tfp.substrates.numpy.distributions.GaussianProcessRegressionModel

Posterior predictive distribution in a conjugate GP regression model.

Inherits From: `GaussianProcess`, `Distribution`

This class represents the distribution over function values at a set of points in some index set, conditioned on noisy observations at some other set of points. More specifically, we assume a Gaussian process prior, `f ~ GP(m, k)` with IID normal noise on observations of function values. In this model posterior inference can be done analytically. This `Distribution` is parameterized by

• the mean and covariance functions of the GP prior,
• the set of (noisy) observations and index points to which they correspond,
• the set of index points at which the resulting posterior predictive distribution over function values is defined,
• the observation noise variance,
• jitter, to compensate for numerical instability of Cholesky decomposition,

in addition to the usual params like `validate_args` and `allow_nan_stats`.

Mathematical Details

Gaussian process regression (GPR) assumes a Gaussian process (GP) prior and a normal likelihood as a generative model for data. Given GP mean function `m`, covariance kernel `k`, and observation noise variance `v`, we have

``````  f ~ GP(m, k)

iid
(y[i] | f, x[i])  ~  Normal(f(x[i]), v),   i = 1, ... , N
``````

where `y[i]` are the noisy observations of function values at points `x[i]`.

In practice, `f` is an infinite object (eg, a function over `R^n`) which can't be realized on a finite machine, but fortunately the marginal distribution over function values at a finite set of points is just a multivariate normal with mean and covariance given by the mean and covariance functions applied at our finite set of points (see [Rasmussen and Williams, 2006][1] for a more extensive discussion of these facts).

We spell out the generative model in detail below, but first, a digression on notation. In what follows we drop the indices on vectorial objects such as `x[i]`, it being implied that we are generally considering finite collections of index points and corresponding function values and noisy observations thereof. Thus `x` should be considered to stand for a collection of index points (indeed, themselves often vectorial). Furthermore:

• `f(x)` refers to the collection of function values at the index points in the collection `x`",
• `m(t)` refers to the collection of values of the mean function at the index points in the collection `t`, and
• `k(x, t)` refers to the matrix whose entries are values of the kernel function `k` at all pairs of index points from `x` and `t`.

With these conventions in place, we may write

``````  (f(x) | x) ~ MVN(m(x), k(x, x))

(y | f(x), x) ~ Normal(f(x), v)
``````

When we condition on observed data `y` at the points `x`, we can derive the posterior distribution over function values `f(x)` at those points. We can then compute the posterior predictive distribution over function values `f(t)` at a new set of points `t`, conditional on those observed data.

``````  (f(t) | t, x, y) ~ MVN(loc, cov)

where

loc = m(t) + k(t, x) @ inv(k(x, x) + v * I) @ (y - m(x))
cov = k(t, t) - k(t, x) @ inv(k(x, x) + v * I) @ k(x, t)
``````

where `I` is the identity matrix of appropriate dimension. Finally, the distribution over noisy observations at the new set of points `t` is obtained by adding IID noise from `Normal(0., observation_noise_variance)`.

