|---| | [ View source on GitHub](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/sts/components/seasonal.py#L287-L507) | Seasonal state space model with effects constrained to sum to zero. Inherits From: [`LinearGaussianStateSpaceModel`](https://www.tensorflow.org/probability/api_docs/python/tfp/distributions/LinearGaussianStateSpaceModel), [`AutoCompositeTensorDistribution`](https://www.tensorflow.org/probability/api_docs/python/tfp/distributions/AutoCompositeTensorDistribution), [`Distribution`](https://www.tensorflow.org/probability/api_docs/python/tfp/distributions/Distribution), [`AutoCompositeTensor`](https://www.tensorflow.org/probability/api_docs/python/tfp/experimental/AutoCompositeTensor) tfp.sts.ConstrainedSeasonalStateSpaceModel( num_timesteps, num_seasons, drift_scale, initial_state_prior, observation_noise_scale=0.0001, num_steps_per_season=1, name=None, **linear_gaussian_ssm_kwargs ) See `SeasonalStateSpaceModel` for background. #### Mathematical details The constrained model implements a reparameterization of the naive `SeasonalStateSpaceModel`. Instead of directly representing the seasonal effects in the latent space, the latent space of the constrained model represents the difference between each effect and the mean effect. The following discussion assumes familiarity with the mathematical details of `SeasonalStateSpaceModel`. *Reparameterization and constraints* : let the seasonal effects at a given timestep be `E = [e_1, ..., e_N]`. The difference between each effect `e_i` and the mean effect is `z_i = e_i - sum_i(e_i)/N`. By itself, this transformation is not invertible because recovering the absolute effects requires that we know the mean as well. To fix this, we'll define `z_N = sum_i(e_i)/N` as the mean effect. It's easy to see that this is invertible: given the mean effect and the differences of the first `N - 1` effects from the mean, it's easy to solve for all `N` effects. Formally, we've defined the invertible linear reparameterization `Z = R E`, where R = [1 - 1/N, -1/N, ..., -1/N, -1/N -1/N, 1 - 1/N, ..., -1/N, -1/N ... -1/N, -1/N, ..., 1/ - 1/N, -1/N 1/N, 1/N, ..., 1/N, 1/N] E = [e_1, ..., e_(N-1), z_N] represents the change of basis from 'effect coordinates' E to 'residual coordinates' Z. The `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)


| ## Attributes || |---|---| | `allow_nan_stats` | Python `bool` describing behavior when a stat is undefined.
Stats return +/- infinity when it makes sense. E.g., the variance of a Cauchy distribution is infinity. However, sometimes the statistic is undefined, e.g., if a distribution's pdf does not achieve a maximum within the support of the distribution, the mode is undefined. If the mean is undefined, then by definition the variance is undefined. E.g. the mean for Student's T for df = 1 is undefined (no clear way to say it is either + or - infinity), so the variance = E\[(X - mean)\*\*2\] is also undefined. | | `batch_shape` | Shape of a single sample from a single event index as a `TensorShape`.
May be partially defined or unknown. The batch dimensions are indexes into independent, non-identical parameterizations of this distribution. | | `drift_scale` | Standard deviation of the drift in effects between seasonal cycles. | | `dtype` | The `DType` of `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`. | | `name_scope` | Returns a [`tf.name_scope`](https://www.tensorflow.org/api_docs/python/tf/name_scope) instance for this class. | | `non_trainable_variables` | Sequence of non-trainable variables owned by this module and its submodules.
> [!NOTE] > **Note:** this method uses reflection to find variables on the current instance and submodules. For performance reasons you may wish to cache the result of calling this method if you don't expect the return value to change. | | `num_seasons` | Number of seasons. | | `num_steps_per_season` | Number of steps in each season. | | `num_timesteps` |
| | `observation_matrix` |

