## Class `ConstrainedSeasonalStateSpaceModel`

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

Inherits From: `LinearGaussianStateSpaceModel`

Defined in `python/sts/seasonal.py`

.

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 - 1/N, ..., -1/N,
...
1/N, 1/N, ..., 1/N]
```

represents the change of basis from 'effect coordinates' E to
'residual coordinates' Z. The `Z`

s 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. The
```

N```
th effect is represented only implicitly, as
the nonzero mean of the first
```

N - 1```
effects. Although the computational
represention is not symmetric across all
```

N`effects, we derived the`

ConstrainedSeasonalStateSpaceModel```
by starting with a symmetric
representation and imposing only a symmetric constraint (the zero-sum
constraint), so the probability model remains symmetric over all
```

N`
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)
```

`__init__`

```
__init__(
num_timesteps,
num_seasons,
drift_scale,
initial_state_prior,
observation_noise_scale=0.0001,
num_steps_per_season=1,
initial_step=0,
validate_args=False,
allow_nan_stats=True,
name=None
)
```

Build a seasonal effect state space model with a zero-sum constraint.

{seasonal_init_args}

## Properties

`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.

#### Returns:

: Python`allow_nan_stats`

`bool`

.

`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.

#### Returns:

:`batch_shape`

`TensorShape`

, possibly unknown.

`drift_scale`

Standard deviation of the drift in effects between seasonal cycles.

`dtype`

The `DType`

of `Tensor`

s handled by this `Distribution`

.

`event_shape`

Shape of a single sample from a single batch as a `TensorShape`

.

May be partially defined or unknown.

#### Returns:

:`event_shape`

`TensorShape`

, possibly unknown.

`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.

`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`

.

#### Returns:

An instance of `ReparameterizationType`

.

`validate_args`

Python `bool`

indicating possibly expensive checks are enabled.

## Methods

`__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.normal([5, 3, 2, 2])
cov = tf.matmul(x, x, transpose_b=True)
chol = tf.cholesky(cov)
loc = tf.random.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 from the [] operator`slices`

#### Returns:

: A new`dist`

`tfd.Distribution`

instance with sliced parameters.

`__iter__`

```
__iter__()
```

`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:

: Means of the per-timestep filtered marginal distributions p(z`filtered_means`

*t | x*{:t}), as a Tensor of shape`sample_shape(x) + batch_shape + [num_timesteps, latent_size]`

.: Covariances of the per-timestep filtered marginal distributions p(z`filtered_covs`

*t | x*{:t}), as a Tensor of shape`batch_shape + [num_timesteps, latent_size, latent_size]`

.: Means of the per-timestep predictive distributions over latent states, p(z`predicted_means`

*{t+1} | x*{:t}), as a Tensor of shape`sample_shape(x) + batch_shape + [num_timesteps, latent_size]`

.: Covariances of the per-timestep predictive distributions over latent states, p(z`predicted_covs`

*{t+1} | x*{:t}), as a Tensor of shape`batch_shape + [num_timesteps, latent_size, latent_size]`

.

#### Returns:

: Means of the smoothed marginal distributions p(z`posterior_means`

*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.: Covariances of the smoothed marginal distributions p(z`posterior_covs`

*t | x*{1:T}), as a Tensor of shape`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 1-D `Tensor`

.

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

#### Args:

: name to give to the op`name`

#### 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`

.: Python`name`

`str`

prepended to names of ops created by this function.: Named arguments forwarded to subclass implementation.`**kwargs`

#### Returns:

: a`cdf`

`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:

: String/value dictionary of initialization arguments to override with new values.`**override_parameters_kwargs`

#### Returns:

: A new instance of`distribution`

`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 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:

: Python`name`

`str`

prepended to names of ops created by this function.: Named arguments forwarded to subclass implementation.`**kwargs`

#### Returns:

: Floating-point`covariance`

`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.: Python`name`

`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 1-D int32 `Tensor`

.

#### Args:

: name to give to the op`name`

#### Returns:

:`event_shape`

`Tensor`

.

`forward_filter`

```
forward_filter(
x,
mask=None
)
```

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:

: a float-type`x`

`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.: optional bool-type`mask`

`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`

.

