneous
time dependency of ``transition_dist`` and ``observation_dist``. However,
because time is included in this distribution's event_shape, the
homogeneous+homogeneous case will have a broadcastable event_shape with
``num_steps = 1``, allowing :meth:`log_prob` to work with arbitrary length
data::
event_shape = (1, obs_dim) # homogeneous + homogeneous case
**References:**
[1] Simo Sarkka, Angel F. Garcia-Fernandez (2019)
"Temporal Parallelization of Bayesian Filters and Smoothers"
https://arxiv.org/pdf/1905.13002.pdf
:ivar int hidden_dim: The dimension of the hidden state.
:ivar int obs_dim: The dimension of the observed state.
:param ~torch.distributions.MultivariateNormal initial_dist: A distribution
over initial states. This should have batch_shape broadcastable to
``self.batch_shape``. This should have event_shape ``(hidden_dim,)``.
:param ~torch.distributions.MultivariateNormal transition_dist: 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).
:param ~torch.distributions.MultivariateNormal observation_dist: 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 = {}
def __init__(self, initial_dist, transition_dist, observation_dist, validate_args=None):
assert isinstance(initial_dist, torch.distributions.MultivariateNormal)
assert isinstance(transition_dist, torch.distributions.MultivariateNormal)
assert isinstance(observation_dist, torch.distributions.MultivariateNormal)
hidden_dim = initial_dist.event_shape[0]
assert transition_dist.event_shape[0] == hidden_dim + hidden_dim
obs_dim = observation_dist.event_shape[0] - hidden_dim
shape = broadcast_shape(initial_dist.batch_shape + (1,),
transition_dist.batch_shape,
observation_dist.batch_shape)
batch_shape, time_shape = shape[:-1], shape[-1:]
event_shape = time_shape + (obs_dim,)
super().__init__(batch_shape, event_shape, validate_args=validate_args)
self.hidden_dim = hidden_dim
self.obs_dim = obs_dim
self._init = mvn_to_gaussian(initial_dist)
self._trans = mvn_to_gaussian(transition_dist)
self._obs = mvn_to_gaussian(observation_dist)
[docs] def expand(self, batch_shape, _instance=None):
new = self._get_checked_instance(GaussianMRF, _instance)
batch_shape = torch.Size(broadcast_shape(self.batch_shape, batch_shape))
new.hidden_dim = self.hidden_dim
new.obs_dim = self.obs_dim
# We only need to expand one of the inputs, since batch_shape is determined
# by broadcasting all three. To save computation in _sequential_gaussian_tensordot(),
# we expand only _init, which is applied only after _sequential_gaussian_tensordot().
new._init = self._init.expand(batch_shape)
new._trans = self._trans
new._obs = self._obs
super(GaussianMRF, new).__init__(batch_shape, self.event_shape, validate_args=False)
new._validate_args = self.__dict__.get('_validate_args')
return new
[docs] def log_prob(self, value):
# We compute a normalized distribution as p(obs,hidden) / p(hidden).
logp_oh = self._trans
logp_h = self._trans
# Combine observation and transition factors.
logp_oh += self._obs.condition(value).event_pad(left=self.hidden_dim)
logp_h += self._obs.marginalize(right=self.obs_dim).event_pad(left=self.hidden_dim)
# Concatenate p(obs,hidden) and p(hidden) into a single Gaussian.
batch_dim = 1 + max(len(self._init.batch_shape) + 1, len(logp_oh.batch_shape))
batch_shape = (1,) * (batch_dim - len(logp_oh.batch_shape)) + logp_oh.batch_shape
logp = Gaussian.cat([logp_oh.expand(batch_shape),
logp_h.expand(batch_shape)])
# Eliminate time dimension.
logp = _sequential_gaussian_tensordot(logp)
# Combine initial factor.
logp = gaussian_tensordot(self._init, logp, dims=self.hidden_dim)
# Marginalize out final state.
logp_oh, logp_h = logp.event_logsumexp()
return logp_oh - logp_h # = log( p(obs,hidden) / p(hidden) )
```