| | `observation_noise` |

| | `observation_noise_scale` | Standard deviation of the observation noise. | | `parameters` | Dictionary of parameters used to instantiate this `Distribution`. | | `reparameterization_type` | Describes how samples from the distribution are reparameterized.
Currently this is one of the static instances `tfd.FULLY_REPARAMETERIZED` or `tfd.NOT_REPARAMETERIZED`. | | `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 list(a.submodules) == [b, c] True list(b.submodules) == [c] True list(c.submodules) == [] True
| | `trainable_variables` | Sequence of trainable variables owned by this module and its submodules.
> [!NOTE] > **Note:** this method uses reflection to find variables on the current instance and submodules. For performance reasons you may wish to cache the result of calling this method if you don't expect the return value to change. | | `transition_matrix` |
| | `transition_noise` |

| | `validate_args` | Python `bool` indicating possibly expensive checks are enabled. | | `variables` | Sequence of variables owned by this module and its submodules.
> [!NOTE] > **Note:** this method uses reflection to find variables on the current instance and submodules. For performance reasons you may wish to cache the result of calling this method if you don't expect the return value to change. | ## Methods ### `backward_smoothing_pass` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/linear_gaussian_ssm.py#L531-L622) backward_smoothing_pass( filtered_means, filtered_covs, predicted_means, predicted_covs ) Run the backward pass in Kalman smoother. The backward smoothing is using Rauch, Tung and Striebel smoother as as discussed in section 18.3.2 of Kevin P. Murphy, 2012, Machine Learning: A Probabilistic Perspective, The MIT Press. The inputs are returned by `forward_filter` function.


| Args || |---|---| | `filtered_means` | Means of the per-timestep filtered marginal distributions p(z\[t\] \| x\[:t\]), as a Tensor of shape `sample_shape(x) + batch_shape + [num_timesteps, latent_size]`. | | `filtered_covs` | Covariances of the per-timestep filtered marginal distributions p(z\[t\] \| x\[:t\]), as a Tensor of shape `sample_shape(x) + batch_shape + [num_timesteps, latent_size, latent_size]`. | | `predicted_means` | Means of the per-timestep predictive distributions over latent states, p(z\[t+1\] \| x\[:t\]), as a Tensor of shape `sample_shape(x) + batch_shape + [num_timesteps, latent_size]`. | | `predicted_covs` | Covariances of the per-timestep predictive distributions over latent states, p(z\[t+1\] \| x\[:t\]), as a Tensor of shape `sample_shape(x) + batch_shape + [num_timesteps, latent_size, latent_size]`. |



| Returns || |---|---| | `posterior_means` | Means of the smoothed marginal distributions p(z\[t\] \| x\[1:T\]), as a Tensor of shape `sample_shape(x) + batch_shape + [num_timesteps, latent_size]`, which is of the same shape as filtered_means. | | `posterior_covs` | Covariances of the smoothed marginal distributions p(z\[t\] \| x\[1:T\]), as a Tensor of shape `sample_shape(x) + batch_shape + [num_timesteps, latent_size, latent_size]`. which is of the same shape as filtered_covs. |
### `batch_shape_tensor` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L992-L1030) 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](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L1411-L1429) 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](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L896-L924) copy( **override_parameters_kwargs ) Creates a deep copy of the distribution. > [!NOTE] > **Note:** the copy distribution may continue to depend on the original initialization arguments.


| 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](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L1648-L1686) 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](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L1701-L1724) 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`](https://www.tensorflow.org/probability/api_docs/python/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` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L1524-L1527) entropy( name='entropy', **kwargs ) Shannon entropy in nats. ### `event_shape_tensor` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L1099-L1125) 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`. |
### `experimental_default_event_space_bijector` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L1768-L1797) experimental_default_event_space_bijector( *args, **kwargs ) Bijector mapping the reals (R\*\*n) to the event space of the distribution. Distributions with continuous support may implement `_default_event_space_bijector` which returns a subclass of [`tfp.bijectors.Bijector`](https://www.tensorflow.org/probability/api_docs/python/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()`](https://www.tensorflow.org/probability/api_docs/python/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`](https://www.tensorflow.org/probability/api_docs/python/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` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L1799-L1847) @classmethod experimental_fit( value, sample_ndims=1, validate_args=False, **init_kwargs ) Instantiates a distribution that maximizes the likelihood of `x`.


