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# tfp.sts.SemiLocalLinearTrendStateSpaceModel

## Class `SemiLocalLinearTrendStateSpaceModel`

State space model for a semi-local linear trend.

Inherits From: `LinearGaussianStateSpaceModel`

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

The semi-local linear trend model is a special case of a linear Gaussian SSM, in which the latent state posits a `level` and `slope`. The `level` evolves via a Gaussian random walk centered at the current `slope`, while the `slope` follows a first-order autoregressive (AR1) process with mean `slope_mean`:

``````level[t] = level[t-1] + slope[t-1] + Normal(0., level_scale)
slope[t] = (slope_mean +
autoregressive_coef * (slope[t-1] - slope_mean) +
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_mean`, `slope_scale`, `autoregressive_coef`, 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 semi-local 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., autoregressive_coef]]
transition_noise ~ N(loc=slope_mean - autoregressive_coef * slope_mean,
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:

``````semilocal_trend_model = SemiLocalLinearTrendStateSpaceModel(
num_timesteps=50,
level_scale=0.5,
slope_mean=0.2,
slope_scale=0.5,
autoregressive_coef=0.9,
initial_state_prior=tfd.MultivariateNormalDiag(scale_diag=[1., 1.]))

y = semilocal_trend_model.sample() # y has shape [50, 1]
lp = semilocal_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:

``````semilocal_trend_model = SemiLocalLinearTrendStateSpaceModel(
num_timesteps=50,
level_scale=tf.ones(),
slope_mean=0.2,
slope_scale=0.5,
autoregressive_coef=0.9,
initial_state_prior=tfd.MultivariateNormalDiag(
scale_diag=tf.ones([10, 10, 2])))

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

## `__init__`

``````__init__(
num_timesteps,
level_scale,
slope_mean,
slope_scale,
autoregressive_coef,
initial_state_prior,
observation_noise_scale=0.0,
initial_step=0,
validate_args=False,
allow_nan_stats=True,
name=None
)
``````

Build a state space model implementing a semi-local linear trend.

#### Args:

• `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_mean`: Scalar (any additional dimensions are treated as batch dimensions) `float` `Tensor` indicating the expected long-term mean of the latent slope.
• `slope_scale`: Scalar (any additional dimensions are treated as batch dimensions) `float` `Tensor` indicating the standard deviation of the slope transitions.
• `autoregressive_coef`: Scalar (any additional dimensions are treated as batch dimensions) `float` `Tensor` defining the AR1 process on the latent slope.
• `initial_state_prior`: instance of `tfd.MultivariateNormal` representing the prior distribution on latent states; must have event shape ``.
• `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: "SemiLocalLinearTrendStateSpaceModel".

## Properties

### `allow_nan_stats`

Python `bool` describing behavior when a stat is undefined.

Stats return +/- infinity when it makes sense. E.g., the variance of a Cauchy distribution is infinity. However, sometimes the statistic is undefined, e.g., if a distribution's pdf does not achieve a maximum within the support of the distribution, the mode is undefined. If the mean is undefined, then by definition the variance is undefined. E.g. the mean for Student's T for df = 1 is undefined (no clear way to say it is either + or - infinity), so the variance = E[(X - mean)**2] is also undefined.

#### Returns:

• `allow_nan_stats`: Python `bool`.

### `batch_shape`

Shape of a single sample from a single event index as a `TensorShape`.

May be partially defined or unknown.

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

#### Returns:

• `batch_shape`: `TensorShape`, possibly unknown.

### `dtype`

The `DType` of `Tensor`s handled by this `Distribution`.

### `event_shape`

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

May be partially defined or unknown.

#### Returns:

• `event_shape`: `TensorShape`, possibly unknown.

### `name`

Name prepended to all ops created by this `Distribution`.

### `parameters`

Dictionary of parameters used to instantiate this `Distribution`.

### `reparameterization_type`

Describes how samples from the distribution are reparameterized.

Currently this is one of the static instances `tfd.FULLY_REPARAMETERIZED` or `tfd.NOT_REPARAMETERIZED`.

#### Returns:

An instance of `ReparameterizationType`.

### `validate_args`

Python `bool` indicating possibly expensive checks are enabled.

## Methods

### `__getitem__`

``````__getitem__(slices)
``````

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

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

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

#### Args:

• `slices`: slices from the [] operator

#### Returns:

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

### `__iter__`

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

### `backward_smoothing_pass`

