# tfp.sts.LocalLinearTrendStateSpaceModel

State space model for a local linear trend.

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 local linear trend model is a special case of a linear Gaussian SSM, in which the latent state posits a `level` and `slope`, each evolving via a Gaussian random walk:

``````level[t] = level[t-1] + slope[t-1] + Normal(0., level_scale)
slope[t] = slope[t-1] + Normal(0., slope_scale)
``````

The latent state is the two-dimensional tuple `[level, slope]`. The `level` is observed at each timestep.

The parameters `level_scale`, `slope_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 and of the `initial_state_prior`.

#### Mathematical Details

The linear trend model implements a `tfp.distributions.LinearGaussianStateSpaceModel` with `latent_size = 2` and `observation_size = 1`, following the transition model:

``````transition_matrix = [[1., 1.]
[0., 1.]]
transition_noise ~ N(loc=0., scale=diag([level_scale, slope_scale]))
``````

which implements the evolution of `[level, slope]` described above, and the observation model:

``````observation_matrix = [[1., 0.]]
observation_noise ~ N(loc=0, scale=observation_noise_scale)
``````

which picks out the first latent component, i.e., the `level`, as the observation at each timestep.

#### Examples

A simple model definition:

``````linear_trend_model = LocalLinearTrendStateSpaceModel(
num_timesteps=50,
level_scale=0.5,
slope_scale=0.5,
initial_state_prior=tfd.MultivariateNormalDiag(scale_diag=[1., 1.]))

y = linear_trend_model.sample() # y has shape [50, 1]
lp = linear_trend_model.log_prob(y) # log_prob is scalar
``````

Passing additional parameter dimensions constructs a batch of models. The overall batch shape is the broadcast batch shape of the parameters:

``````linear_trend_model = LocalLinearTrendStateSpaceModel(
num_timesteps=50,
level_scale=tf.ones([10]),
slope_scale=0.5,
initial_state_prior=tfd.MultivariateNormalDiag(
scale_diag=tf.ones([10, 10, 2])))

y = linear_trend_model.sample(5) # y has shape [5, 10, 10, 50, 1]
lp = linear_trend_model.log_prob(y) # has shape [5, 10, 10]
``````

`num_timesteps` Scalar `int` `Tensor` number of timesteps to model with this distribution.
`level_scale` Scalar (any additional dimensions are treated as batch dimensions) `float` `Tensor` indicating the standard deviation of the level transitions.
`slope_scale` Scalar (any additional dimensions are treated as batch dimensions) `float` `Tensor` indicating the standard deviation of the slope transitions.
`initial_state_prior` instance of `tfd.MultivariateNormal` representing the prior distribution on latent states; must have event shape `[2]`.
`observation_noise_scale` Scalar (any additional dimensions are treated as batch dimensions) `float` `Tensor` indicating the standard deviation of the observation noise.
`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: "LocalLinearTrendStateSpaceModel".

`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.

`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.

`final_step` DEPRECATED FUNCTION

`initial_step`

`latent_size` DEPRECATED FUNCTION

`level_scale` Standard deviation of the level transitions.
`name` Name prepended to all ops created by this `Distribution`.
`name_scope` Returns a `tf.name_scope` instance for this class.
`num_timesteps`

`observation_matrix`

`observation_noise`

`observation_noise_scale` Standard deviation of the observation noise.
`observation_size`

`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`.

`slope_scale` Standard deviation of the slope transitions.
`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.

`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.

## Methods

### `backward_smoothing_pass`

View source

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`

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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`

View source

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`.

### `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`.

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`

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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.

View source

### `latents_to_observations`

View source

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 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`.

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`.

View source

Mean.

View source

Mode.

View source

### `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.

### `posterior_marginals`

View source

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`.

### `posterior_sample`

View source

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.
`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`.
`seed` Python `int` random seed.
`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

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`

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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` Python integer or `tfp.util.SeedStream` instance, for seeding PRNG.
`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

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.

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.

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`

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`s and `tf.Tensor`s whose names included the module name:

````mod = MyModule()`
`mod(tf.ones([1, 2]))`
`<tf.Tensor: shape=(1, 3), dtype=float32, numpy=..., dtype=float32)>`
`mod.w`
`<tf.Variable 'my_module/Variable:0' shape=(2, 3) dtype=float32,`
`numpy=..., dtype=float32)>`
```

Args
`method` The method to wrap.

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

### `__getitem__`

View source

Slices the batch axes of this distribution, returning a new instance.

``````b = tfd.Bernoulli(logits=tf.zeros([3, 5, 7, 9]))
b.batch_shape  # => [3, 5, 7, 9]
b2 = b[:, tf.newaxis, ..., -2:, 1::2]
b2.batch_shape  # => [3, 1, 5, 2, 4]

x = tf.random.normal([5, 3, 2, 2])
cov = tf.matmul(x, x, transpose_b=True)
chol = tf.cholesky(cov)
loc = tf.random.normal([4, 1, 3, 1])
mvn = tfd.MultivariateNormalTriL(loc, chol)
mvn.batch_shape  # => [4, 5, 3]
mvn.event_shape  # => [2]
mvn2 = mvn[:, 3:, ..., ::-1, tf.newaxis]
mvn2.batch_shape  # => [4, 2, 3, 1]
mvn2.event_shape  # => [2]
``````

Args
`slices` slices from the [] operator

Returns
`dist` A new `tfd.Distribution` instance with sliced parameters.

View source