Observation distribution from a linear Gaussian state space model.
Inherits From: AutoCompositeTensorDistribution
, Distribution
oryx.distributions.LinearGaussianStateSpaceModel(
num_timesteps,
transition_matrix,
transition_noise,
observation_matrix,
observation_noise,
initial_state_prior,
initial_step=0,
mask=None,
experimental_parallelize=False,
validate_args=False,
allow_nan_stats=True,
name='LinearGaussianStateSpaceModel'
)
A linear Gaussian state space model, sometimes called a Kalman filter, posits
a latent state vector z[t]
of dimension latent_size
that evolves
over time following linear Gaussian transitions,
z[t+1] = F * z[t] + N(b; Q) # latent state
x[t] = H * z[t] + N(c; R) # observed series
for transition matrix F
, transition bias b
and covariance matrix
Q
, and observation matrix H
, bias c
and covariance matrix R
. At each
timestep, the model generates an observable vector x[t]
, a noisy projection
of the latent state. The transition and observation models may be fixed or
may vary between timesteps.
This Distribution represents the marginal distribution on
observations, p(x)
. The marginal log_prob
is implemented by
Kalman filtering [1], and sample
by an efficient forward
recursion. Both operations require time linear in T
, the total
number of timesteps.
Shapes
The event shape is [num_timesteps, observation_size]
, where
observation_size
is the dimension of each observation x[t]
.
The observation and transition models must return consistent
shapes.
This implementation supports vectorized computation over a batch of models. All of the parameters (prior distribution, transition and observation operators and noise models) must have a consistent batch shape.
Timevarying processes
Any of the modeldefining parameters (prior distribution, transition
and observation operators and noise models) may be specified as a
callable taking an integer timestep t
and returning a
timedependent value. The dimensionality (latent_size
and
observation_size
) must be the same at all timesteps.
Importantly, the timestep is passed as a Tensor
, not a Python
integer, so any conditional behavior must occur inside the
TensorFlow graph. For example, suppose we want to use a different
transition model on even days than odd days. It does not work to
write
def transition_matrix(t):
if t % 2 == 0:
return even_day_matrix
else:
return odd_day_matrix
since the value of t
is not fixed at graphconstruction
time. Instead we need to write
def transition_matrix(t):
return tf.cond(tf.equal(tf.mod(t, 2), 0),
lambda : even_day_matrix,
lambda : odd_day_matrix)
so that TensorFlow can switch between operators appropriately at runtime.
Examples
Consider a simple tracking model, in which a twodimensional latent state
represents the position of a vehicle, and at each timestep we
see a noisy observation of this position (e.g., a GPS reading). The
vehicle is assumed to move by a random walk with standard deviation
step_std
at each step, and observation noise level std
. We build
the marginal distribution over noisy observations as a state space model:
tfd = tfp.distributions
ndims = 2
step_std = 1.0
noise_std = 5.0
model = tfd.LinearGaussianStateSpaceModel(
num_timesteps=100,
transition_matrix=tf.linalg.LinearOperatorIdentity(ndims),
transition_noise=tfd.MultivariateNormalDiag(
scale_diag=step_std**2 * tf.ones([ndims])),
observation_matrix=tf.linalg.LinearOperatorIdentity(ndims),
observation_noise=tfd.MultivariateNormalDiag(
scale_diag=noise_std**2 * tf.ones([ndims])),
initial_state_prior=tfd.MultivariateNormalDiag(
scale_diag=tf.ones([ndims])))
using the identity matrix for the transition and observation operators. We can then use this model to generate samples, compute marginal likelihood of observed sequences, and perform posterior inference.
x = model.sample(5) # Sample from the prior on sequences of observations.
lp = model.log_prob(x) # Marginal likelihood of a (batch of) observations.
# Compute the filtered posterior on latent states given observations,
# and extract the mean and covariance for the current (final) timestep.
_, filtered_means, filtered_covs, _, _, _, _ = model.forward_filter(x)
current_location_posterior = tfd.MultivariateNormalTriL(
loc=filtered_means[..., 1, :],
scale_tril=tf.linalg.cholesky(filtered_covs[..., 1, :, :]))
# Run a smoothing recursion to extract posterior marginals for locations
# at previous timesteps.
posterior_means, posterior_covs = model.posterior_marginals(x)
initial_location_posterior = tfd.MultivariateNormalTriL(
loc=posterior_means[..., 0, :],
scale_tril=tf.linalg.cholesky(posterior_covs[..., 0, :, :]))
*
Args  

num_timesteps

Integer Tensor total number of timesteps.