#### Returns:

: Per-timestep log marginal likelihoods`log_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].`

: Means of the per-timestep filtered marginal distributions p(z`filtered_means`

*t | x*{:t}), as a Tensor of shape`sample_shape(x) + batch_shape + [num_timesteps, latent_size]`

.: Covariances of the per-timestep filtered marginal distributions p(z`filtered_covs`

*t | x*{:t}), as a Tensor of shape`sample_shape(mask) + batch_shape + [num_timesteps, latent_size, latent_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`

.: Means of the per-timestep predictive distributions over latent states, p(z`predicted_means`

*{t+1} | x*{:t}), as a Tensor of shape`sample_shape(x) + batch_shape + [num_timesteps, latent_size]`

.: Covariances of the per-timestep predictive distributions over latent states, p(z`predicted_covs`

*{t+1} | x*{:t}), as a Tensor of shape`sample_shape(mask) + batch_shape + [num_timesteps, latent_size, latent_size]`

. Note that the covariances depend only on the model and the mask, not on the data, so this may have fewer dimensions than`predicted_means`

.: Means of the per-timestep predictive distributions over observations, p(x`observation_means`

*{t} | x*{:t-1}), as a Tensor of shape`sample_shape(x) + batch_shape + [num_timesteps, observation_size]`

.: Covariances of the per-timestep predictive distributions over observations, p(x`observation_covs`

*{t} | x*{:t-1}), 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`observation_means`

.

`is_scalar_batch`

```
is_scalar_batch(name='is_scalar_batch')
```

Indicates that `batch_shape == []`

.

#### Args:

: Python`name`

`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:

: Python`name`

`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 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.: Python`name`

`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.

`latents_to_observations`

```
latents_to_observations(
latent_means,
latent_covs
)
```

Push latent means and covariances forward through the observation model.

#### Args:

: float`latent_means`

`Tensor`

of shape`[..., num_timesteps, latent_size]`

: float`latent_covs`

`Tensor`

of shape`[..., num_timesteps, latent_size, latent_size]`

.

#### Returns:

: float`observation_means`

`Tensor`

of shape`[..., num_timesteps, observation_size]`

: float`observation_covs`

`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`

.: Python`name`

`str`

prepended to names of ops created by this function.: Named arguments forwarded to subclass implementation.`**kwargs`

#### Returns:

: a`logcdf`

`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 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`

.

#### Args:

:`value`

`float`

or`double`

`Tensor`

.: Python`name`

`str`

prepended to names of ops created by this function.: Named arguments forwarded to subclass implementation.`**kwargs`

#### Returns:

: a`log_prob`

`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`

.: Python`name`

`str`

prepended to names of ops created by this function.: Named arguments forwarded to subclass implementation.`**kwargs`

#### 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.

`param_shapes`

```
param_shapes(
cls,
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 to prepend ops with.`name`

#### Returns:

`dict`

of parameter name to `Tensor`

shapes.

`param_static_shapes`

```
param_static_shapes(
cls,
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:

: if`ValueError`

`sample_shape`

is a`TensorShape`

and is not fully defined.

`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(x)
```

where `x`

is an observation sequence.

#### Args:

: a float-type`x`

`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.: optional bool-type`mask`

`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`

.

#### Returns:

: Means of the per-timestep smoothed distributions over latent states, p(x`smoothed_means`

*{t} | x*{:T}), as a Tensor of shape`sample_shape(x) + batch_shape + [num_timesteps, observation_size]`

.: Covariances of the per-timestep smoothed distributions over latent states, p(x`smoothed_covs`

*{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`

.

`prob`

```
prob(
value,
name='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`

.

#### Args:

:`value`

`float`

or`double`

`Tensor`

.: Python`name`

`str`

prepended to names of ops created by this function.: Named arguments forwarded to subclass implementation.`**kwargs`

#### Returns:

: a`prob`

`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`

.: Python`name`

`str`

prepended to names of ops created by this function.: Named arguments forwarded to subclass implementation.`**kwargs`

#### Returns:

: a`quantile`

`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:

: 0D or 1D`sample_shape`

`int32`

`Tensor`

. Shape of the generated samples.: Python integer seed for RNG`seed`

: name to give to the op.`name`

: Named arguments forwarded to subclass implementation.`**kwargs`

#### Returns:

: a`samples`

`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:

: Python`name`

`str`

prepended to names of ops created by this function.: Named arguments forwarded to subclass implementation.`**kwargs`

#### Returns:

: Floating-point`stddev`

`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`

.: Python`name`

`str`

prepended to names of ops created by this function.: Named arguments forwarded to subclass implementation.`**kwargs`

#### Returns:

`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:

: Python`name`

`str`

prepended to names of ops created by this function.: Named arguments forwarded to subclass implementation.`**kwargs`

#### Returns:

: Floating-point`variance`

`Tensor`

with shape identical to`batch_shape + event_shape`

, i.e., the same shape as`self.mean()`

.