``````backward_smoothing_pass(
filtered_means,
filtered_covs,
predicted_means,
predicted_covs
)
``````

Run the backward pass in Kalman smoother.

The backward smoothing is using Rauch, Tung and Striebel smoother as as discussed in section 18.3.2 of Kevin P. Murphy, 2012, Machine Learning: A Probabilistic Perspective, The MIT Press. The inputs are returned by `forward_filter` function.

#### Args:

• `filtered_means`: Means of the per-timestep filtered marginal distributions p(zt | 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(zt | 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(zt | 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(zt | x{1:T}), as a Tensor of shape `batch_shape + [num_timesteps, latent_size, latent_size]`. which is of the same shape as filtered_covs.

### `batch_shape_tensor`

``````batch_shape_tensor(name='batch_shape_tensor')
``````

Shape of a single sample from a single event index as a 1-D `Tensor`.

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

#### Args:

• `name`: name to give to the op

#### Returns:

• `batch_shape`: `Tensor`.

### `cdf`

``````cdf(
value,
name='cdf',
**kwargs
)
``````

Cumulative distribution function.

Given random variable `X`, the cumulative distribution function `cdf` is:

``````cdf(x) := P[X <= x]
``````

#### Args:

• `value`: `float` or `double` `Tensor`.
• `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`

``````copy(**override_parameters_kwargs)
``````

Creates a deep copy of the distribution.

#### Args:

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

#### Returns:

• `distribution`: A new instance of `type(self)` initialized from the union of self.parameters and override_parameters_kwargs, i.e., `dict(self.parameters, **override_parameters_kwargs)`.

### `covariance`

``````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`

``````cross_entropy(
other,
name='cross_entropy'
)
``````

Computes the (Shannon) cross entropy.

Denote this distribution (`self`) by `P` and the `other` distribution by `Q`. Assuming `P, Q` are absolutely continuous with respect to one another and permit densities `p(x) dr(x)` and `q(x) dr(x)`, (Shannon) cross entropy is defined as:

``````H[P, Q] = E_p[-log q(X)] = -int_F p(x) log q(x) dr(x)
``````

where `F` denotes the support of the random variable `X ~ P`.

#### Returns:

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

### `entropy`

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

Shannon entropy in nats.

### `event_shape_tensor`

``````event_shape_tensor(name='event_shape_tensor')
``````

Shape of a single sample from a single batch as a 1-D int32 `Tensor`.

#### Args:

• `name`: name to give to the op

#### Returns:

• `event_shape`: `Tensor`.

### `forward_filter`

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

Run a Kalman filter over a provided sequence of outputs.

Note that the returned values `filtered_means`, `predicted_means`, and `observation_means` depend on the observed time series `x`, while the corresponding covariances are independent of the observed series; i.e., they depend only on the model itself. This means that the mean values have shape ```concat([sample_shape(x), batch_shape, [num_timesteps, {latent/observation}_size]])```, while the covariances have shape ```concat[(batch_shape, [num_timesteps, {latent/observation}_size, {latent/observation}_size]])```, which does not depend on the sample shape.

#### Args:

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

#### Returns:

• `log_likelihoods`: Per-timestep log marginal likelihoods ```log p(x_t | x_{:t-1})``` evaluated at the input `x`, as a `Tensor` of shape `sample_shape(x) + batch_shape + [num_timesteps].`
• `filtered_means`: Means of the per-timestep filtered marginal distributions p(zt | 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(zt | 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`

``````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`

``````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`

``````kl_divergence(
other,
name='kl_divergence'
)
``````

Computes the Kullback--Leibler divergence.

Denote this distribution (`self`) by `p` and the `other` distribution by `q`. Assuming `p, q` are absolutely continuous with respect to reference measure `r`, the KL divergence is defined as:

``````KL[p, q] = E_p[log(p(X)/q(X))]
= -int_F p(x) log q(x) dr(x) + int_F p(x) log p(x) dr(x)
= H[p, q] - H[p]
``````

where `F` denotes the support of the random variable `X ~ p`, `H[., .]` denotes (Shannon) cross entropy, and `H[.]` denotes (Shannon) entropy.

#### Returns:

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

### `latents_to_observations`

``````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`

``````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`

``````log_prob(
value,
name='log_prob',
**kwargs
)
``````

Log probability density/mass function.