transition_matrix

A transition operator, represented by a Tensor or
LinearOperator of shape [latent_size, latent_size] , or by a
callable taking as argument a scalar integer Tensor t and
returning a Tensor or LinearOperator representing the transition
operator from latent state at time t to time t + 1 .

transition_noise

An instance of
tfd.MultivariateNormalLinearOperator with event shape
[latent_size] , representing the mean and covariance of the
transition noise model, or a callable taking as argument a
scalar integer Tensor t and returning such a distribution
representing the noise in the transition from time t to time t + 1 .

observation_matrix

An observation operator, represented by a Tensor
or LinearOperator of shape [observation_size, latent_size] ,
or by a callable taking as argument a scalar integer Tensor
t and returning a timestepspecific Tensor or
LinearOperator.

observation_noise

An instance of
tfd.MultivariateNormalLinearOperator with event shape
[observation_size] , representing the mean and covariance of
the observation noise model, or a callable taking as argument
a scalar integer Tensor t and returning a timestepspecific
noise model.

initial_state_prior

An instance of MultivariateNormalLinearOperator
representing the prior distribution on latent states; must
have event shape [latent_size] .

initial_step

optional int specifying the time of the first
modeled timestep. This is added as an offset when passing
timesteps t to (optional) callables specifying
timestepspecific transition and observation models.

mask

Optional default missingness mask used for density and posterior
inference calculations (any method that takes a mask argument).
Booltype Tensor with rightmost dimension
[num_timesteps] ; True values specify that the value of x
at that timestep is masked, i.e., not conditioned on.
Default value: None .

experimental_parallelize

If True , use parallel message passing
algorithms from tfp.experimental.parallel_filter to perform operations
in O(log num_timesteps) sequential steps. The overall FLOP and memory
cost may be larger than for the sequential implementations, though
only by a constant factor.
Default value: False .

validate_args

Python bool , default False . Whether to validate input
with asserts. If validate_args is False , and the inputs are
invalid, correct behavior is not guaranteed.

allow_nan_stats

Python bool , default True . If False , raise an
exception if a statistic (e.g. mean/mode/etc...) is undefined for any
batch member If True , batch members with valid parameters leading to
undefined statistics will return NaN for this statistic.

name

The name to give Ops created by the initializer. 
Methods
backward_smoothing_pass
backward_smoothing_pass(
filtered_means, filtered_covs, predicted_means, predicted_covs
)
Run the backward pass in Kalman smoother.
The backward smoothing is using Rauch, Tung and Striebel smoother as
as discussed in section 18.3.2 of Kevin P. Murphy, 2012, Machine Learning:
A Probabilistic Perspective, The MIT Press. The inputs are returned by
forward_filter
function.
Args  

filtered_means

Means of the pertimestep filtered marginal
distributions p(z[t]  x[:t]), as a Tensor of shape
sample_shape(x) + batch_shape + [num_timesteps, latent_size] .

filtered_covs

Covariances of the pertimestep filtered marginal
distributions p(z[t]  x[:t]), as a Tensor of shape
sample_shape(x) + batch_shape + [num_timesteps, latent_size,
latent_size] .

predicted_means

Means of the pertimestep predictive
distributions over latent states, p(z[t+1]  x[:t]), as a
Tensor of shape sample_shape(x) + batch_shape +
[num_timesteps, latent_size] .

predicted_covs

Covariances of the pertimestep predictive
distributions over latent states, p(z[t+1]  x[:t]), as a
Tensor of shape sample_shape(x) + batch_shape +
[num_timesteps, latent_size, latent_size] .