| Args || |---|---| | `value` | a `Tensor` valid sample from this distribution family. | | `sample_ndims` | Positive `int` Tensor number of leftmost dimensions of `value` that index i.i.d. samples. Default value: `1`. | | `validate_args` | Python `bool`, default `False`. When `True`, distribution parameters are checked for validity despite possibly degrading runtime performance. When `False`, invalid inputs may silently render incorrect outputs. Default value: `False`. | | `**init_kwargs` | Additional keyword arguments passed through to `cls.__init__`. These take precedence in case of collision with the fitted parameters; for example, `tfd.Normal.experimental_fit([1., 1.], scale=20.)` returns a Normal distribution with `scale=20.` rather than the maximum likelihood parameter `scale=0.`. |



| Returns || |---|---| | `maximum_likelihood_instance` | instance of `cls` with parameters that maximize the likelihood of `value`. |
### `experimental_local_measure` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L1872-L1914) experimental_local_measure( value, backward_compat=False, **kwargs ) Returns a log probability density together with a `TangentSpace`. A `TangentSpace` allows us to calculate the correct push-forward density when we apply a transformation to a `Distribution` on a strict submanifold of R\^n (typically via a `Bijector` in the `TransformedDistribution` subclass). The density correction uses the basis of the tangent space.


| Args || |---|---| | `value` | `float` or `double` `Tensor`. | | `backward_compat` | `bool` specifying whether to fall back to returning `FullSpace` as the tangent space, and representing R\^n with the standard basis. | | `**kwargs` | Named arguments forwarded to subclass implementation. |



| Returns || |---|---| | `log_prob` | a `Tensor` representing the log probability density, of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`. | | `tangent_space` | a `TangentSpace` object (by default `FullSpace`) representing the tangent space to the manifold at `value`. |



| Raises || |---|---| | UnspecifiedTangentSpaceError if `backward_compat` is False and the `_experimental_tangent_space` attribute has not been defined. ||
### `experimental_sample_and_log_prob` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L1229-L1259) experimental_sample_and_log_prob( sample_shape=(), seed=None, name='sample_and_log_prob', **kwargs ) Samples from this distribution and returns the log density of the sample. The default implementation simply calls `sample` and `log_prob`: def _sample_and_log_prob(self, sample_shape, seed, **kwargs): x = self.sample(sample_shape=sample_shape, seed=seed, **kwargs) return x, self.log_prob(x, **kwargs) However, some subclasses may provide more efficient and/or numerically stable implementations.


| Args || |---|---| | `sample_shape` | integer `Tensor` desired shape of samples to draw. Default value: `()`. | | `seed` | PRNG seed; see [`tfp.random.sanitize_seed`](https://www.tensorflow.org/probability/api_docs/python/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` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/linear_gaussian_ssm.py#L839-L912) forward_filter( x, mask=None, final_step_only=False ) Run a Kalman filter over a provided sequence of outputs. Note that the returned values `filtered_means`, `predicted_means`, and `observation_means` depend on the observed time series `x`, while the corresponding covariances are independent of the observed series; i.e., they depend only on the model itself. This means that the mean values have shape `concat([sample_shape(x), batch_shape, [num_timesteps, {latent/observation}_size]])`, while the covariances have shape `concat[(batch_shape, [num_timesteps, {latent/observation}_size, {latent/observation}_size]])`, which does not depend on the sample shape.


