## Class `AdditiveStateSpaceModel`

A state space model representing a sum of component state space models.

Inherits From: `LinearGaussianStateSpaceModel`

A state space model (SSM) posits a set of latent (unobserved) variables that evolve over time with dynamics specified by a probabilistic transition model `p(z[t+1] | z[t])`. At each timestep, we observe a value sampled from an observation model conditioned on the current state, `p(x[t] | z[t])`. The special case where both the transition and observation models are Gaussians with mean specified as a linear function of the inputs, is known as a linear Gaussian state space model and supports tractable exact probabilistic calculations; see `tfp.distributions.LinearGaussianStateSpaceModel` for details.

The `AdditiveStateSpaceModel` represents a sum of component state space models. Each of the `N` components describes a random process generating a distribution on observed time series `x1[t], x2[t], ..., xN[t]`. The additive model represents the sum of these processes, `y[t] = x1[t] + x2[t] + ... + xN[t] + eps[t]`, where `eps[t] ~ N(0, observation_noise_scale)` is an observation noise term.

#### Mathematical Details

The additive model concatenates the latent states of its component models. The generative process runs each component's dynamics in its own subspace of latent space, and then observes the sum of the observation models from the components.

Formally, the transition model is linear Gaussian:

``````p(z[t+1] | z[t]) ~ Normal(loc = transition_matrix.matmul(z[t]),
cov = transition_cov)
``````

where each `z[t]` is a latent state vector concatenating the component state vectors, `z[t] = [z1[t], z2[t], ..., zN[t]]`, so it has size `latent_size = sum([c.latent_size for c in components])`.

The transition matrix is the block-diagonal composition of transition matrices from the component processes:

``````transition_matrix =
[[ c0.transition_matrix,  0.,                   ..., 0.                   ],
[ 0.,                    c1.transition_matrix, ..., 0.                   ],
[ ...                    ...                   ...                       ],
[ 0.,                    0.,                   ..., cN.transition_matrix ]]
``````

and the noise covariance is similarly the block-diagonal composition of component noise covariances:

``````transition_cov =
[[ c0.transition_cov, 0.,                ..., 0.                ],
[ 0.,                c1.transition_cov, ..., 0.                ],
[ ...                ...                     ...               ],
[ 0.,                0.,                ..., cN.transition_cov ]]
``````

The observation model is also linear Gaussian,

``````p(y[t] | z[t]) ~ Normal(loc = observation_matrix.matmul(z[t]),
stddev = observation_noise_scale)
``````

This implementation assumes scalar observations, so `observation_matrix` has shape `[1, latent_size]`. The additive observation matrix simply concatenates the observation matrices from each component:

``````observation_matrix =
concat([c0.obs_matrix, c1.obs_matrix, ..., cN.obs_matrix], axis=-1)
``````

The effect is that each component observation matrix acts on the dimensions of latent state corresponding to that component, and the overall expected observation is the sum of the expected observations from each component.

If `observation_noise_scale` is not explicitly specified, it is also computed by summing the noise variances of the component processes:

``````observation_noise_scale = sqrt(sum([
c.observation_noise_scale**2 for c in components]))
``````

#### Examples

To construct an additive state space model combining a local linear trend and day-of-week seasonality component (note, the `StructuralTimeSeries` classes, e.g., `Sum`, provide a higher-level interface for this construction, which will likely be preferred by most users):

``````  num_timesteps = 30
local_ssm = tfp.sts.LocalLinearTrendStateSpaceModel(
num_timesteps=num_timesteps,
level_scale=0.5,
slope_scale=0.1,
initial_state_prior=tfd.MultivariateNormalDiag(
loc=[0., 0.], scale_diag=[1., 1.]))
day_of_week_ssm = tfp.sts.SeasonalStateSpaceModel(
num_timesteps=num_timesteps,
num_seasons=7,
initial_state_prior=tfd.MultivariateNormalDiag(
loc=tf.zeros(), scale_diag=tf.ones()))
component_ssms=[local_ssm, day_of_week_ssm],
observation_noise_scale=0.1)

print(y.shape)
# => []
``````

## `__init__`

``````__init__(
component_ssms,
constant_offset=0.0,
observation_noise_scale=None,
initial_state_prior=None,
initial_step=0,
validate_args=False,
allow_nan_stats=True,
name=None
)
``````

Build a state space model representing the sum of component models.

#### Args:

• `component_ssms`: Python `list` containing one or more `tfd.LinearGaussianStateSpaceModel` instances. The components will in general implement different time-series models, with possibly different `latent_size`, but they must have the same `dtype`, event shape (`num_timesteps` and `observation_size`), and their batch shapes must broadcast to a compatible batch shape.
• `constant_offset`: scalar `float` `Tensor`, or batch of scalars, specifying a constant value added to the sum of outputs from the component models. This allows the components to model the shifted series `observed_time_series - constant_offset`. Default value: `0.`
• `observation_noise_scale`: Optional scalar `float` `Tensor` indicating the standard deviation of the observation noise. May contain additional batch dimensions, which must broadcast with the batch shape of elements in `component_ssms`. If `observation_noise_scale` is specified for the `AdditiveStateSpaceModel`, the observation noise scales of component models are ignored. If `None`, the observation noise scale is derived by summing the noise variances of the component models, i.e., ```observation_noise_scale = sqrt(sum( [ssm.observation_noise_scale**2 for ssm in component_ssms]))```.
• `initial_state_prior`: Optional instance of `tfd.MultivariateNormal` representing a prior distribution on the latent state at time `initial_step`. If `None`, defaults to the independent priors from component models, i.e., `[component.initial_state_prior for component in component_ssms]`. Default value: `None`.
• `initial_step`: Optional scalar `int` `Tensor` specifying the starting timestep. Default value: 0.
• `validate_args`: Python `bool`. Whether to validate input with asserts. If `validate_args` is `False`, and the inputs are invalid, correct behavior is not guaranteed. Default value: `False`.
• `allow_nan_stats`: Python `bool`. 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. Default value: `True`.
• `name`: Python `str` name prefixed to ops created by this class. Default value: "AdditiveStateSpaceModel".

