# oryx.distributions.StudentTProcess

Marginal distribution of a Student's T process at finitely many points.

A Student's T process (TP) is an indexed collection of random variables, any finite collection of which are jointly Multivariate Student's T. While this definition applies to finite index sets, it is typically implicit that the index set is infinite; in applications, it is often some finite dimensional real or complex vector space. In such cases, the TP may be thought of as a distribution over (real- or complex-valued) functions defined over the index set.

Just as Student's T distributions are fully specified by their degrees of freedom, location and scale, a Student's T process can be completely specified by a degrees of freedom parameter, mean function and covariance function. Let `S` denote the index set and `K` the space in which each indexed random variable takes its values (again, often R or C). The mean function is then a map `m: S -> K`, and the covariance function, or kernel, is a positive-definite function `k: (S x S) -> K`. The properties of functions drawn from a TP are entirely dictated (up to translation) by the form of the kernel function.

This `Distribution` represents the marginal joint distribution over function values at a given finite collection of points `[x[1], ..., x[N]]` from the index set `S`. By definition, this marginal distribution is just a multivariate Student's T distribution, whose mean is given by the vector `[ m(x[1]), ..., m(x[N]) ]` and whose covariance matrix is constructed from pairwise applications of the kernel function to the given inputs:

``````    | k(x[1], x[1])    k(x[1], x[2])  ...  k(x[1], x[N]) |
| k(x[2], x[1])    k(x[2], x[2])  ...  k(x[2], x[N]) |
|      ...              ...                 ...      |
| k(x[N], x[1])    k(x[N], x[2])  ...  k(x[N], x[N]) |
``````

For this to be a valid covariance matrix, it must be symmetric and positive definite; hence the requirement that `k` be a positive definite function (which, by definition, says that the above procedure will yield PD matrices).

Note also we use a parameterization as suggested in [1], which requires `df` to be greater than 2. This allows for the covariance for any finite dimensional marginal of the TP (a multivariate Student's T distribution) to just be the PD matrix generated by the kernel.

#### Mathematical Details

The probability density function (pdf) is a multivariate Student's T whose parameters are derived from the TP's properties:

``````pdf(x; df, index_points, mean_fn, kernel) = MultivariateStudentT(df, loc, K)
K = (df - 2) / df  * (kernel.matrix(index_points, index_points) +
observation_noise_variance * eye(N))
loc = (x - mean_fn(index_points))^T @ K @ (x - mean_fn(index_points))
``````

where:

• `df` is the degrees of freedom parameter for the TP.
• `index_points` are points in the index set over which the TP is defined,
• `mean_fn` is a callable mapping the index set to the TP's mean values,
• `kernel` is `PositiveSemidefiniteKernel`-like and represents the covariance function of the TP,
• `observation_noise_variance` is a term added to the diagonal of the kernel matrix. In the limit of `df` to `inf`, this represents the observation noise of a gaussian likelihood.
• `eye(N)` is an N-by-N identity matrix.

