Posterior predictive of a variational Gaussian process.
Inherits From: GaussianProcess
, Distribution
oryx.distributions.VariationalGaussianProcess(
kernel, index_points, inducing_index_points,
variational_inducing_observations_loc, variational_inducing_observations_scale,
mean_fn=None, observation_noise_variance=None, predictive_noise_variance=None,
jitter=1e06, validate_args=False, allow_nan_stats=False,
name='VariationalGaussianProcess'
)
This distribution implements the variational Gaussian process (VGP), as
described in [Titsias, 2009][1] and [Hensman, 2013][2]. The VGP is an
inducing pointbased approximation of an exact GP posterior
(see Mathematical Details, below). Ultimately, this Distribution class
represents a marginal distrbution over function values at a
collection of index_points
. It is parameterized by
 a kernel function,
 a mean function,
 the (scalar) observation noise variance of the normal likelihood,
 a set of index points,
 a set of inducing index points, and
 the parameters of the (fullrank, Gaussian) variational posterior distribution over function values at the inducing points, conditional on some observations.
A VGP is "trained" by selecting any kernel parameters, the locations of the
inducing index points, and the variational parameters. [Titsias, 2009][1] and
[Hensman, 2013][2] describe a variational lower bound on the marginal log
likelihood of observed data, which this class offers through the
variational_loss
method (this is the negative lower bound, for convenience
when plugging into a TF Optimizer's minimize
function).
Training may be done in minibatches.
[Titsias, 2009][1] describes a closed form for the optimal variational
parameters, in the case of sufficiently small observational data (ie,
small enough to fit in memory but big enough to warrant approximating the GP
posterior). A method to compute these optimal parameters in terms of the full
observational data set is provided as a staticmethod,
optimal_variational_posterior
. It returns a
MultivariateNormalLinearOperator
instance with optimal location and
scale parameters.
Mathematical Details
Notation
We will in general be concerned about three collections of index points, and it'll be good to give them names:
x[1], ..., x[N]
: observation index points  locations of our observed data.z[1], ..., z[M]
: inducing index points  locations of the "summarizing" inducing pointst[1], ..., t[P]
: predictive index points  locations where we are making posterior predictions based on observations and the variational parameters.
To lighten notation, we'll use X, Z, T
to denote the above collections.
Similarly, we'll denote by f(X)
the collection of function values at each of
the x[i]
, and by Y
, the collection of (noisy) observed data at each x[i].
We'll denote kernel matrices generated from pairs of index points as
K_tt,
K_xt,
K_tz`, etc, e.g.,
 k(t[1], z[1]) k(t[1], z[2]) ... k(t[1], z[M]) 
K_tz =  k(t[2], z[1]) k(t[2], z[2]) ... k(t[2], z[M]) 
 ... ... ... 
 k(t[P], z[1]) k(t[P], z[2]) ... k(t[P], z[M]) 
Preliminaries
A Gaussian process is an indexed collection of random variables, any finite
collection of which are jointly Gaussian. Typically, the index set is some
finitedimensional, real vector space, and indeed we make this assumption in
what follows. The GP may then be thought of as a distribution over functions
on the index set. Samples from the GP are functions on the whole index set;
these can't be represented in finite compute memory, so one typically works
with the marginals at a finite collection of index points. The properties of
the GP are entirely determined by its mean function m
and covariance
function k
. The generative process, assuming a meanzero normal likelihood
with stddev sigma
, is
f ~ GP(m, k)
Y  f(X) ~ Normal(f(X), sigma), i = 1, ... , N
In finite terms (ie, marginalizing out all but a finite number of f(X)'sigma), we can write
f(X) ~ MVN(loc=m(X), cov=K_xx)
Y  f(X) ~ Normal(f(X), sigma), i = 1, ... , N
Posterior inference is possible in analytical closed form but becomes intractible as data sizes get large. See [Rasmussen, 2006][3] for details.
The VGP
The VGP is an inducing pointbased approximation of an exact GP posterior, where two approximating assumptions have been made:
 function values at noninducing points are mutually independent conditioned on function values at the inducing points,
the (expensive) posterior over function values at inducing points conditional on observations is replaced with an arbitrary (learnable) fullrank Gaussian distribution,
q(f(Z)) = MVN(loc=m, scale=S),
where
m
andS
are parameters to be chosen by optimizing an evidence lower bound (ELBO).
The posterior predictive distribution becomes
q(f(T)) = integral df(Z) p(f(T)  f(Z)) q(f(Z))
= MVN(loc = A @ m, scale = B^(1/2))
where
A = K_tz @ K_zz^1
B = K_tt  A @ (K_zz  S S^T) A^T
The approximate posterior predictive distribution q(f(T))
is what the
VariationalGaussianProcess
class represents.