Examples

Draw joint samples from the posterior predictive distribution in a GP

regression model

``````import numpy as np
from tensorflow_probability.python.internal.backend.numpy.compat import v2 as tf
import tensorflow_probability as tfp; tfp = tfp.substrates.numpy

tfb = tfp.bijectors
tfd = tfp.distributions
psd_kernels = tfp.math.psd_kernels

# Generate noisy observations from a known function at some random points.
observation_noise_variance = .5
f = lambda x: np.sin(10*x[..., 0]) * np.exp(-x[..., 0]**2)
observation_index_points = np.random.uniform(-1., 1., 50)[..., np.newaxis]
observations = (f(observation_index_points) +
np.random.normal(0., np.sqrt(observation_noise_variance)))

index_points = np.linspace(-1., 1., 100)[..., np.newaxis]

kernel = psd_kernels.MaternFiveHalves()

gprm = tfd.GaussianProcessRegressionModel(
kernel=kernel,
index_points=index_points,
observation_index_points=observation_index_points,
observations=observations,
observation_noise_variance=observation_noise_variance)

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

Above, we have used the kernel with default parameters, which are unlikely to be good. Instead, we can train the kernel hyperparameters on the data, as in the next example.

Optimize model parameters via maximum marginal likelihood

Here we learn the kernel parameters as well as the observation noise variance using gradient descent on the 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)

observation_index_points = np.random.uniform(-1., 1., 50)[..., np.newaxis]
observations = f(observation_index_points) + np.random.normal(0., .05, 50)

# Define a kernel with trainable parameters. Note we use TransformedVariable
# to apply a positivity constraint.
amplitude = tfp.util.TransformedVariable(
1., tfb.Exp(), dtype=tf.float64, name='amplitude')
length_scale = tfp.util.TransformedVariable(
1., tfb.Exp(), dtype=tf.float64, name='length_scale')

observation_noise_variance = tfp.util.TransformedVariable(
np.exp(-5), tfb.Exp(), name='observation_noise_variance')

# We'll use an unconditioned GP to train the kernel parameters.
gp = tfd.GaussianProcess(
kernel=kernel,
index_points=observation_index_points,
observation_noise_variance=observation_noise_variance)

@tf.function
def optimize():
loss = -gp.log_prob(observations)
return loss

# We can construct the posterior at a new set of `index_points` using the same
# kernel (with the same parameters, which we'll optimize below).
index_points = np.linspace(-1., 1., 100)[..., np.newaxis]
gprm = tfd.GaussianProcessRegressionModel(
kernel=kernel,
index_points=index_points,
observation_index_points=observation_index_points,
observations=observations,
observation_noise_variance=observation_noise_variance)

# First train the model, then draw and plot posterior samples.
for i in range(1000):
neg_log_likelihood_ = optimize()
if i % 100 == 0:
print("Step {}: NLL = {}".format(i, neg_log_likelihood_))

print("Final NLL = {}".format(neg_log_likelihood_))

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

import matplotlib.pyplot as plt
plt.scatter(np.squeeze(observation_index_points), observations)
plt.plot(np.stack([index_points[:, 0]]*10).T, samples.T, c='r', alpha=.2)
``````
Marginalization of model hyperparameters

Here we use TensorFlow Probability's MCMC functionality to perform marginalization of the model hyperparameters: kernel params as well as observation noise variance.

``````f = lambda x: np.sin(10*x[..., 0]) * np.exp(-x[..., 0]**2)
observation_index_points = np.random.uniform(-1., 1., 25)[..., np.newaxis]
observations = np.random.normal(f(observation_index_points), .05)

gaussian_process_model = tfd.JointDistributionSequential([
tfd.LogNormal(np.float64(0.), np.float64(1.)),
tfd.LogNormal(np.float64(0.), np.float64(1.)),
tfd.LogNormal(np.float64(0.), np.float64(1.)),
lambda noise_variance, length_scale, amplitude: tfd.GaussianProcess(
index_points=observation_index_points,
observation_noise_variance=noise_variance)
])

initial_chain_states = [
1e-1 * tf.ones([], dtype=np.float64, name='init_amplitude'),
1e-1 * tf.ones([], dtype=np.float64, name='init_length_scale'),
1e-1 * tf.ones([], dtype=np.float64, name='init_obs_noise_variance')
]

# Since HMC operates over unconstrained space, we need to transform the
# samples so they live in real-space.
unconstraining_bijectors = [
tfp.bijectors.Softplus(),
tfp.bijectors.Softplus(),
tfp.bijectors.Softplus(),
]

def unnormalized_log_posterior(*args):
return gaussian_process_model.log_prob(*args, x=observations)

num_results = 200
@tf.function
def run_mcmc():
return tfp.mcmc.sample_chain(
num_results=num_results,
num_burnin_steps=500,
num_steps_between_results=3,
current_state=initial_chain_states,
kernel=tfp.mcmc.TransformedTransitionKernel(
inner_kernel = tfp.mcmc.HamiltonianMonteCarlo(
target_log_prob_fn=unnormalized_log_posterior,
step_size=[np.float64(.15)],
num_leapfrog_steps=3),
bijector=unconstraining_bijectors),
trace_fn=lambda _, pkr: pkr.inner_results.is_accepted)
[
amplitudes,
length_scales,
observation_noise_variances
], is_accepted = run_mcmc()

print("Acceptance rate: {}".format(np.mean(is_accepted)))

# Now we can sample from the posterior predictive distribution at a new set
# of index points.
index_points = np.linspace(-1., 1., 200)[..., np.newaxis]
gprm = tfd.GaussianProcessRegressionModel(
# Batch of `num_results` kernels parameterized by the MCMC samples.
index_points=index_points,
observation_index_points=observation_index_points,
observations=observations,
observation_noise_variance=observation_noise_variances)
samples = gprm.sample()

# Plot posterior samples and their mean, target function, and observations.
plt.plot(np.stack([index_points[:, 0]]*num_results).T,
samples.numpy().T,
c='r',
alpha=.01)
plt.plot(index_points[:, 0], np.mean(samples, axis=0), c='k')
plt.plot(index_points[:, 0], f(index_points))
plt.scatter(observation_index_points[:, 0], observations)
``````

References

[1]: Carl Rasmussen, Chris Williams. Gaussian Processes For Machine Learning, 2006.

