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tfp.distributions.StudentTProcess

Class StudentTProcess

Inherits From: MultivariateStudentTLinearOperator

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) +
     jitter * 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,
  • jitter is added to the diagonal to ensure positive definiteness up to machine precision (otherwise Cholesky-decomposition is prone to failure),
  • eye(N) is an N-by-N identity matrix.

Examples

Draw joint samples from a TP prior
import numpy as np
import tensorflow as tf
import tensorflow_probability as tfp

tfd = tfp.distributions
psd_kernels = tfp.positive_semidefinite_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(df=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 = noise_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=tf.get_variable('amplitude', np.float32)
length_scale=tf.get_variable('length_scale', np.float32)

# Define a kernel with trainable parameters.
kernel = psd_kernels.ExponentiatedQuadratic(
    amplitude=tf.nn.softplus(amplitude),
    length_scale=tf.nn.softplus(length_scale))

tp = tfp.StudentTProcess(df=3., kernel, observed_index_points)
neg_log_likelihood = -tp.log_prob(observed_values)

optimize = tf.train.AdamOptimize().minimize(neg_log_likelihood)

with tf.Session() as sess:
  sess.run(tf.global_variables_initializer())

  for i in range(1000):
    _, nll_ = sess.run([optimize, nll])
    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

__init__

__init__(
    df,
    kernel,
    index_points,
    mean_fn=None,
    jitter=1e-06,
    validate_args=False,
    allow_nan_stats=False,
    name='StudentTProcess'
)

Instantiate a StudentTProcess Distribution.

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: float Tensor representing finite (batch of) vector(s) of points in the index set over which the TP is defined. Shape has the form [b1, ..., bB, e, f1, ..., fF] where F is the number of feature dimensions and must equal kernel.feature_ndims 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 Tensor of shape [b1, ..., bB, f1, ..., fF] and returns a Tensor whose shape is broadcastable with [b1, ..., bB]. Default value: None implies constant zero function.
  • jitter: float scalar Tensor added to the diagonal of the covariance matrix to ensure positive definiteness of the covariance matrix. Default value: 1e-6.
  • 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.

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.

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.

Returns:

  • event_shape: TensorShape, possibly unknown.

index_points

jitter

kernel

loc

The location parameter of the distribution.

loc applies an elementwise shift to the distribution.

X ~ MultivariateT(loc=0, scale=1)   # Identity scale, zero shift.
Y = scale @ X + loc

Returns:

The loc Tensor.

mean_fn

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.

scale

The scale parameter of the distribution.

scale applies an affine scale to the distribution.

X ~ MultivariateT(loc=0, scale=1)   # Identity scale, zero shift.
Y = scale @ X + loc

Returns:

The scale LinearOperator.

validate_args

Python bool indicating possibly expensive checks are enabled.

Methods

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'
)

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.

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')

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.

Additional documentation from MultivariateStudentTLinearOperator:

The covariance for Multivariate Student's t equals

scale @ scale.T * df / (df - 2), when df > 2
infinity, when 1 < df <= 2
NaN, when df <= 1

If self.allow_nan_stats=False, then an exception will be raised rather than returning NaN.

Args:

  • name: Python str prepended to names of ops created by this function.

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.

Args:

Returns:

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

entropy

entropy(name='entropy')

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.

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.

Args:

Returns:

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

log_cdf

log_cdf(
    value,
    name='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.

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'
)

Log probability density/mass function.

Args:

  • value: float or double Tensor.
  • name: Python str prepended to names of ops created by this function.

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'
)

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.

Returns:

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

mean

mean(name='mean')

Mean.

Additional documentation from MultivariateStudentTLinearOperator:

The mean of Student's T equals loc if df > 1, otherwise it is NaN. If self.allow_nan_stats=False, then an exception will be raised rather than returning NaN.

mode

mode(name='mode')

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.

prob

prob(
    value,
    name='prob'
)

Probability density/mass function.

Args:

  • value: float or double Tensor.
  • name: Python str prepended to names of ops created by this function.

Returns:

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

quantile

quantile(
    value,
    name='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.

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'
)

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.

Returns:

  • samples: a Tensor with prepended dimensions sample_shape.

stddev

stddev(name='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.

Additional documentation from MultivariateStudentTLinearOperator:

The standard deviation for Student's T equals

sqrt(diag(scale @ scale.T)) * df / (df - 2), when df > 2
infinity, when 1 < df <= 2
NaN, when df <= 1

Args:

  • name: Python str prepended to names of ops created by this function.

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'
)

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.

Returns:

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

variance

variance(name='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.

Additional documentation from MultivariateStudentTLinearOperator:

The variance for Student's T equals

diag(scale @ scale.T) * df / (df - 2), when df > 2
infinity, when 1 < df <= 2
NaN, when df <= 1

If self.allow_nan_stats=False, then an exception will be raised rather than returning NaN.

Args:

  • name: Python str prepended to names of ops created by this function.

Returns:

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