tfp.substrates.jax.distributions.StudentTProcessRegressionModel

StudentTProcessRegressionModel.

Inherits From: StudentTProcess, AutoCompositeTensorDistribution, Distribution

View aliases

Main aliases

tfp.experimental.substrates.jax.distributions.StudentTProcessRegressionModel

tfp.substrates.jax.distributions.StudentTProcessRegressionModel(
    df,
    kernel,
    index_points=None,
    observation_index_points=None,
    observations=None,
    observation_noise_variance=0.0,
    predictive_noise_variance=None,
    mean_fn=None,
    cholesky_fn=None,
    marginal_fn=None,
    always_yield_multivariate_student_t=None,
    validate_args=False,
    allow_nan_stats=False,
    name='StudentTProcessRegressionModel',
    _conditional_kernel=None,
    _conditional_mean_fn=None
)

This is an analogue of the GaussianProcessRegressionModel, except we use a Student-T process here (i.e. this represents the posterior predictive of a Student-T process, where we assume that the kernel hyperparameters and degrees of freedom are constants, and hence do optimizations in the constructor).

Specifically, we assume a Student-T Process prior, f ~ StP(df, m, k) along with a noise model whose mathematical implementation is similar to a Gaussian Process albeit whose interpretation is very different.

In particular, this noise model takes the following form:

  e ~ MVT(df + n, 0, 1 + v / df * (1 + f^T k(t, t)^-1 f^T))

which can be interpreted as Multivariate T noise whose covariance depends on the data fit term.

Using the conventions in the GaussianProcessRegressionModel class, we have the posterior predictive distribution as:

  (f(t) | t, x, f(x)) ~ MVT(df + n, loc, cov)

  where

  n is the number of observation points
  b = (y - loc)^T @ inv(k(x, x) + v * I) @ (y - loc)
  v = observation noise variance
  loc = k(t, x) @ inv(k(x, x) + v * I) @ (y - loc)
  cov = (df + b - 2) / (df + n - 2) * (
    k(t, t) - k(t, x) @ inv(k(x, x) + v * I) @ k(x, t))

Note that the posterior predictive mean is the same as a Gaussian Process, but the covariance is multiplied by a term that takes in to account observations.

This distribution does precomputaiton in the constructor by assuming observation_index_points, observations, kernel and observation_noise_variance are fixed, and that mean_fn is the zero function. We do these precomputations in a non-tape safe way.

References

[1]: Amar Shah, Andrew Gordon Wilson, Zoubin Ghahramani. Student-t Processes as Alternatives to Gaussian Processes https://arxiv.org/abs/1402.4306

[2]: Qingtao Tang, Yisen Wang, Shu-Tao Xia Student-T Process Regression with Dependent Student-t Noise https://www.ijcai.org/proceedings/2017/393

Args
df	Positive Floating-point Tensor representing the degrees of freedom. Must be greather than 2.
kernel	PositiveSemidefiniteKernel-like instance representing the StP's covariance function.
index_points	(Nested) Tensor representing finite collection, or batch of collections, of points in the index set over which the STP is defined. 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 an e-dimensional multivariate Student T. The batch shape must be broadcastable with kernel.batch_shape and any batch dims yielded by mean_fn.
observation_index_points	(Nested) Tensor representing finite collection, or batch of collections, of points in the index set for which some data has been observed. 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. [b1, ..., bB, e] must be broadcastable with the shape of observations, and [b1, ..., bB] must be broadcastable with the shapes of all other batched parameters (kernel.batch_shape, index_points, etc). The default value is None, which corresponds to the empty set of observations, and simply results in the prior predictive model (a STP with noise of variance predictive_noise_variance).
observations	float Tensor representing collection, or batch of collections, of observations corresponding to observation_index_points. Shape has the form [b1, ..., bB, e], which must be brodcastable with the batch and example shapes of observation_index_points. The batch shape [b1, ..., bB] must be broadcastable with the shapes of all other batched parameters (kernel.batch_shape, index_points, etc.).
observation_noise_variance	float Tensor representing the variance of the noise in the Normal likelihood distribution of the model. May be batched, in which case the batch shape must be broadcastable with the shapes of all other batched parameters (kernel.batch_shape, index_points, etc.). Default value: 0.
predictive_noise_variance	float Tensor representing the variance in the posterior predictive model. If None, we simply re-use observation_noise_variance for the posterior predictive noise. If set explicitly, however, we use this value. This allows us, for example, to omit predictive noise variance (by setting this to zero) to obtain noiseless posterior predictions of function values, conditioned on noisy observations.
mean_fn	Python callable that acts on index_points to produce a collection, or batch of collections, 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 the constant zero function.
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.
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_with_jitter_fn.
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: 'StudentTProcessRegressionModel'.
_conditional_kernel	Internal parameter -- do not use.
_conditional_mean_fn	Internal parameter -- do not use.

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.
cholesky_fn
df
dtype	The DType of Tensors 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_cholesky
observation_index_points
observation_noise_variance
observations
parameters	Dictionary of parameters used to instantiate this Distribution.
predictive_noise_variance
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

View source

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

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

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copy(
    **override_parameters_kwargs
)

Creates a deep copy of the distribution.

