tfp.substrates.jax.sts.ConstrainedSeasonalStateSpaceModel

Seasonal state space model with effects constrained to sum to zero.

Inherits From: LinearGaussianStateSpaceModel, AutoCompositeTensorDistribution, Distribution

tfp.substrates.jax.sts.ConstrainedSeasonalStateSpaceModel(
    num_timesteps,
    num_seasons,
    drift_scale,
    initial_state_prior,
    observation_noise_scale=0.0001,
    num_steps_per_season=1,
    name=None,
    **linear_gaussian_ssm_kwargs
)

See SeasonalStateSpaceModel for background.

Mathematical details

The constrained model implements a reparameterization of the naive SeasonalStateSpaceModel. Instead of directly representing the seasonal effects in the latent space, the latent space of the constrained model represents the difference between each effect and the mean effect. The following discussion assumes familiarity with the mathematical details of SeasonalStateSpaceModel.

Reparameterization and constraints: let the seasonal effects at a given timestep be E = [e_1, ..., e_N]. The difference between each effect e_i and the mean effect is z_i = e_i - sum_i(e_i)/N. By itself, this transformation is not invertible because recovering the absolute effects requires that we know the mean as well. To fix this, we'll define z_N = sum_i(e_i)/N as the mean effect. It's easy to see that this is invertible: given the mean effect and the differences of the first N - 1 effects from the mean, it's easy to solve for all N effects. Formally, we've defined the invertible linear reparameterization Z = R E, where

R = [1 - 1/N, -1/N,    ..., -1/N,      -1/N
     -1/N,    1 - 1/N, ..., -1/N,      -1/N
     ...
     -1/N,    -1/N,    ...,  1/ - 1/N, -1/N
     1/N,     1/N,     ...,  1/N,       1/N]
E = [e_1, ..., e_(N-1), z_N]

represents the change of basis from 'effect coordinates' E to 'residual coordinates' Z. The Zs form the latent space of the ConstrainedSeasonalStateSpaceModel.

To constrain the mean effect z_N to zero, we fix the prior to zero, p(z_N) ~ N(0., 0), and after the transition at each timestep we project z_N back to zero. Note that this projection is linear: to set the Nth dimension to zero, we simply multiply by the identity matrix with a missing element in the bottom right, i.e., Z_constrained = P Z, where P = eye(N) - scatter((N-1, N-1), 1).

Model: concretely, suppose a naive seasonal effect model has initial state prior N(m, S), transition matrix F and noise covariance Q, and observation matrix H. Then the corresponding constrained seasonal effect model has initial state prior N(P R m, P R S R' P'), transition matrix P R F R^-1 and noise covariance F R Q R' F', and observation matrix H R^-1, where the change-of-basis matrix R and constraint projection matrix P are as defined above. This follows directly from applying the reparameterization Z = R E, and then enforcing the zero-sum constraint on the prior and transition noise covariances.

In practice, because the sum of effects z_N is constrained to be zero, it will never contribute a term to any linear operation on the latent space, so we can drop that dimension from the model entirely. ConstrainedSeasonalStateSpaceModel does this, so that it implements the N - 1 dimension latent space z_1, ..., z_[N-1].

Note that since we constrained the mean effect to be zero, the latent z_i's now recover their interpretation as the actual effects, z_i = e_i for i =1, ..., N - 1, even though they were originally defined as residuals. TheNth effect is represented only implicitly, as the nonzero mean of the firstN - 1effects. Although the computational represention is not symmetric across allNeffects, we derived theConstrainedSeasonalStateSpaceModelby starting with a symmetric representation and imposing only a symmetric constraint (the zero-sum constraint), so the probability model remains symmetric over allN` seasonal effects.