#### Examples

##### Draw joint samples from a TP prior
``````import numpy as np
from tensorflow_probability.python.internal.backend.jax.compat import v2 as tf
from tensorflow_probability.substrates import numpy as tfp

tfd = tfp.distributions
psd_kernels = tfp.math.psd_kernels

num_points = 100
# Index points should be a collection (100, here) of feature vectors. In this
# example, we're using 1-d vectors, so we just need to reshape the output from
# np.linspace, to give a shape of (100, 1).
index_points = np.expand_dims(np.linspace(-1., 1., num_points), -1)

# Define a kernel with default parameters.

tp = tfd.StudentTProcess(3., kernel, index_points)

samples = tp.sample(10)
# ==> 10 independently drawn, joint samples at `index_points`

noisy_tp = tfd.StudentTProcess(
df=3.,
kernel=kernel,
index_points=index_points)
noisy_samples = noisy_tp.sample(10)
# ==> 10 independently drawn, noisy joint samples at `index_points`
``````
##### Optimize kernel parameters via maximum marginal likelihood.
``````# Suppose we have some data from a known function. Note the index points in
# general have shape `[b1, ..., bB, f1, ..., fF]` (here we assume `F == 1`),
# so we need to explicitly consume the feature dimensions (just the last one
# here).
f = lambda x: np.sin(10*x[..., 0]) * np.exp(-x[..., 0]**2)
observed_index_points = np.expand_dims(np.random.uniform(-1., 1., 50), -1)
# Squeeze to take the shape from [50, 1] to [50].
observed_values = f(observed_index_points)

amplitude = tfp.util.TransformedVariable(
1., tfp.bijectors.Softplus(), dtype=np.float64, name='amplitude')
length_scale = tfp.util.TransformedVariable(
1., tfp.bijectors.Softplus(), dtype=np.float64, name='length_scale')

# Define a kernel with trainable parameters.
amplitude=amplitude,
length_scale=length_scale)

tp = tfd.StudentTProcess(3., kernel, observed_index_points)

@tf.function
def optimize():
loss = -tp.log_prob(observed_values)
return loss

for i in range(1000):
nll = optimize()
if i % 100 == 0:
print("Step {}: NLL = {}".format(i, nll))
print("Final NLL = {}".format(nll))
``````

#### References

[1]: Amar Shah, Andrew Gordon Wilson, and Zoubin Ghahramani. Student-t Processes as Alternatives to Gaussian Processes. In Artificial Intelligence and Statistics, 2014. https://www.cs.cmu.edu/~andrewgw/tprocess.pdf

`df` Positive Floating-point `Tensor` representing the degrees of freedom. Must be greater than 2.
`kernel` `PositiveSemidefiniteKernel`-like instance representing the TP's covariance function.
`index_points` (Nested) `float` `Tensor` representing finite (batch of) vector(s) of points in the index set over which the TP is defined. Shape (or shape of each nested component) has the form `[b1, ..., bB, e, f1, ..., fF]` where `F` is the number of feature dimensions and must equal `kernel.feature_ndims` (or its corresponding nested component) and `e` is the number (size) of index points in each batch. Ultimately this distribution corresponds to a `e`-dimensional multivariate Student's T. The batch shape must be broadcastable with `kernel.batch_shape` and any batch dims yielded by `mean_fn`.
`mean_fn` Python `callable` that acts on `index_points` to produce a (batch of) vector(s) of mean values at `index_points`. Takes a (nested) `Tensor` of shape `[b1, ..., bB, e, f1, ..., fF]` and returns a `Tensor` whose shape is broadcastable with `[b1, ..., bB, e]`. Default value: `None` implies constant zero function.
`observation_noise_variance` `float` `Tensor` representing (batch of) scalar variance(s) of the noise in the Normal likelihood distribution of the model. If batched, the batch shape must be broadcastable with the shapes of all other batched parameters (`kernel.batch_shape`, `index_points`, etc.). Default value: `0.`
`marginal_fn` A Python callable that takes a location, covariance matrix, optional `validate_args`, `allow_nan_stats` and `name` arguments, and returns a multivariate Student T subclass of `tfd.Distribution`. Default value: `None`, in which case a Cholesky-factorizing function is created using `make_cholesky_factored_marginal_fn` and the `jitter` argument.
`cholesky_fn` Callable which takes a single (batch) matrix argument and returns a Cholesky-like lower triangular factor. Default value: `None`, in which case `make_cholesky_with_jitter_fn` is used with the `jitter` parameter. At most one of `cholesky_fn` and `marginal_fn` should be set.
`jitter` `float` scalar `Tensor` added to the diagonal of the covariance matrix to ensure positive definiteness of the covariance matrix. This argument is ignored if `cholesky_fn` is set. Default value: `1e-6`.
`always_yield_multivariate_student_t` Deprecated and ignored.
`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`.
`allow_nan_stats` Python `bool`, default `True`. When `True`, statistics (e.g., mean, mode, variance) use the value "`NaN`" to indicate the result is undefined. When `False`, an exception is raised if one or more of the statistic's batch members are undefined. Default value: `False`.
`name` Python `str` name prefixed to Ops created by this class. Default value: "StudentTProcess".