Returns  

posterior_means

Means of the smoothed marginal distributions
p(z[t]  x[1:T]), as a Tensor of shape
sample_shape(x) + batch_shape + [num_timesteps, latent_size] ,
which is of the same shape as filtered_means.

posterior_covs

Covariances of the smoothed marginal distributions
p(z[t]  x[1:T]), as a Tensor of shape
sample_shape(x) + batch_shape + [num_timesteps, latent_size,
latent_size] . which is of the same shape as filtered_covs.

batch_shape_tensor
batch_shape_tensor(
name='batch_shape_tensor'
)
Shape of a single sample from a single event index as a 1D Tensor
.
The batch dimensions are indexes into independent, nonidentical parameterizations of this distribution.
Args  

name

name to give to the op 
Returns  

batch_shape

Tensor .

cdf
cdf(
value, name='cdf', **kwargs
)
Cumulative distribution function.
Given random variable X
, the cumulative distribution function cdf
is:
cdf(x) := P[X <= x]
Args  

value

float or double Tensor .

name

Python str prepended to names of ops created by this function.

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

cdf

a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .

copy
copy(
**override_parameters_kwargs
)
Creates a deep copy of the distribution.
Args  

**override_parameters_kwargs

String/value dictionary of initialization arguments to override with new values. 
Returns  

distribution

A new instance of type(self) initialized from the union
of self.parameters and override_parameters_kwargs, i.e.,
dict(self.parameters, **override_parameters_kwargs) .

covariance
covariance(
name='covariance', **kwargs
)
Covariance.
Covariance is (possibly) defined only for nonscalarevent distributions.
For example, for a lengthk
, vectorvalued distribution, it is calculated
as,
Cov[i, j] = Covariance(X_i, X_j) = E[(X_i  E[X_i]) (X_j  E[X_j])]
where Cov
is a (batch of) k x k
matrix, 0 <= (i, j) < k
, and E
denotes expectation.
Alternatively, for nonvector, multivariate distributions (e.g.,
matrixvalued, Wishart), Covariance
shall return a (batch of) matrices
under some vectorization of the events, i.e.,
Cov[i, j] = Covariance(Vec(X)_i, Vec(X)_j) = [as above]
where Cov
is a (batch of) k' x k'
matrices,
0 <= (i, j) < k' = reduce_prod(event_shape)
, and Vec
is some function
mapping indices of this distribution's event dimensions to indices of a
lengthk'
vector.
Args  

name

Python str prepended to names of ops created by this function.

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

covariance

Floatingpoint Tensor with shape [B1, ..., Bn, k', k']
where the first n dimensions are batch coordinates and
k' = reduce_prod(self.event_shape) .

cross_entropy
cross_entropy(
other, name='cross_entropy'
)
Computes the (Shannon) cross entropy.
Denote this distribution (self
) by P
and the other
distribution by
Q
. Assuming P, Q
are absolutely continuous with respect to
one another and permit densities p(x) dr(x)
and q(x) dr(x)
, (Shannon)
cross entropy is defined as:
H[P, Q] = E_p[log q(X)] = int_F p(x) log q(x) dr(x)
where F
denotes the support of the random variable X ~ P
.
Args  

other

tfp.distributions.Distribution instance.

name

Python str prepended to names of ops created by this function.

Returns  

cross_entropy

self.dtype Tensor with shape [B1, ..., Bn]
representing n different calculations of (Shannon) cross entropy.

entropy
entropy(
name='entropy', **kwargs
)
Shannon entropy in nats.
event_shape_tensor
event_shape_tensor(
name='event_shape_tensor'
)
Shape of a single sample from a single batch as a 1D int32 Tensor
.
Args  

name

name to give to the op 
Returns  

event_shape

Tensor .

experimental_default_event_space_bijector
experimental_default_event_space_bijector(
*args, **kwargs
)
Bijector mapping the reals (R**n) to the event space of the distribution.
Distributions with continuous support may implement
_default_event_space_bijector
which returns a subclass of
tfp.bijectors.Bijector
that maps R**n to the distribution's event space.
For example, the default bijector for the Beta
distribution
is tfp.bijectors.Sigmoid()
, which maps the real line to [0, 1]
, the
support of the Beta
distribution. The default bijector for the
CholeskyLKJ
distribution is tfp.bijectors.CorrelationCholesky
, which
maps R^(k * (k1) // 2) to the submanifold of k x k lower triangular
matrices with ones along the diagonal.
The purpose of experimental_default_event_space_bijector
is
to enable gradient descent in an unconstrained space for Variational
Inference and Hamiltonian Monte Carlo methods. Some effort has been made to
choose bijectors such that the tails of the distribution in the
unconstrained space are between Gaussian and Exponential.
For distributions with discrete event space, or for which TFP currently
lacks a suitable bijector, this function returns None
.
Args  

*args

Passed to implementation _default_event_space_bijector .

