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# tfp.substrates.numpy.sts.DynamicLinearRegressionStateSpaceModel

State space model for a dynamic linear regression from provided covariates.

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 dynamic linear regression model is a special case of a linear Gaussian SSM and a generalization of typical (static) linear regression. The model represents regression `weights` with a latent state which evolves via a Gaussian random walk:

``````weights[t] ~ Normal(weights[t-1], drift_scale)
``````

The latent state (the weights) has dimension `num_features`, while the parameters `drift_scale` and `observation_noise_scale` are each (a batch of) scalars. The batch shape of this `Distribution` is the broadcast batch shape of these parameters, the `initial_state_prior`, and the `design_matrix`. `num_features` is determined from the last dimension of `design_matrix` (equivalent to the number of columns in the design matrix in linear regression).

#### Mathematical Details

The dynamic linear regression model implements a `tfp.distributions.LinearGaussianStateSpaceModel` with ```latent_size = num_features``` and `observation_size = 1` following the transition model:

``````transition_matrix = eye(num_features)
transition_noise ~ Normal(0, diag([drift_scale]))
``````

which implements the evolution of `weights` described above. The observation model is:

``````observation_matrix[t] = design_matrix[t]
observation_noise ~ Normal(0, observation_noise_scale)
``````

#### Examples

Given `series1`, `series2` as `Tensors` each of shape `[num_timesteps]` representing covariate time series, we create a dynamic regression model which conditions on these via the following:

``````dynamic_regression_ssm = DynamicLinearRegressionStateSpaceModel(
num_timesteps=42,
design_matrix=tf.stack([series1, series2], axis=-1),
drift_scale=3.14,
initial_state_prior=tfd.MultivariateNormalDiag(scale_diag=[1., 2.]),
observation_noise_scale=1.)

y = dynamic_regression_ssm.sample()  # shape [42, 1]
lp = dynamic_regression_ssm.log_prob(y)  # scalar
``````

Passing additional parameter and `initial_state_prior` dimensions constructs a batch of models, consider the following:

``````dynamic_regression_ssm = DynamicLinearRegressionStateSpaceModel(
num_timesteps=42,
design_matrix=tf.stack([series1, series2], axis=-1),
drift_scale=[3.14, 1.],
initial_state_prior=tfd.MultivariateNormalDiag(scale_diag=[1., 2.]),
observation_noise_scale=[1., 2.])

y = dynamic_regression_ssm.sample(3)  # shape [3, 2, 42, 1]
lp = dynamic_regression_ssm.log_prob(y)  # shape [3, 2]
``````

Which (effectively) constructs two independent state space models; the first with `drift_scale = 3.14` and `observation_noise_scale = 1.`, the second with `drift_scale = 1.` and `observation_noise_scale = 2.`. We then sample from each of the models three times and calculate the log probability of each of the samples under each of the models.

Similarly, it is also possible to add batch dimensions via the `design_matrix`.

`num_timesteps` Scalar `int` `Tensor` number of timesteps to model with this distribution.
`design_matrix` float `Tensor` of shape ```concat([batch_shape, [num_timesteps, num_features]])```.
`drift_scale` Scalar (any additional dimensions are treated as batch dimensions) `float` `Tensor` indicating the standard deviation of the latent state transitions.
`initial_state_prior` instance of `tfd.MultivariateNormal` representing the prior distribution on latent states. Must have event shape `[num_features]`.
`observation_noise_scale` Scalar (any additional dimensions are treated as batch dimensions) `float` `Tensor` indicating the standard deviation of the observation noise. Default value: `0.`.
`name` Python `str` name prefixed to ops created by this class. Default value: 'DynamicLinearRegressionStateSpaceModel'.
`**linear_gaussian_ssm_kwargs` Optional additional keyword arguments to to the base `tfd.LinearGaussianStateSpaceModel` constructor.

`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 weights at each timestep.
`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.

`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_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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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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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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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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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.

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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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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Shannon entropy in nats.

### `event_shape_tensor`

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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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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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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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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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Samples from this distribution and returns the log density of the sample.

The default implementation simply calls `sample` and `log_prob`:

``````def _sample_and_log_prob(self, sample_shape, seed, **kwargs):
x = self.sample(sample_shape=sample_shape, seed=seed, **kwargs)
return x, self.log_prob(x, **kwargs)
``````

However, some subclasses may provide more efficient and/or numerically stable implementations.

Args
`sample_shape` integer `Tensor` desired shape of samples to draw. Default value: `()`.
`seed` PRNG seed; see `tfp.random.sanitize_seed` for details. Default value: `None`.
`name` name to give to the op. Default value: `'sample_and_log_prob'`.
`**kwargs` Named arguments forwarded to subclass implementation.

Returns
`samples` a `Tensor`, or structure of `Tensor`s, with prepended dimensions `sample_shape`.
`log_prob` a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`.

### `forward_filter`

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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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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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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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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.

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### `latents_to_observations`

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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 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 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.

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`.

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Mean.

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Mode.

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### `param_shapes`

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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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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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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.ParameterProperties`dict mapping constructor argument names to`ParameterProperties` instances.

Raises
`NotImplementedError` if the distribution class does not implement `_parameter_properties`.

### `posterior_marginals`

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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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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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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 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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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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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.

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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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.

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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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.

View source

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