`ValueError` if `mean_fn` is not `None` and is not callable.

`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.

`cholesky_fn`

`df`

`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_shard_axis_names` The list or structure of lists of active shard axis names.
`index_points`

`jitter`

`kernel`

`marginal_fn`

`mean_fn`

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

`parameters` Dictionary of parameters used to instantiate this `Distribution`.
`reparameterization_type` Describes how samples from the distribution are reparameterized.

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

`trainable_variables`

`validate_args` Python `bool` indicating possibly expensive checks are enabled.
`variables`

## Methods

### `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`

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`

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.

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`

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`

Shannon entropy in nats.

### `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`

Bijector mapping the reals (R**n) to the event space of the distribution.

Distributions with continuous support may implement `_default_event_space_bijector` which returns a subclass of `tfp.bijectors.Bijector` that maps R**n to the distribution's event space. For example, the default bijector for the `Beta` distribution is `tfp.bijectors.Sigmoid()`, which maps the real line to `[0, 1]`, the support of the `Beta` distribution. The default bijector for the `CholeskyLKJ` distribution is `tfp.bijectors.CorrelationCholesky`, which maps R^(k * (k-1) // 2) to the submanifold of k x k lower triangular matrices with ones along the diagonal.

The purpose of `experimental_default_event_space_bijector` is to enable gradient descent in an unconstrained space for Variational Inference and Hamiltonian Monte Carlo methods. Some effort has been made to choose bijectors such that the tails of the distribution in the unconstrained space are between Gaussian and Exponential.

For distributions with discrete event space, or for which TFP currently lacks a suitable bijector, this function returns `None`.

Args
`*args` Passed to implementation `_default_event_space_bijector`.
`**kwargs` Passed to implementation `_default_event_space_bijector`.

Returns
`event_space_bijector` `Bijector` instance or `None`.

### `experimental_fit`

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`

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`

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

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

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

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

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

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

### `get_marginal_distribution`

Compute the marginal over function values at `index_points`.

Args
`index_points` `float` (nested) `Tensor` representing finite (batch of) vector(s) of points in the index set over which the STP is defined. Shape (or the shape of each nested component) has the form `[b1, ..., bB, e, f1, ..., fF]` where `F` is the number of feature dimensions and must equal `kernel.feature_ndims` (or its corresponding nested component) and `e` is the number (size) of index points in each batch. Ultimately this distribution corresponds to a `e`-dimensional multivariate student t. The batch shape must be broadcastable with `kernel.batch_shape` and any batch dims yielded by `mean_fn`.

Returns
`marginal` a Student T distribution with vector event shape.

### `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`

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`

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.

### `log_cdf`

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 probability density/mass function.

Additional documentation from `StudentTProcess`:

##### `kwargs`:
• `index_points`: optional `float` `Tensor` representing a finite (batch of) of points in the index set over which this STP is defined. The shape (or shape of each nested component) has the form `[b1, ..., bB, e,f1, ..., fF]` where `F` is the number of feature dimensions and must equal `self.kernel.feature_ndims` (or its corresponding nested component) and `e` is the number of index points in each batch. Ultimately, this distribution corresponds to an `e`-dimensional multivariate Student T. The batch shape must be broadcastable with `kernel.batch_shape` and any batch dims yieldedby `mean_fn`. If not specified, `self.index_points` is used. Default value: `None`.
• `is_missing`: optional `bool` `Tensor` of shape `[..., e]`, where `e` is the number of index points in each batch. Represents a batch of Boolean masks. When `is_missing` is not `None`, the returned log-prob is for the marginal distribution, in which all dimensions for which `is_missing` is `True` have been marginalized out. The batch dimensions of `is_missing` must broadcast with the sample and batch dimensions of `value` and of this `Distribution`. 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.