Examples

A constrained state-space model with day-of-week seasonality on hourly data:

day_of_week = ConstrainedSeasonalStateSpaceModel(
  num_timesteps=30,
  num_seasons=7,
  drift_scale=0.1,
  initial_state_prior=tfd.MultivariateNormalDiag(
    scale_diag=tf.ones([7-1], dtype=tf.float32)),
  num_steps_per_season=24)

A model with basic month-of-year seasonality on daily data, demonstrating seasons of varying length:

month_of_year = ConstrainedSeasonalStateSpaceModel(
  num_timesteps=2 * 365,  # 2 years
  num_seasons=12,
  drift_scale=0.1,
  initial_state_prior=tfd.MultivariateNormalDiag(
    scale_diag=tf.ones([12-1], dtype=tf.float32)),
  num_steps_per_season=[31, 28, 31, 30, 30, 31, 31, 31, 30, 31, 30, 31],
  initial_step=22)

Note that we've used initial_step=22 to denote that the model begins on January 23 (steps are zero-indexed). This version works over time periods not involving a leap year. A general implementation of month-of-year seasonality would require additional logic:

num_days_per_month = np.array(
  [[31, 28, 31, 30, 30, 31, 31, 31, 30, 31, 30, 31],
   [31, 29, 31, 30, 30, 31, 31, 31, 30, 31, 30, 31],  # year with leap day
   [31, 28, 31, 30, 30, 31, 31, 31, 30, 31, 30, 31],
   [31, 28, 31, 30, 30, 31, 31, 31, 30, 31, 30, 31]])

month_of_year = ConstrainedSeasonalStateSpaceModel(
  num_timesteps=4 * 365 + 2,  # 8 years with leap days
  num_seasons=12,
  drift_scale=0.1,
  initial_state_prior=tfd.MultivariateNormalDiag(
    scale_diag=tf.ones([12-1], dtype=tf.float32)),
  num_steps_per_season=num_days_per_month,
  initial_step=22)

allow_nan_stats Python bool describing behavior when a stat is undefined.

Stats return +/- infinity when it makes sense. E.g., the variance of a Cauchy distribution is infinity. However, sometimes the statistic is undefined, e.g., if a distribution's pdf does not achieve a maximum within the support of the distribution, the mode is undefined. If the mean is undefined, then by definition the variance is undefined. E.g. the mean for Student's T for df = 1 is undefined (no clear way to say it is either + or - infinity), so the variance = E[(X - mean)**2] is also undefined.

batch_shape Shape of a single sample from a single event index as a TensorShape.

May be partially defined or unknown.

The batch dimensions are indexes into independent, non-identical parameterizations of this distribution.

drift_scale Standard deviation of the drift in effects between seasonal cycles.
dtype The DType of Tensors handled by this Distribution.
event_shape Shape of a single sample from a single batch as a TensorShape.

May be partially defined or unknown.

experimental_parallelize

experimental_shard_axis_names The list or structure of lists of active shard axis names.
initial_state_prior

initial_step

mask

name Name prepended to all ops created by this Distribution.
num_seasons Number of seasons.
num_steps_per_season Number of steps in each season.
num_timesteps

observation_matrix

observation_noise

observation_noise_scale Standard deviation of the observation noise.
parameters Dictionary of parameters used to instantiate this Distribution.
reparameterization_type Describes how samples from the distribution are reparameterized.

Currently this is one of the static instances tfd.FULLY_REPARAMETERIZED or tfd.NOT_REPARAMETERIZED.

trainable_variables

transition_matrix

transition_noise

validate_args Python bool indicating possibly expensive checks are enabled.
variables

Methods

backward_smoothing_pass

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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 per-timestep 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 per-timestep 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 per-timestep 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 per-timestep 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

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batch_shape_tensor(
    name='batch_shape_tensor'
)

Shape of a single sample from a single event index as a 1-D Tensor.

The batch dimensions are indexes into independent, non-identical parameterizations of this distribution.

Args
name name to give to the op

Returns
batch_shape Tensor.

cdf

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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

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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

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covariance(
    name=&#x27;covariance', **kwargs
)

Covariance.