`kernel` `PositiveSemidefiniteKernel`-like instance representing the GP's covariance function.
`index_points` `float` `Tensor` representing finite collection, or batch of collections, of points in the index set over which the GP 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 an `e`-dimensional multivariate normal. The batch shape must be broadcastable with `kernel.batch_shape` and any batch dims yielded by `mean_fn`.
`observation_index_points` `float` `Tensor` representing finite collection, or batch of collections, of points in the index set for which some data has been observed. 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. `[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 GP 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 GP with noise of variance `predictive_noise_variance`).
`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 `Tensor` of shape `[b1, ..., bB, f1, ..., fF]` and returns a `Tensor` whose shape is broadcastable with `[b1, ..., bB]`. 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.
`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_normal` If `False` (the default), we produce a scalar `Normal` distribution when the number of `index_points` is statically known to be `1`. If `True`, we avoid this behavior, ensuring that the event shape will retain the `1` from `index_points`.
`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: 'GaussianProcessRegressionModel'.
`_conditional_kernel` Internal parameter -- do not use.
`_conditional_mean_fn` Internal parameter -- do not use.

`ValueError` if either

• only one of `observations` and `observation_index_points` is given, or
• `mean_fn` is not `None` and 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`

`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_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

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

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`

View source

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`

View source

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`

View source

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`.

`other` types with built-in registrations: `MultivariateNormalDiag`, `MultivariateNormalDiagPlusLowRank`, `MultivariateNormalFullCovariance`, `MultivariateNormalLinearOperator`, `MultivariateNormalTriL`, `Normal`

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

Shannon entropy in nats.

`event_shape_tensor`

View source

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`

View source

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

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`

View source

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`

View source

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`

View source

Compute the marginal of this GP over function values at `index_points`.

Args
`index_points` `float` `Tensor` representing finite (batch of) vector(s) of points in the index set over which the GP 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 normal. The batch shape must be broadcastable with `kernel.batch_shape` and any batch dims yielded by `mean_fn`.

Returns
`marginal` a `Normal` or `MultivariateNormalLinearOperator` distribution, according to whether `index_points` consists of one or many index points, respectively.

`is_scalar_batch`

View source

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

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

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.

`other` types with built-in registrations: `MultivariateNormalDiag`, `MultivariateNormalDiagPlusLowRank`, `MultivariateNormalFullCovariance`, `MultivariateNormalLinearOperator`, `MultivariateNormalTriL`, `Normal`

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 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 probability density/mass function.

Additional documentation from `GaussianProcess`:

`kwargs`:
• `index_points`: optional `float` `Tensor` representing a finite (batch of) of points in the index set over which this GP is defined. The shape has the form `[b1, ..., bB, e, f1, ..., fF]` where `F` is the number of feature dimensions and must equal `self.kernel.feature_ndims` and `e` is the number of index points in each batch. Ultimately, this distribution corresponds to an `e`-dimensional multivariate normal. The batch shape must be broadcastable with `kernel.batch_shape` and any batch dims yielded by `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.

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`.

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Mean.

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Mode.

`param_shapes`

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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`

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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`

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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`

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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 `GaussianProcessRegressionModel.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` `Tensor` representing finite collection, or batch of collections, of points in the index set over which the GP 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 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
`gprm` An instance of `Distribution` that represents the posterior predictive.

`precompute_regression_model`

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Returns a GaussianProcessRegressionModel 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 the `kernel`, `observations` and `observation_index_points`.

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

Args
`kernel` `PositiveSemidefiniteKernel`-like instance representing the GP's covariance function.
`observation_index_points` `float` `Tensor` representing finite collection, or batch of collections, of points in the index set for which some data has been observed. 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. `[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 GP 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 GP with noise of variance `predictive_noise_variance`).
`observations_mask` Deprecated. Prefer `observations_is_missing`. `bool` `Tensor` of shape `[..., e]`, representing a batch of boolean masks. When `observation_masks` is not `None`, the returned distribution is conditioned only on the observations for which the corresponding elements of `observations_masks` are `True`.
`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` `float` `Tensor` representing finite collection, or batch of collections, of points in the index set over which the GP 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 an `e`-dimensional multivariate normal. 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 `Tensor` of shape `[b1, ..., bB, f1, ..., fF]` and returns a `Tensor` whose shape is broadcastable with `[b1, ..., bB]`. 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.
`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: 'PrecomputedGaussianProcessRegressionModel'.

Returns An instance of `GaussianProcessRegressionModel` with precomputed quantities associated with observations.

`prob`

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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 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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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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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.

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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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.

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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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.

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