tfp.distributions.StudentTProcess

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

Inherits From: AutoCompositeTensorDistribution, Distribution, AutoCompositeTensor

tfp.distributions.StudentTProcess(
    df,
    kernel,
    index_points=None,
    mean_fn=None,
    observation_noise_variance=0.0,
    marginal_fn=None,
    cholesky_fn=None,
    jitter=1e-06,
    always_yield_multivariate_student_t=None,
    validate_args=False,
    allow_nan_stats=False,
    name='StudentTProcess'
)

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
import tensorflow.compat.v2 as tf
import tensorflow_probability 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.
kernel = psd_kernels.ExponentiatedQuadratic()

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.
kernel = psd_kernels.ExponentiatedQuadratic(
    amplitude=amplitude,
    length_scale=length_scale)

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

optimizer = tf.optimizers.Adam()

@tf.function
def optimize():
  with tf.GradientTape() as tape:
    loss = -tp.log_prob(observed_values)
  grads = tape.gradient(loss, tp.trainable_variables)
  optimizer.apply_gradients(zip(grads, tp.trainable_variables))
  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

Args
`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".

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

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 `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`.
`name_scope`	Returns a `tf.name_scope` instance for this class.
`non_trainable_variables`	Sequence of non-trainable variables owned by this module and its submodules. Note: this method uses reflection to find variables on the current instance and submodules. For performance reasons you may wish to cache the result of calling this method if you don't expect the return value to change.
`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`.
`submodules`	Sequence of all sub-modules. Submodules are modules which are properties of this module, or found as properties of modules which are properties of this module (and so on). `a = tf.Module()` `b = tf.Module()` `c = tf.Module()` `a.b = b` `b.c = c` `list(a.submodules) == [b, c]` `True` `list(b.submodules) == [c]` `True` `list(c.submodules) == []` `True`
`trainable_variables`	Sequence of trainable variables owned by this module and its submodules. Note: this method uses reflection to find variables on the current instance and submodules. For performance reasons you may wish to cache the result of calling this method if you don't expect the return value to change.
`validate_args`	Python `bool` indicating possibly expensive checks are enabled.
`variables`	Sequence of variables owned by this module and its submodules. Note: this method uses reflection to find variables on the current instance and submodules. For performance reasons you may wish to cache the result of calling this method if you don't expect the return value to change.

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.

Args
`other`	`tfp.distributions.Distribution` instance.
`name`	Python `str` prepended to names of ops created by this function.

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

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.`.

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`.

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`.

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

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.

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.

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.

tfp.distributions.StudentTProcess Stay organized with collections Save and categorize content based on your preferences.

Mathematical Details

Examples

Draw joint samples from a TP prior

Optimize kernel parameters via maximum marginal likelihood.

References

Args

Raises

Attributes

Methods

batch_shape_tensor

cdf

copy

covariance

cross_entropy

entropy

event_shape_tensor

experimental_default_event_space_bijector

experimental_fit

experimental_local_measure

experimental_sample_and_log_prob

get_marginal_distribution

is_scalar_batch

is_scalar_event

kl_divergence

log_cdf

log_prob

kwargs:

log_survival_function

mean

mode

param_shapes

param_static_shapes

parameter_properties

posterior_predictive

prob

quantile

sample

stddev

survival_function

unnormalized_log_prob

variance

with_name_scope

__getitem__

__iter__

tfp.distributions.StudentTProcess

`batch_shape_tensor`

`cdf`

`copy`

`covariance`

`cross_entropy`

`entropy`

`event_shape_tensor`

`experimental_default_event_space_bijector`

`experimental_fit`

`experimental_local_measure`

`experimental_sample_and_log_prob`

`get_marginal_distribution`

`is_scalar_batch`

`is_scalar_event`

`kl_divergence`

`log_cdf`

`log_prob`

`kwargs`:

`log_survival_function`

`mean`

`mode`

`param_shapes`

`param_static_shapes`

`parameter_properties`

`posterior_predictive`

`prob`

`quantile`

`sample`

`stddev`

`survival_function`

`unnormalized_log_prob`

`variance`

`with_name_scope`

`getitem`

`iter`