| Args || |---|---| | `x` | a float-type `Tensor` with rightmost dimensions `[num_timesteps, observation_size]` matching `self.event_shape`. Additional dimensions must match or be broadcastable to `self.batch_shape`; any further dimensions are interpreted as a sample shape. | | `mask` | optional bool-type `Tensor` with rightmost dimension `[num_timesteps]`; `True` values specify that the value of `x` at that timestep is masked, i.e., not conditioned on. Additional dimensions must match or be broadcastable to `self.batch_shape`; any further dimensions must match or be broadcastable to the sample shape of `x`. Default value: `None` (falls back to `self.mask`). | | `final_step_only` | optional Python `bool`. If `True`, the `num_timesteps` dimension is omitted from all return values and only the value from the final timestep is returned (in this case, `log_likelihoods` will be the *cumulative* log marginal likelihood). This may be significantly more efficient than returning all values (although note that no efficiency gain is expected when `self.experimental_parallelize=True`). Default value: `False`. |



| Returns || |---|---| | `log_likelihoods` | Per-timestep log marginal likelihoods `log p(x[t] | x[:t-1])` evaluated at the input `x`, as a `Tensor` of shape `sample_shape(x) + batch_shape + [num_timesteps].` If `final_step_only` is `True`, this will instead be the *cumulative* log marginal likelihood at the final step. | | `filtered_means` | Means of the per-timestep filtered marginal distributions p(z\[t\] \| x\[:t\]), as a Tensor of shape `sample_shape(x) + batch_shape + [num_timesteps, latent_size]`. | | `filtered_covs` | Covariances of the per-timestep filtered marginal distributions p(z\[t\] \| x\[:t\]), as a Tensor of shape `sample_shape(x) + batch_shape + [num_timesteps, latent_size, latent_size]`. Since posterior covariances do not depend on observed data, some implementations may return a Tensor whose shape omits the initial `sample_shape(x)`. | | `predicted_means` | Means of the per-timestep predictive distributions over latent states, p(z\[t+1\] \| x\[:t\]), as a Tensor of shape `sample_shape(x) + batch_shape + [num_timesteps, latent_size]`. | | `predicted_covs` | Covariances of the per-timestep predictive distributions over latent states, p(z\[t+1\] \| x\[:t\]), as a Tensor of shape `sample_shape(x) + batch_shape + [num_timesteps, latent_size, latent_size]`. Since posterior covariances do not depend on observed data, some implementations may return a Tensor whose shape omits the initial `sample_shape(x)`. | | `observation_means` | Means of the per-timestep predictive distributions over observations, p(x\[t\] \| x\[:t-1\]), as a Tensor of shape `sample_shape(x) + batch_shape + [num_timesteps, observation_size]`. | | `observation_covs` | Covariances of the per-timestep predictive distributions over observations, p(x\[t\] \| x\[:t-1\]), as a Tensor of shape `sample_shape(x) + batch_shape + [num_timesteps, observation_size, observation_size]`. Since posterior covariances do not depend on observed data, some implementations may return a Tensor whose shape omits the initial `sample_shape(x)`. |
### `is_scalar_batch` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L1156-L1168) 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](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L1142-L1154) 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](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L1730-L1761) 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`](https://www.tensorflow.org/probability/api_docs/python/tfp/distributions/Distribution) instance. | | `name` | Python `str` prepended to names of ops created by this function. |