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

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

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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 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

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

Shannon entropy in nats.

event_shape_tensor

View source

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

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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 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

View source

@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

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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

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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 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 Tensors, 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

View source

get_marginal_distribution(
    index_points=None
)

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

View source

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

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

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 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

View source

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

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

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

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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

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

Mean.

mode

View source

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

Mode.

param_shapes

View source

@classmethod
param_shapes(
    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

View source

@classmethod
param_static_shapes(
    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.

parameter_properties

View source

@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.

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.ParameterPropertiesdict mapping constructor argument names toParameterProperties` instances.

Raises
NotImplementedError	if the distribution class does not implement _parameter_properties.

posterior_predictive

View source

posterior_predictive(
    observations, predictive_index_points=None, **kwargs
)

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.

Warning: This method assumes index_points is the only varying parameter (i.e. is a Variable / changes after initialization) and hence is not tape-safe.

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.

precompute_regression_model

View source

@staticmethod
precompute_regression_model(
    df,
    kernel,
    observation_index_points,
    observations,
    observations_is_missing=None,
    index_points=None,
    observation_noise_variance=0.0,
    predictive_noise_variance=None,
    mean_fn=None,
    cholesky_fn=None,
    always_yield_multivariate_student_t=None,
    validate_args=False,
    allow_nan_stats=False,
    name='PrecomputedStudentTProcessRegressionModel',
    _precomputed_divisor_matrix_cholesky=None,
    _precomputed_solve_on_observation=None
)

Returns a StudentTProcessRegressionModel with precomputed quantities.

This differs from the constructor by precomputing quantities associated with observations in a non-tape safe way. index_points is the only parameter that is allowed to vary (i.e. is a Variable / changes after initialization).

Specifically:

• We make observation_index_points and observations mandatory parameters.
• We precompute kernel(observation_index_points, observation_index_points) along with any other associated quantities relating to df, kernel, observations and observation_index_points.

A typical usecase would be optimizing kernel hyperparameters for a StudenTProcess, and computing the posterior predictive with respect to those optimized hyperparameters and observation / index-points pairs.

Warning: This method assumes index_points is the only varying parameter (i.e. is a Variable / changes after initialization) and hence is not tape-safe.

Args
df	Positive Floating-point Tensor representing the degrees of freedom. Must be greather than 2.
kernel	PositiveSemidefiniteKernel-like instance representing the StP's covariance function.
observation_index_points	(Nested) float Tensor representing finite collection, or batch of collections, of points in the index set for which some data has been observed. 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. [b1, ..., bB, e] must be broadcastable with the shape of observations, and [b1, ..., bB] must be broadcastable with the shapes of all other batched parameters (kernel.batch_shape, index_points, etc). The default value is None, which corresponds to the empty set of observations, and simply results in the prior predictive model (a StP with noise of variance predictive_noise_variance).
observations	float Tensor representing collection, or batch of collections, of observations corresponding to observation_index_points. Shape has the form [b1, ..., bB, e], which must be brodcastable with the batch and example shapes of observation_index_points. The batch shape [b1, ..., bB] must be broadcastable with the shapes of all other batched parameters (kernel.batch_shape, index_points, etc.). The default value is None, which corresponds to the empty set of observations, and simply results in the prior predictive model (a StP with noise of variance predictive_noise_variance).
observations_is_missing	bool Tensor of shape [..., e], representing a batch of boolean masks. When observations_is_missing is not None, the returned distribution is conditioned only on the observations for which the corresponding elements of observations_is_missing are True.
index_points	(Nested) float Tensor representing finite collection, or batch of collections, of points in the index set over which the StP 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 an e-dimensional multivariate Student T. The batch shape must be broadcastable with kernel.batch_shape and any batch dims yielded by mean_fn.
observation_noise_variance	float Tensor representing the variance of the noise in the Normal likelihood distribution of the model. May be batched, in which case the batch shape must be broadcastable with the shapes of all other batched parameters (kernel.batch_shape, index_points, etc.). Default value: 0.
predictive_noise_variance	float Tensor representing the variance in the posterior predictive model. If None, we simply re-use observation_noise_variance for the posterior predictive noise. If set explicitly, however, we use this value. This allows us, for example, to omit predictive noise variance (by setting this to zero) to obtain noiseless posterior predictions of function values, conditioned on noisy observations.
mean_fn	Python callable that acts on index_points to produce a collection, or batch of collections, 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 the constant zero function.
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.
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: 'PrecomputedStudentTProcessRegressionModel'.
_precomputed_divisor_matrix_cholesky	Internal parameter -- do not use.
_precomputed_solve_on_observation	Internal parameter -- do not use.

Returns An instance of StudentTProcessRegressionModel with precomputed quantities associated with observations.

prob

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

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

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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

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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 for details.
name	name to give to the op.
**kwargs	Named arguments forwarded to subclass implementation.

Returns
samples	a Tensor with prepended dimensions sample_shape.

stddev

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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

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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

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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

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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().

__getitem__

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__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.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__

View source

__iter__()