**kwargs

Passed to implementation _default_event_space_bijector .

Returns  

event_space_bijector

Bijector instance or None .

experimental_fit
@classmethod
experimental_fit( value, sample_ndims=1, validate_args=False, **init_kwargs )
Instantiates a distribution that maximizes the likelihood of x
.
Args  

value

a Tensor valid sample from this distribution family.

sample_ndims

Positive int Tensor number of leftmost dimensions of
value that index i.i.d. samples.
Default value: 1 .

validate_args

Python bool , default False . When True , distribution
parameters are checked for validity despite possibly degrading runtime
performance. When False , invalid inputs may silently render incorrect
outputs.
Default value: False .

**init_kwargs

Additional keyword arguments passed through to
cls.__init__ . These take precedence in case of collision with the
fitted parameters; for example,
tfd.Normal.experimental_fit([1., 1.], scale=20.) returns a Normal
distribution with scale=20. rather than the maximum likelihood
parameter scale=0. .

Returns  

maximum_likelihood_instance

instance of cls with parameters that
maximize the likelihood of value .

experimental_local_measure
experimental_local_measure(
value, backward_compat=False, **kwargs
)
Returns a log probability density together with a TangentSpace
.
A TangentSpace
allows us to calculate the correct pushforward
density when we apply a transformation to a Distribution
on
a strict submanifold of R^n (typically via a Bijector
in the
TransformedDistribution
subclass). The density correction uses
the basis of the tangent space.
Args  

value

float or double Tensor .

backward_compat

bool specifying whether to fall back to returning
FullSpace as the tangent space, and representing R^n with the standard
basis.

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

log_prob

a Tensor representing the log probability density, of shape
sample_shape(x) + self.batch_shape with values of type self.dtype .

tangent_space

a TangentSpace object (by default FullSpace )
representing the tangent space to the manifold at value .

Raises  

UnspecifiedTangentSpaceError if backward_compat is False and
the _experimental_tangent_space attribute has not been defined.

experimental_sample_and_log_prob
experimental_sample_and_log_prob(
sample_shape=(), seed=None, name='sample_and_log_prob', **kwargs
)
Samples from this distribution and returns the log density of the sample.
The default implementation simply calls sample
and log_prob
:
def _sample_and_log_prob(self, sample_shape, seed, **kwargs):
x = self.sample(sample_shape=sample_shape, seed=seed, **kwargs)
return x, self.log_prob(x, **kwargs)
However, some subclasses may provide more efficient and/or numerically stable implementations.
Args  

sample_shape

integer Tensor desired shape of samples to draw.
Default value: () .

seed

PRNG seed; see tfp.random.sanitize_seed for details.
Default value: None .

name

name to give to the op.
Default value: 'sample_and_log_prob' .

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

samples

a Tensor , or structure of Tensor s, with prepended dimensions
sample_shape .

log_prob

a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .

forward_filter
forward_filter(
x, mask=None, final_step_only=False
)
Run a Kalman filter over a provided sequence of outputs.
Note that the returned values filtered_means
, predicted_means
, and
observation_means
depend on the observed time series x
, while the
corresponding covariances are independent of the observed series; i.e., they
depend only on the model itself. This means that the mean values have shape
concat([sample_shape(x), batch_shape, [num_timesteps,
{latent/observation}_size]])
, while the covariances have shape
concat[(batch_shape, [num_timesteps, {latent/observation}_size,
{latent/observation}_size]])
, which does not depend on the sample shape.
Args  

x

a floattype Tensor with rightmost dimensions
[num_timesteps, observation_size] matching
self.event_shape . Additional dimensions must match or be
broadcastable to self.batch_shape ; any further dimensions
are interpreted as a sample shape.

mask

optional booltype Tensor with rightmost dimension
[num_timesteps] ; True values specify that the value of x
at that timestep is masked, i.e., not conditioned on. Additional
dimensions must match or be broadcastable to self.batch_shape ; any
further dimensions must match or be broadcastable to the sample
shape of x .
Default value: None (falls back to self.mask ).

final_step_only

optional Python bool . If True , the num_timesteps
dimension is omitted from all return values and only the value from the
final timestep is returned (in this case, log_likelihoods will
be the cumulative log marginal likelihood). This may be significantly
more efficient than returning all values (although note that no
efficiency gain is expected when self.experimental_parallelize=True ).
Default value: False .