| Returns || |---|---| | `kl_divergence` | `self.dtype` `Tensor` with shape `[B1, ..., Bn]` representing `n` different calculations of the Kullback-Leibler divergence. |
### `latent_size_tensor` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/linear_gaussian_ssm.py#L510-L512) latent_size_tensor() ### `latents_to_observations` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/linear_gaussian_ssm.py#L1342-L1382) 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](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L1373-L1395) 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](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L1275-L1287) 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` (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` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L1452-L1476) 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](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L1533-L1536) mean( name='mean', **kwargs ) Mean. ### `mode` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L1692-L1695) mode( name='mode', **kwargs ) Mode. ### `observation_size_tensor` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/linear_gaussian_ssm.py#L521-L523) observation_size_tensor() ### `param_shapes` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L701-L727) @classmethod param_shapes( sample_shape, name='DistributionParamShapes' ) Shapes of parameters given the desired shape of a call to `sample()`. (deprecated) > [!WARNING] > **Deprecated:** THIS FUNCTION IS DEPRECATED. It will be removed after 2021-03-01. Instructions for updating: The `param_shapes` method of `tfd.Distribution` is deprecated; use `parameter_properties` instead. 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](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L729-L769) @classmethod param_static_shapes( sample_shape ) param_shapes with static (i.e. `TensorShape`) shapes. (deprecated) > [!WARNING] > **Deprecated:** THIS FUNCTION IS DEPRECATED. It will be removed after 2021-03-01. Instructions for updating: The `param_static_shapes` method of `tfd.Distribution` is deprecated; use `parameter_properties` instead. 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` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L655-L699) @classmethod parameter_properties( dtype=tf.float32, num_classes=None ) Returns a dict mapping constructor arg names to property annotations. This dict should include an entry for each of the distribution's `Tensor`-valued constructor arguments. Distribution subclasses are not required to implement `_parameter_properties`, so this method may raise `NotImplementedError`. Providing a `_parameter_properties` implementation enables several advanced features, including: - Distribution batch slicing (`sliced_distribution = distribution[i:j]`). - Automatic inference of `_batch_shape` and `_batch_shape_tensor`, which must otherwise be computed explicitly. - Automatic instantiation of the distribution within TFP's internal property tests. - Automatic construction of 'trainable' instances of the distribution using appropriate bijectors to avoid violating parameter constraints. This enables the distribution family to be used easily as a surrogate posterior in variational inference. In the future, parameter property annotations may enable additional functionality; for example, returning Distribution instances from [`tf.vectorized_map`](https://www.tensorflow.org/api_docs/python/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` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/linear_gaussian_ssm.py#L1065-L1130) posterior_marginals( x, mask=None ) Run a Kalman smoother to return posterior mean and cov. Note that the returned values `smoothed_means` depend on the observed time series `x`, while the `smoothed_covs` are independent of the observed series; i.e., they depend only on the model itself. This means that the mean values have shape `concat([sample_shape(x), batch_shape, [num_timesteps, {latent/observation}_size]])`, while the covariances have shape `concat[(batch_shape, [num_timesteps, {latent/observation}_size, {latent/observation}_size]])`, which does not depend on the sample shape. This function only performs smoothing. If the user wants the intermediate values, which are returned by filtering pass `forward_filter`, one could get it by: (log_likelihoods, filtered_means, filtered_covs, predicted_means, predicted_covs, observation_means, observation_covs) = model.forward_filter(x) smoothed_means, smoothed_covs = model.backward_smoothing_pass( filtered_means, filtered_covs, predicted_means, predicted_covs) where `x` is an observation sequence.


| Args || |---|---| | `x` | a float-type `Tensor` with rightmost dimensions `[num_timesteps, observation_size]` matching `self.event_shape`. Additional dimensions must match or be broadcastable to `self.batch_shape`; any further dimensions are interpreted as a sample shape. | | `mask` | optional bool-type `Tensor` with rightmost dimension `[num_timesteps]`; `True` values specify that the value of `x` at that timestep is masked, i.e., not conditioned on. Additional dimensions must match or be broadcastable to `self.batch_shape`; any further dimensions must match or be broadcastable to the sample shape of `x`. Default value: `None` (falls back to `self.mask`). |



| Returns || |---|---| | `smoothed_means` | Means of the per-timestep smoothed distributions over latent states, p(z\[t\] \| x\[:T\]), as a Tensor of shape `sample_shape(x) + batch_shape + [num_timesteps, observation_size]`. | | `smoothed_covs` | Covariances of the per-timestep smoothed distributions over latent states, p(z\[t\] \| x\[:T\]), as a Tensor of shape `sample_shape(mask) + batch_shape + [num_timesteps, observation_size, observation_size]`. Note that the covariances depend only on the model and the mask, not on the data, so this may have fewer dimensions than `filtered_means`. |
### `posterior_sample` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/linear_gaussian_ssm.py#L1132-L1218) posterior_sample( x, sample_shape=(), mask=None, seed=None, name=None ) Draws samples from the posterior over latent trajectories. This method uses Durbin-Koopman sampling \[1\], an efficient algorithm to sample from the posterior latents of a linear Gaussian state space model. The cost of drawing a sample is equal to the cost of drawing a prior sample (`.sample(sample_shape)`), plus the cost of Kalman smoothing ( `.posterior_marginals(...)` on both the observed time series and the prior sample. This method is significantly more efficient in graph mode, because it uses only the posterior means and can elide the unneeded calculation of marginal covariances. \[1\] Durbin, J. and Koopman, S.J. A simple and efficient simulation smoother for state space time series analysis. *Biometrika* 89(3):603-615, 2002. https://www.jstor.org/stable/4140605