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.

Mode.

### `param_shapes`

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

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

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

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

Returns
`dict` of parameter name to `TensorShape`.

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

### `parameter_properties`

Returns a dict mapping constructor arg names to property annotations.

This dict should include an entry for each of the distribution's `Tensor`-valued constructor arguments.

Distribution subclasses are not required to implement `_parameter_properties`, so this method may raise `NotImplementedError`. Providing a `_parameter_properties` implementation enables several advanced features, including:

• Distribution batch slicing (`sliced_distribution = distribution[i:j]`).
• Automatic inference of `_batch_shape` and `_batch_shape_tensor`, which must otherwise be computed explicitly.
• Automatic instantiation of the distribution within TFP's internal property tests.
• Automatic construction of 'trainable' instances of the distribution using appropriate bijectors to avoid violating parameter constraints. This enables the distribution family to be used easily as a surrogate posterior in variational inference.

In the future, parameter property annotations may enable additional functionality; for example, returning Distribution instances from `tf.vectorized_map`.

Args
`dtype` Optional float `dtype` to assume for continuous-valued parameters. Some constraining bijectors require advance knowledge of the dtype because certain constants (e.g., `tfb.Softplus.low`) must be instantiated with the same dtype as the values to be transformed.
`num_classes` Optional `int` `Tensor` number of classes to assume when inferring the shape of parameters for categorical-like distributions. Otherwise ignored.

Returns
`parameter_properties` A `str ->`tfp.python.internal.parameter_properties.ParameterProperties`dict mapping constructor argument names to`ParameterProperties` instances.

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

### `posterior_predictive`

Return the posterior predictive distribution associated with this distribution.

Returns the posterior predictive distribution `p(Y' | X, Y, X')` where:

• `X'` is `predictive_index_points`
• `X` is `self.index_points`.
• `Y` is `observations`.

This is equivalent to using the `StudentTProcessRegressionModel.precompute_regression_model` method.

Args
`observations` `float` `Tensor` representing collection, or batch of collections, of observations corresponding to `self.index_points`. Shape has the form `[b1, ..., bB, e]`, which must be broadcastable with the batch and example shapes of `self.index_points`. The batch shape `[b1, ..., bB]` must be broadcastable with the shapes of all other batched parameters
`predictive_index_points` `float` (nested) `Tensor` representing finite collection, or batch of collections, of points in the index set over which the TP is defined. Shape (or shape of each nested component) has the form `[b1, ..., bB, e, f1, ..., fF]` where `F` is the number of feature dimensions and must equal `kernel.feature_ndims` (or its corresponding nested component) and `e` is the number (size) of predictive index points in each batch. The batch shape must be broadcastable with this distributions `batch_shape`. Default value: `None`.
`**kwargs` Any other keyword arguments to pass / override.

Returns
`stprm` An instance of `Distribution` that represents the posterior predictive.

### `prob`

Probability density/mass function.

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 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`

Generate samples of the specified shape.

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

Args
`sample_shape` 0D or 1D `int32` `Tensor`. Shape of the generated samples.
`seed` PRNG seed; see `tfp.random.sanitize_seed` for details.
`name` name to give to the op.
`**kwargs` Named arguments forwarded to subclass implementation.

Returns
`samples` a `Tensor` with prepended dimensions `sample_shape`.

### `stddev`

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.

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`

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`

Variance.

Variance is defined as,

``````Var = E[(X - E[X])**2]
``````

where `X` is the random variable associated with this distribution, `E` denotes expectation, and `Var.shape = batch_shape + event_shape`.

Args
`name` Python `str` prepended to names of ops created by this function.
`**kwargs` Named arguments forwarded to subclass implementation.

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

### `__getitem__`

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

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

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

Args
`slices` slices from the [] operator

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

### `__iter__`

[]
[]
{ "last_modified": "Last updated 2024-05-23 UTC.", "state": "" }