Returns  

log_likelihoods

Pertimestep log marginal likelihoods log
p(x[t]  x[:t1]) evaluated at the input x , as a Tensor
of shape sample_shape(x) + batch_shape + [num_timesteps].
If final_step_only is True , this will instead be the
cumulative log marginal likelihood at the final step.

filtered_means

Means of the pertimestep filtered marginal
distributions p(z[t]  x[:t]), as a Tensor of shape
sample_shape(x) + batch_shape + [num_timesteps, latent_size] .

filtered_covs

Covariances of the pertimestep filtered marginal
distributions p(z[t]  x[:t]), as a Tensor of shape
sample_shape(x) + batch_shape + [num_timesteps, latent_size,
latent_size] . Since posterior covariances do not depend on observed
data, some implementations may return a Tensor whose shape omits the
initial sample_shape(x) .

predicted_means

Means of the pertimestep predictive
distributions over latent states, p(z[t+1]  x[:t]), as a
Tensor of shape sample_shape(x) + batch_shape +
[num_timesteps, latent_size] .

predicted_covs

Covariances of the pertimestep predictive
distributions over latent states, p(z[t+1]  x[:t]), as a
Tensor of shape sample_shape(x) + batch_shape +
[num_timesteps, latent_size, latent_size] . Since posterior covariances
do not depend on observed data, some implementations may return a
Tensor whose shape omits the initial sample_shape(x) .

observation_means

Means of the pertimestep predictive
distributions over observations, p(x[t]  x[:t1]), as a
Tensor of shape sample_shape(x) + batch_shape +
[num_timesteps, observation_size] .

observation_covs

Covariances of the pertimestep predictive
distributions over observations, p(x[t]  x[:t1]), as a
Tensor of shape sample_shape(x) + batch_shape + [num_timesteps,
observation_size, observation_size] . Since posterior covariances
do not depend on observed data, some implementations may return a
Tensor whose shape omits the initial sample_shape(x) .

is_scalar_batch
is_scalar_batch(
name='is_scalar_batch'
)
Indicates that batch_shape == []
.
Args  

name

Python str prepended to names of ops created by this function.

Returns  

is_scalar_batch

bool scalar Tensor .

is_scalar_event
is_scalar_event(
name='is_scalar_event'
)
Indicates that event_shape == []
.
Args  

name

Python str prepended to names of ops created by this function.

Returns  

is_scalar_event

bool scalar Tensor .

kl_divergence
kl_divergence(
other, name='kl_divergence'
)
Computes the KullbackLeibler divergence.
Denote this distribution (self
) by p
and the other
distribution by
q
. Assuming p, q
are absolutely continuous with respect to reference
measure r
, the KL divergence is defined as:
KL[p, q] = E_p[log(p(X)/q(X))]
= int_F p(x) log q(x) dr(x) + int_F p(x) log p(x) dr(x)
= H[p, q]  H[p]
where F
denotes the support of the random variable X ~ p
, H[., .]
denotes (Shannon) cross entropy, and H[.]
denotes (Shannon) entropy.
Args  

other

tfp.distributions.Distribution instance.

name

Python str prepended to names of ops created by this function.

Returns  

kl_divergence

self.dtype Tensor with shape [B1, ..., Bn]
representing n different calculations of the KullbackLeibler
divergence.

latent_size_tensor
latent_size_tensor()
latents_to_observations
latents_to_observations(
latent_means, latent_covs
)
Push latent means and covariances forward through the observation model.
Args  

latent_means

float Tensor of shape [..., num_timesteps, latent_size]

latent_covs

float Tensor of shape
[..., num_timesteps, latent_size, latent_size] .

Returns  

observation_means

float Tensor of shape
[..., num_timesteps, observation_size]

observation_covs

float Tensor of shape
[..., num_timesteps, observation_size, observation_size]

log_cdf
log_cdf(
value, name='log_cdf', **kwargs
)
Log cumulative distribution function.
Given random variable X
, the cumulative distribution function cdf
is:
log_cdf(x) := Log[ P[X <= x] ]
Often, a numerical approximation can be used for log_cdf(x)
that yields
a more accurate answer than simply taking the logarithm of the cdf
when
x << 1
.
Args  

value

float or double Tensor .

name

Python str prepended to names of ops created by this function.