Additional documentation from `LinearGaussianStateSpaceModel`:

##### `kwargs`:
• `mask`: optional bool-type `Tensor` with rightmost dimension `[num_timesteps]`; `True` values specify that the value of `x` at that timestep is masked, i.e., not conditioned on. Additional dimensions must match or be broadcastable to `self.batch_shape`; any further dimensions must match or be broadcastable to the sample shape of `x`. Default value: `None`.

#### Args:

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

#### Returns:

• `log_prob`: a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`.

### `log_survival_function`

``````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`

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

Mean.

### `mode`

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

Mode.

### `param_shapes`

``````param_shapes(
cls,
sample_shape,
name='DistributionParamShapes'
)
``````

Shapes of parameters given the desired shape of a call to `sample()`.

This is a class method that describes what key/value arguments are required to instantiate the given `Distribution` so that a particular shape is returned for that instance's call to `sample()`.

Subclasses should override class method `_param_shapes`.

#### Args:

• `sample_shape`: `Tensor` or python list/tuple. Desired shape of a call to `sample()`.
• `name`: name to prepend ops with.

#### Returns:

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

### `param_static_shapes`

``````param_static_shapes(
cls,
sample_shape
)
``````

param_shapes with static (i.e. `TensorShape`) shapes.

This is a class method that describes what key/value arguments are required to instantiate the given `Distribution` so that a particular shape is returned for that instance's call to `sample()`. Assumes that the sample's shape is known statically.

Subclasses should override class method `_param_shapes` to return constant-valued tensors when constant values are fed.

#### Args:

• `sample_shape`: `TensorShape` or python list/tuple. Desired shape of a call to `sample()`.

#### Returns:

`dict` of parameter name to `TensorShape`.

#### Raises:

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

### `posterior_marginals`

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

Run a Kalman smoother to return posterior mean and cov.

Note that the returned values `smoothed_means` depend on the observed time series `x`, while the `smoothed_covs` are independent of the observed series; i.e., they depend only on the model itself. This means that the mean values have shape ```concat([sample_shape(x), batch_shape, [num_timesteps, {latent/observation}_size]])```, while the covariances have shape ```concat[(batch_shape, [num_timesteps, {latent/observation}_size, {latent/observation}_size]])```, which does not depend on the sample shape.

This function only performs smoothing. If the user wants the intermediate values, which are returned by filtering pass `forward_filter`, one could get it by:

``````(log_likelihoods,
filtered_means, filtered_covs,
predicted_means, predicted_covs,
observation_means, observation_covs) = model.forward_filter(x)
smoothed_means, smoothed_covs = model.backward_smoothing_pass(x)
``````

where `x` is an observation sequence.

#### Args:

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

#### Returns:

• `smoothed_means`: Means of the per-timestep smoothed distributions over latent states, p(x{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(x{t} | x{:T}), as a Tensor of shape ```sample_shape(mask) + batch_shape + [num_timesteps, observation_size, observation_size]```. Note that the covariances depend only on the model and the mask, not on the data, so this may have fewer dimensions than `filtered_means`.

### `prob`

``````prob(
value,
name='prob',
**kwargs
)
``````

Probability density/mass function.

Additional documentation from `LinearGaussianStateSpaceModel`:

##### `kwargs`:
• `mask`: optional bool-type `Tensor` with rightmost dimension `[num_timesteps]`; `True` values specify that the value of `x` at that timestep is masked, i.e., not conditioned on. Additional dimensions must match or be broadcastable to `self.batch_shape`; any further dimensions must match or be broadcastable to the sample shape of `x`. Default value: `None`.

#### Args:

• `value`: `float` or `double` `Tensor`.
• `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`

``````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`

``````sample(
sample_shape=(),
seed=None,
name='sample',
**kwargs
)
``````

Generate samples of the specified shape.

Note that a call to `sample()` without arguments will generate a single sample.

#### Args:

• `sample_shape`: 0D or 1D `int32` `Tensor`. Shape of the generated samples.
• `seed`: Python integer seed for RNG
• `name`: name to give to the op.
• `**kwargs`: Named arguments forwarded to subclass implementation.

#### Returns:

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

### `stddev`

``````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`

``````survival_function(
value,
name='survival_function',
**kwargs
)
``````

Survival function.

Given random variable `X`, the survival function is defined:

``````survival_function(x) = P[X > x]
= 1 - P[X <= x]
= 1 - cdf(x).
``````

#### Args:

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

#### Returns:

`Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`.

### `variance`

``````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()`.