Covariance is (possibly) defined only for non-scalar-event distributions.

For example, for a length-k, vector-valued 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 non-vector, multivariate distributions (e.g., matrix-valued, 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 length-k' vector.

Args
name Python str prepended to names of ops created by this function.
**kwargs Named arguments forwarded to subclass implementation.

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

cross_entropy

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cross_entropy(
    other, name=&#x27;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

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entropy(
    name=&#x27;entropy', **kwargs
)

Shannon entropy in nats.

event_shape_tensor

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event_shape_tensor(
    name=&#x27;event_shape_tensor'
)

Shape of a single sample from a single batch as a 1-D int32 Tensor.

Args
name name to give to the op

Returns
event_shape Tensor.

experimental_default_event_space_bijector

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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 * (k-1) // 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

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@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

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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 push-forward 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

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experimental_sample_and_log_prob(
    sample_shape=(), seed=None, name=&#x27;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 Tensors, 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

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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 float-type 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 bool-type 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 Per-timestep log marginal likelihoods log p(x[t] | x[:t-1]) 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 per-timestep 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 per-timestep 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 per-timestep 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 per-timestep 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 per-timestep predictive distributions over observations, p(x[t] | x[:t-1]), as a Tensor of shape sample_shape(x) + batch_shape + [num_timesteps, observation_size].
observation_covs Covariances of the per-timestep predictive distributions over observations, p(x[t] | x[:t-1]), 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

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is_scalar_batch(
    name=&#x27;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

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is_scalar_event(
    name=&#x27;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

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kl_divergence(
    other, name=&#x27;kl_divergence'
)

Computes the Kullback--Leibler 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 Kullback-Leibler divergence.

latent_size_tensor

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latent_size_tensor()

latents_to_observations

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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

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log_cdf(
    value, name=&#x27;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

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log_prob(
    value, name=&#x27;log_prob', **kwargs
)

Log probability density/mass function.

Additional documentation from LinearGaussianStateSpaceModel:

kwargs:
  • mask: optional bool-type 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).

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

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log_survival_function(
    value, name=&#x27;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

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mean(
    name=&#x27;mean', **kwargs
)

Mean.

mode

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mode(
    name=&#x27;mode', **kwargs
)

Mode.

observation_size_tensor

View source

observation_size_tensor()

param_shapes

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@classmethod
param_shapes(
    sample_shape, name=&#x27;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

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@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 constant-valued 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

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@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 continuous-valued 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 categorical-like distributions. Otherwise ignored.

Returns
parameter_properties A str ->tfp.python.internal.parameter_properties.ParameterPropertiesdict mapping constructor argument names toParameterProperties` instances.

Raises
NotImplementedError if the distribution class does not implement _parameter_properties.

posterior_marginals

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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 float-type 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 bool-type 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 per-timestep 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 per-timestep 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

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posterior_sample(
    x, sample_shape=(), mask=None, seed=None, name=None
)

Draws samples from the posterior over latent trajectories.

This method uses Durbin-Koopman 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):603-615, 2002. https://www.jstor.org/stable/4140605

Args
x a float-type 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 bool-type 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

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prob(
    value, name=&#x27;prob', **kwargs
)

Probability density/mass function.

Additional documentation from LinearGaussianStateSpaceModel:

kwargs:
  • mask: optional bool-type 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).

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

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quantile(
    value, name=&#x27;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

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sample(
    sample_shape=(), seed=None, name=&#x27;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

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stddev(
    name=&#x27;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 Floating-point Tensor with shape identical to batch_shape + event_shape, i.e., the same shape as self.mean().

survival_function

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survival_function(
    value, name=&#x27;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

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unnormalized_log_prob(
    value, name=&#x27;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

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variance(
    name=&#x27;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 Floating-point Tensor with shape identical to batch_shape + event_shape, i.e., the same shape as self.mean().

__getitem__

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__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__

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__iter__()