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

logcdf

a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .

log_prob
log_prob(
value, name='log_prob', **kwargs
)
Log probability density/mass function.
Additional documentation from LinearGaussianStateSpaceModel
:
kwargs
:
mask
: optional booltypeTensor
with rightmost dimension[num_timesteps]
;True
values specify that the value ofx
at that timestep is masked, i.e., not conditioned on. Additional dimensions must match or be broadcastable toself.batch_shape
; any further dimensions must match or be broadcastable to the sample shape ofx
. Default value:None
(falls back toself.mask
).
Args  

value

float or double Tensor .

name

Python str prepended to names of ops created by this function.

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

log_prob

a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .

log_survival_function
log_survival_function(
value, name='log_survival_function', **kwargs
)
Log survival function.
Given random variable X
, the survival function is defined:
log_survival_function(x) = Log[ P[X > x] ]
= Log[ 1  P[X <= x] ]
= Log[ 1  cdf(x) ]
Typically, different numerical approximations can be used for the log
survival function, which are more accurate than 1  cdf(x)
when x >> 1
.
Args  

value

float or double Tensor .

name

Python str prepended to names of ops created by this function.

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

Tensor of shape sample_shape(x) + self.batch_shape with values of type
self.dtype .

mean
mean(
name='mean', **kwargs
)
Mean.
mode
mode(
name='mode', **kwargs
)
Mode.
observation_size_tensor
observation_size_tensor()
param_shapes
@classmethod
param_shapes( sample_shape, name='DistributionParamShapes' )
Shapes of parameters given the desired shape of a call to sample()
.
This is a class method that describes what key/value arguments are required
to instantiate the given Distribution
so that a particular shape is
returned for that instance's call to sample()
.
Subclasses should override class method _param_shapes
.
Args  

sample_shape

Tensor or python list/tuple. Desired shape of a call to
sample() .

name

name to prepend ops with. 
Returns  

dict of parameter name to Tensor shapes.

param_static_shapes
@classmethod
param_static_shapes( sample_shape )
param_shapes with static (i.e. TensorShape
) shapes.
This is a class method that describes what key/value arguments are required
to instantiate the given Distribution
so that a particular shape is
returned for that instance's call to sample()
. Assumes that the sample's
shape is known statically.
Subclasses should override class method _param_shapes
to return
constantvalued tensors when constant values are fed.
Args  

sample_shape

TensorShape or python list/tuple. Desired shape of a call
to sample() .

Returns  

dict of parameter name to TensorShape .

Raises  

ValueError

if sample_shape is a TensorShape and is not fully defined.

parameter_properties
@classmethod
parameter_properties( dtype=tf.float32, num_classes=None )
Returns a dict mapping constructor arg names to property annotations.
This dict should include an entry for each of the distribution's
Tensor
valued constructor arguments.
Distribution subclasses are not required to implement
_parameter_properties
, so this method may raise NotImplementedError
.
Providing a _parameter_properties
implementation enables several advanced
features, including:
 Distribution batch slicing (
sliced_distribution = distribution[i:j]
).  Automatic inference of
_batch_shape
and_batch_shape_tensor
, which must otherwise be computed explicitly.  Automatic instantiation of the distribution within TFP's internal property tests.
 Automatic construction of 'trainable' instances of the distribution using appropriate bijectors to avoid violating parameter constraints. This enables the distribution family to be used easily as a surrogate posterior in variational inference.
In the future, parameter property annotations may enable additional
functionality; for example, returning Distribution instances from
tf.vectorized_map
.
Args  

dtype

Optional float dtype to assume for continuousvalued parameters.
Some constraining bijectors require advance knowledge of the dtype
because certain constants (e.g., tfb.Softplus.low ) must be
instantiated with the same dtype as the values to be transformed.

num_classes

Optional int Tensor number of classes to assume when
inferring the shape of parameters for categoricallike distributions.
Otherwise ignored.