#### Raises:

• `ValueError`: if components have different `num_timesteps`.

## 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:

• `allow_nan_stats`: Python `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.

### `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`.

### `name_scope`

Returns a `tf.name_scope` instance for this class.

### `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`.

### `submodules`

Sequence of all sub-modules.

Submodules are modules which are properties of this module, or found as properties of modules which are properties of this module (and so on).

``````a = tf.Module()
b = tf.Module()
c = tf.Module()
a.b = b
b.c = c
assert list(a.submodules) == [b, c]
assert list(b.submodules) == [c]
assert list(c.submodules) == []
``````

#### Returns:

A sequence of all submodules.

### `trainable_variables`

Sequence of variables owned by this module and it's submodules.

#### Returns:

A sequence of variables for the current module (sorted by attribute name) followed by variables from all submodules recursively (breadth first).

### `validate_args`

Python `bool` indicating possibly expensive checks are enabled.

### `variables`

Sequence of variables owned by this module and it's submodules.

#### Returns:

A sequence of variables for the current module (sorted by attribute name) followed by variables from all submodules recursively (breadth first).

## Methods

### `__getitem__`

View source

``````__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  # => 
mvn2 = mvn[:, 3:, ..., ::-1, tf.newaxis]
mvn2.batch_shape  # => [4, 2, 3, 1]
mvn2.event_shape  # => 
``````

#### Args:

• `slices`: slices from the [] operator

#### Returns:

• `dist`: A new `tfd.Distribution` instance with sliced parameters.

### `__iter__`

View source

``````__iter__()
``````

### `backward_smoothing_pass`

View source

``````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 `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 ```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 `batch_shape + [num_timesteps, latent_size, latent_size]`. which is of the same shape as filtered_covs.

### `batch_shape_tensor`

View source

``````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`

View source

``````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`

View source

``````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`

View source

``````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:

• `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`

View source

``````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`.

#### Returns:

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

### `entropy`

View source

``````entropy(
name='entropy',
**kwargs
)
``````

Shannon entropy in nats.

### `event_shape_tensor`

View source

``````event_shape_tensor(name='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`.

### `forward_filter`

View source

``````forward_filter(
x,
)
``````

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`.

#### 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].`
• `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(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`.
• `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(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`.
• `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(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`

View source

``````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`

View source

``````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`

View source

``````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.

#### Returns:

• `kl_divergence`: `self.dtype` `Tensor` with shape `[B1, ..., Bn]` representing `n` different calculations of the Kullback-Leibler divergence.

### `latents_to_observations`

View source

``````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`

View source

``````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`

View source

``````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`.
• `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`

View source

``````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`

View source

``````mean(
name='mean',
**kwargs
)
``````

Mean.

### `mode`

View source

``````mode(
name='mode',
**kwargs
)
``````

Mode.

### `param_shapes`

View source

``````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`: name to prepend ops with.

#### Returns:

`dict` of parameter name to `Tensor` shapes.

### `param_static_shapes`

View source

``````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:

• `ValueError`: if `sample_shape` is a `TensorShape` and is not fully defined.

### `posterior_marginals`

View source

``````posterior_marginals(
x,
)
``````

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:

• `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`.

#### 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`.

### `prob`

View source

``````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`.
• `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`

View source

``````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`

View source

``````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`: Python integer seed for RNG
• `name`: name to give to the op.
• `**kwargs`: Named arguments forwarded to subclass implementation.

#### Returns:

• `samples`: a `Tensor` with prepended dimensions `sample_shape`.

### `stddev`

View source

``````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`: Floating-point `Tensor` with shape identical to `batch_shape + event_shape`, i.e., the same shape as `self.mean()`.

### `survival_function`

View source

``````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`.

### `variance`

View source

``````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`: Floating-point `Tensor` with shape identical to `batch_shape + event_shape`, i.e., the same shape as `self.mean()`.

### `with_name_scope`

``````with_name_scope(
cls,
method
)
``````

Decorator to automatically enter the module name scope.

``````class MyModule(tf.Module):
@tf.Module.with_name_scope
def __call__(self, x):
if not hasattr(self, 'w'):
self.w = tf.Variable(tf.random.normal([x.shape, 64]))
return tf.matmul(x, self.w)
``````

Using the above module would produce `tf.Variable`s and `tf.Tensor`s whose names included the module name:

``````mod = MyModule()
mod(tf.ones([8, 32]))
# ==> <tf.Tensor: ...>
mod.w
# ==> <tf.Variable ...'my_module/w:0'>
``````

#### Args:

• `method`: The method to wrap.

#### Returns:

The original method wrapped such that it enters the module's name scope.