Model selection in this framework entails choosing the kernel parameters, inducing point locations, and variational parameters. We do this by optimizing a variational lower bound on the marginal log likelihood of observed data. The lower bound takes the following form (see [Titsias, 2009][1] and [Hensman, 2013][2] for details on the derivation):
L(Z, m, S, Y) = (
MVN(loc=(K_zx @ K_zz^1) @ m, scale_diag=sigma).log_prob(Y) 
(Tr(K_xx  K_zx @ K_zz^1 @ K_xz) +
Tr(S @ S^T @ K_zz^1 @ K_zx @ K_xz @ K_zz^1)) / (2 * sigma^2) 
KL(q(f(Z))  p(f(Z))))
where in the final KL term, p(f(Z))
is the GP prior on inducing point
function values. This variational lower bound can be computed on minibatches
of the full data set (X, Y)
. A method to compute the negative variational
lower bound is implemented as VariationalGaussianProcess.variational_loss
.
Optimal variational parameters
As described in [Titsias, 2009][1], a closed form optimum for the variational
location and scale parameters, m
and S
, can be computed when the
observational data are not prohibitively voluminous. The
optimal_variational_posterior
function to computes the optimal variational
posterior distribution over inducing point function values in terms of the GP
parameters (mean and kernel functions), inducing point locations, observation
index points, and observations. Note that the inducing index point locations
must still be optimized even when these parameters are known functions of the
inducing index points. The optimal parameters are computed as follows:
C = sigma^2 (K_zz + K_zx @ K_xz)^1
optimal Gaussian covariance: K_zz @ C @ K_zz
optimal Gaussian location: sigma^2 K_zz @ C @ K_zx @ Y
Usage Examples
Here's an example of defining and training a VariationalGaussianProcess on some toy generated data.
import matplotlib.pyplot as plt
import numpy as np
from tensorflow_probability.python.internal.backend.jax.compat import v2 as tf
import tensorflow_probability as tfp; tfp = tfp.substrates.jax
tf.enable_v2_behavior()
tfb = tfp.bijectors
tfd = tfp.distributions
tfk = tfp.math.psd_kernels
# We'll use double precision throughout for better numerics.
dtype = np.float64
# Generate noisy data from a known function.
f = lambda x: np.exp(x[..., 0]**2 / 20.) * np.sin(1. * x[..., 0])
true_observation_noise_variance_ = dtype(1e1) ** 2
num_training_points_ = 100
x_train_ = np.concatenate(
[np.random.uniform(6., 0., [num_training_points_ // 2 , 1]),
np.random.uniform(1., 10., [num_training_points_ // 2 , 1])],
axis=0).astype(dtype)
y_train_ = (f(x_train_) +
np.random.normal(
0., np.sqrt(true_observation_noise_variance_),
[num_training_points_]).astype(dtype))
# Create kernel with trainable parameters, and trainable observation noise
# variance variable. Each of these is constrained to be positive.
amplitude = tfp.util.TransformedVariable(
1., tfb.Softplus(), dtype=dtype, name='amplitude')
length_scale = tfp.util.TransformedVariable(
1., tfb.Softplus(), dtype=dtype, name='length_scale')
kernel = tfk.ExponentiatedQuadratic(
amplitude=amplitude,
length_scale=length_scale)
observation_noise_variance = tfp.util.TransformedVariable(
1., tfb.Softplus(), dtype=dtype, name='observation_noise_variance')
# Create trainable inducing point locations and variational parameters.
num_inducing_points_ = 20
inducing_index_points = tf.Variable(
np.linspace(5., 5., num_inducing_points_)[..., np.newaxis],
dtype=dtype, name='inducing_index_points')
variational_inducing_observations_loc = tf.Variable(
np.zeros([num_inducing_points_], dtype=dtype),
name='variational_inducing_observations_loc')
variational_inducing_observations_scale = tf.Variable(
np.eye(num_inducing_points_, dtype=dtype),
name='variational_inducing_observations_scale')
# These are the index point locations over which we'll construct the
# (approximate) posterior predictive distribution.
num_predictive_index_points_ = 500
index_points_ = np.linspace(13, 13,
num_predictive_index_points_,
dtype=dtype)[..., np.newaxis]
# Construct our variational GP Distribution instance.
vgp = tfd.VariationalGaussianProcess(
kernel,
index_points=index_points_,
inducing_index_points=inducing_index_points,
variational_inducing_observations_loc=
variational_inducing_observations_loc,
variational_inducing_observations_scale=
variational_inducing_observations_scale,
observation_noise_variance=observation_noise_variance)
# For training, we use some simplistic numpybased minibatching.
batch_size = 64
optimizer = tf.optimizers.Adam(learning_rate=.1)
@tf.function
def optimize(x_train_batch, y_train_batch):
with tf.GradientTape() as tape:
# Create the loss function we want to optimize.
loss = vgp.variational_loss(
observations=y_train_batch,
observation_index_points=x_train_batch,
kl_weight=float(batch_size) / float(num_training_points_))
grads = tape.gradient(loss, vgp.trainable_variables)
optimizer.apply_gradients(zip(grads, vgp.trainable_variables))
return loss
num_iters = 10000
num_logs = 10
for i in range(num_iters):
batch_idxs = np.random.randint(num_training_points_, size=[batch_size])
x_train_batch = x_train_[batch_idxs, ...]