Returns  

parameter_properties

A
str > tfp.python.internal.parameter_properties.ParameterPropertiesdict mapping constructor argument names to ParameterProperties`
instances.

Raises  

NotImplementedError

if the distribution class does not implement
_parameter_properties .

posterior_marginals
posterior_marginals(
x, mask=None
)
Run a Kalman smoother to return posterior mean and cov.
Note that the returned values smoothed_means
depend on the observed
time series x
, while the smoothed_covs
are independent
of the observed series; i.e., they depend only on the model itself.
This means that the mean values have shape concat([sample_shape(x),
batch_shape, [num_timesteps, {latent/observation}_size]])
,
while the covariances have shape concat[(batch_shape, [num_timesteps,
{latent/observation}_size, {latent/observation}_size]])
, which
does not depend on the sample shape.
This function only performs smoothing. If the user wants the
intermediate values, which are returned by filtering pass forward_filter
,
one could get it by:
(log_likelihoods,
filtered_means, filtered_covs,
predicted_means, predicted_covs,
observation_means, observation_covs) = model.forward_filter(x)
smoothed_means, smoothed_covs = model.backward_smoothing_pass(
filtered_means, filtered_covs,
predicted_means, predicted_covs)
where x
is an observation sequence.
Args  

x

a floattype Tensor with rightmost dimensions
[num_timesteps, observation_size] matching
self.event_shape . Additional dimensions must match or be
broadcastable to self.batch_shape ; any further dimensions
are interpreted as a sample shape.

mask

optional booltype Tensor with rightmost dimension
[num_timesteps] ; True values specify that the value of x
at that timestep is masked, i.e., not conditioned on. Additional
dimensions must match or be broadcastable to self.batch_shape ; any
further dimensions must match or be broadcastable to the sample
shape of x .
Default value: None (falls back to self.mask ).

Returns  

smoothed_means

Means of the pertimestep smoothed
distributions over latent states, p(z[t]  x[:T]), as a
Tensor of shape sample_shape(x) + batch_shape +
[num_timesteps, observation_size] .

smoothed_covs

Covariances of the pertimestep smoothed
distributions over latent states, p(z[t]  x[:T]), as a
Tensor of shape sample_shape(mask) + batch_shape + [num_timesteps,
observation_size, observation_size] . Note that the covariances depend
only on the model and the mask, not on the data, so this may have fewer
dimensions than filtered_means .

posterior_sample
posterior_sample(
x, sample_shape=(), mask=None, seed=None, name=None
)
Draws samples from the posterior over latent trajectories.
This method uses DurbinKoopman sampling [1], an efficient algorithm to
sample from the posterior latents of a linear Gaussian state space model.
The cost of drawing a sample is equal to the cost of drawing a prior
sample (.sample(sample_shape)
), plus the cost of Kalman smoothing (
.posterior_marginals(...)
on both the observed time series and the
prior sample. This method is significantly more efficient in graph mode,
because it uses only the posterior means and can elide the unneeded
calculation of marginal covariances.
[1] Durbin, J. and Koopman, S.J. A simple and efficient simulation smoother for state space time series analysis. Biometrika 89(3):603615, 2002. https://www.jstor.org/stable/4140605
Args  

x

a floattype Tensor with rightmost dimensions
[num_timesteps, observation_size] matching
self.event_shape . Additional dimensions must match or be
broadcastable with self.batch_shape .

sample_shape

int Tensor shape of samples to draw.
Default value: () .

mask

optional booltype Tensor with rightmost dimension
[num_timesteps] ; True values specify that the value of x
at that timestep is masked, i.e., not conditioned on. Additional
dimensions must match or be broadcastable with self.batch_shape and
x.shape[:2] .
Default value: None (falls back to self.mask ).

seed

PRNG seed; see tfp.random.sanitize_seed for details.

name

Python str name for ops generated by this method.

Returns  

latent_posterior_sample

Float Tensor of shape
concat([sample_shape, batch_shape, [num_timesteps, latent_size]]) ,
where batch_shape is the broadcast shape of self.batch_shape ,
x.shape[:2] , and mask.shape[:1] , representing n samples from
the posterior over latent states given the observed value x .

prob
prob(
value, name='prob', **kwargs
)
Probability density/mass function.
Additional documentation from LinearGaussianStateSpaceModel
:
kwargs
:
mask
: optional booltypeTensor
with rightmost dimension[num_timesteps]
;True
values specify that the value ofx
at that timestep is masked, i.e., not conditioned on. Additional dimensions must match or be broadcastable toself.batch_shape
; any further dimensions must match or be broadcastable to the sample shape ofx
. Default value:None
(falls back toself.mask
).
Args  

value

float or double Tensor .

name

Python str prepended to names of ops created by this function.