| Args || |---|---| | `x` | a float-type `Tensor` with rightmost dimensions `[num_timesteps, observation_size]` matching `self.event_shape`. Additional dimensions must match or be broadcastable with `self.batch_shape`. | | `sample_shape` | `int` `Tensor` shape of samples to draw. Default value: `()`. | | `mask` | optional bool-type `Tensor` with rightmost dimension `[num_timesteps]`; `True` values specify that the value of `x` at that timestep is masked, i.e., not conditioned on. Additional dimensions must match or be broadcastable with `self.batch_shape` and `x.shape[:-2]`. Default value: `None` (falls back to `self.mask`). | | `seed` | PRNG seed; see [`tfp.random.sanitize_seed`](https://www.tensorflow.org/probability/api_docs/python/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` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L1303-L1315) 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` (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` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L1555-L1573) 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](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L1189-L1205) sample( sample_shape=(), seed=None, name='sample', **kwargs ) Generate samples of the specified shape. Note that a call to `sample()` without arguments will generate a single sample.


| Args || |---|---| | `sample_shape` | 0D or 1D `int32` `Tensor`. Shape of the generated samples. | | `seed` | PRNG seed; see [`tfp.random.sanitize_seed`](https://www.tensorflow.org/probability/api_docs/python/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` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L1613-L1642) 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](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L1498-L1518) survival_function( value, name='survival_function', **kwargs ) Survival function. Given random variable `X`, the survival function is defined: survival_function(x) = P[X > x] = 1 - P[X <= x] = 1 - cdf(x).


| Args || |---|---| | `value` | `float` or `double` `Tensor`. | | `name` | Python `str` prepended to names of ops created by this function. | | `**kwargs` | Named arguments forwarded to subclass implementation. |



| Returns || |---|---| | `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`. ||
### `unnormalized_log_prob` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L1335-L1357) unnormalized_log_prob( value, name='unnormalized_log_prob', **kwargs ) Potentially unnormalized log probability density/mass function. This function is similar to `log_prob`, but does not require that the return value be normalized. (Normalization here refers to the total integral of probability being one, as it should be by definition for any probability distribution.) This is useful, for example, for distributions where the normalization constant is difficult or expensive to compute. By default, this simply calls `log_prob`.


| Args || |---|---| | `value` | `float` or `double` `Tensor`. | | `name` | Python `str` prepended to names of ops created by this function. | | `**kwargs` | Named arguments forwarded to subclass implementation. |



| Returns || |---|---| | `unnormalized_log_prob` | a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`. |
### `variance` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L1579-L1607) 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` @classmethod with_name_scope( 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[1], 3])) return tf.matmul(x, self.w) Using the above module would produce [`tf.Variable`](https://www.tensorflow.org/api_docs/python/tf/Variable)s and [`tf.Tensor`](https://www.tensorflow.org/api_docs/python/tf/Tensor)s whose names included the module name: mod = MyModule() mod(tf.ones([1, 2])) mod.w


| Args || |---|---| | `method` | The method to wrap. |



| Returns || |---|---| | The original method wrapped such that it enters the module's name scope. ||
### `__getitem__` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L825-L852) __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.linalg.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` | slices from the \[\] operator |



| Returns || |---|---| | `dist` | A new `tfd.Distribution` instance with sliced parameters. |
### `__iter__` [View source](https://github.com/tensorflow/probability/blob/v0.25.0/tensorflow_probability/python/distributions/distribution.py#L854-L855) __iter__()