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

prob

a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .

quantile
quantile(
value, name='quantile', **kwargs
)
Quantile function. Aka 'inverse cdf' or 'percent point function'.
Given random variable X
and p in [0, 1]
, the quantile
is:
quantile(p) := x such that P[X <= x] == p
Args  

value

float or double Tensor .

name

Python str prepended to names of ops created by this function.

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

quantile

a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .

sample
sample(
sample_shape=(), seed=None, name='sample', **kwargs
)
Generate samples of the specified shape.
Note that a call to sample()
without arguments will generate a single
sample.
Args  

sample_shape

0D or 1D int32 Tensor . Shape of the generated samples.

seed

PRNG seed; see tfp.random.sanitize_seed for details.

name

name to give to the op. 
**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

samples

a Tensor with prepended dimensions sample_shape .

stddev
stddev(
name='stddev', **kwargs
)
Standard deviation.
Standard deviation is defined as,
stddev = E[(X  E[X])**2]**0.5
where X
is the random variable associated with this distribution, E
denotes expectation, and stddev.shape = batch_shape + event_shape
.
Args  

name

Python str prepended to names of ops created by this function.

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

stddev

Floatingpoint Tensor with shape identical to
batch_shape + event_shape , i.e., the same shape as self.mean() .

survival_function
survival_function(
value, name='survival_function', **kwargs
)
Survival function.
Given random variable X
, the survival function is defined:
survival_function(x) = P[X > x]
= 1  P[X <= x]
= 1  cdf(x).
Args  

value

float or double Tensor .

name

Python str prepended to names of ops created by this function.

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

Tensor of shape sample_shape(x) + self.batch_shape with values of type
self.dtype .

unnormalized_log_prob
unnormalized_log_prob(
value, name='unnormalized_log_prob', **kwargs
)
Potentially unnormalized log probability density/mass function.
This function is similar to log_prob
, but does not require that the
return value be normalized. (Normalization here refers to the total
integral of probability being one, as it should be by definition for any
probability distribution.) This is useful, for example, for distributions
where the normalization constant is difficult or expensive to compute. By
default, this simply calls log_prob
.
Args  

value

float or double Tensor .

name

Python str prepended to names of ops created by this function.

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

unnormalized_log_prob

a Tensor of shape
sample_shape(x) + self.batch_shape with values of type self.dtype .

variance
variance(
name='variance', **kwargs
)
Variance.
Variance is defined as,
Var = E[(X  E[X])**2]
where X
is the random variable associated with this distribution, E
denotes expectation, and Var.shape = batch_shape + event_shape
.
Args  

name

Python str prepended to names of ops created by this function.

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

variance

Floatingpoint Tensor with shape identical to
batch_shape + event_shape , i.e., the same shape as self.mean() .

__getitem__
__getitem__(
slices
)
Slices the batch axes of this distribution, returning a new instance.
b = tfd.Bernoulli(logits=tf.zeros([3, 5, 7, 9]))
b.batch_shape # => [3, 5, 7, 9]
b2 = b[:, tf.newaxis, ..., 2:, 1::2]
b2.batch_shape # => [3, 1, 5, 2, 4]
x = tf.random.stateless_normal([5, 3, 2, 2])
cov = tf.matmul(x, x, transpose_b=True)
chol = tf.linalg.cholesky(cov)
loc = tf.random.stateless_normal([4, 1, 3, 1])
mvn = tfd.MultivariateNormalTriL(loc, chol)
mvn.batch_shape # => [4, 5, 3]
mvn.event_shape # => [2]
mvn2 = mvn[:, 3:, ..., ::1, tf.newaxis]
mvn2.batch_shape # => [4, 2, 3, 1]
mvn2.event_shape # => [2]
Args  

slices

slices from the [] operator 
Returns  

dist

A new tfd.Distribution instance with sliced parameters.

__iter__
__iter__()