y_train_batch = y_train_[batch_idxs]
loss = optimize(x_train_batch, y_train_batch)
if i % (num_iters / num_logs) == 0 or i + 1 == num_iters:
print(i, loss.numpy())
# Generate a plot with
#  the posterior predictive mean
#  training data
#  inducing index points (plotted vertically at the mean of the variational
# posterior over inducing point function values)
#  50 posterior predictive samples
num_samples = 50
samples = vgp.sample(num_samples).numpy()
mean = vgp.mean().numpy()
inducing_index_points_ = inducing_index_points.numpy()
variational_loc = variational_inducing_observations_loc.numpy()
plt.figure(figsize=(15, 5))
plt.scatter(inducing_index_points_[..., 0], variational_loc,
marker='x', s=50, color='k', zorder=10)
plt.scatter(x_train_[..., 0], y_train_, color='#00ff00', zorder=9)
plt.plot(np.tile(index_points_, (num_samples)),
samples.T, color='r', alpha=.1)
plt.plot(index_points_, mean, color='k')
plt.plot(index_points_, f(index_points_), color='b')
Here we use the same data setup, but compute the optimal variational
parameters instead of training them.
# We'll use double precision throughout for better numerics.
dtype = np.float64
# Generate noisy data from a known function.
f = lambda x: np.exp(x[..., 0]**2 / 20.) * np.sin(1. * x[..., 0])
true_observation_noise_variance_ = dtype(1e1) ** 2
num_training_points_ = 1000
x_train_ = np.random.uniform(10., 10., [num_training_points_, 1])
y_train_ = (f(x_train_) +
np.random.normal(
0., np.sqrt(true_observation_noise_variance_),
[num_training_points_]))
# Create kernel with trainable parameters, and trainable observation noise
# variance variable. Each of these is constrained to be positive.
amplitude = tfp.util.TransformedVariable(
1., tfb.Softplus(), dtype=dtype, name='amplitude')
length_scale = tfp.util.TransformedVariable(
1., tfb.Softplus(), dtype=dtype, name='length_scale')
kernel = tfk.ExponentiatedQuadratic(
amplitude=amplitude,
length_scale=length_scale)
observation_noise_variance = tfp.util.TransformedVariable(
1., tfb.Softplus(), dtype=dtype, name='observation_noise_variance')
# Create trainable inducing point locations and variational parameters.
num_inducing_points_ = 10
inducing_index_points = tf.Variable(
np.linspace(10., 10., num_inducing_points_)[..., np.newaxis],
dtype=dtype, name='inducing_index_points')
variational_loc, variational_scale = (
tfd.VariationalGaussianProcess.optimal_variational_posterior(
kernel=kernel,
inducing_index_points=inducing_index_points,
observation_index_points=x_train_,
observations=y_train_,
observation_noise_variance=observation_noise_variance))
# These are the index point locations over which we'll construct the
# (approximate) posterior predictive distribution.
num_predictive_index_points_ = 500
index_points_ = np.linspace(13, 13,
num_predictive_index_points_,
dtype=dtype)[..., np.newaxis]
# Construct our variational GP Distribution instance.
vgp = tfd.VariationalGaussianProcess(
kernel,
index_points=index_points_,
inducing_index_points=inducing_index_points,
variational_inducing_observations_loc=variational_loc,
variational_inducing_observations_scale=variational_scale,
observation_noise_variance=observation_noise_variance)
# For training, we use some simplistic numpybased minibatching.
batch_size = 64
optimizer = tf.optimizers.Adam(learning_rate=.05, beta_1=.5, beta_2=.99)
@tf.function
def optimize(x_train_batch, y_train_batch):
with tf.GradientTape() as tape:
# Create the loss function we want to optimize.
loss = vgp.variational_loss(
observations=y_train_batch,
observation_index_points=x_train_batch,
kl_weight=float(batch_size) / float(num_training_points_))
grads = tape.gradient(loss, vgp.trainable_variables)
optimizer.apply_gradients(zip(grads, vgp.trainable_variables))
return loss
num_iters = 300
num_logs = 10
for i in range(num_iters):
batch_idxs = np.random.randint(num_training_points_, size=[batch_size])
x_train_batch_ = x_train_[batch_idxs, ...]
y_train_batch_ = y_train_[batch_idxs]
loss = optimize(x_train_batch, y_train_batch)
if i % (num_iters / num_logs) == 0 or i + 1 == num_iters:
print(i, loss.numpy())
# Generate a plot with
#  the posterior predictive mean
#  training data
#  inducing index points (plotted vertically at the mean of the
# variational posterior over inducing point function values)
#  50 posterior predictive samples
num_samples = 50
samples_ = vgp.sample(num_samples).numpy()
mean_ = vgp.mean().numpy()
inducing_index_points_ = inducing_index_points.numpy()
variational_loc_ = variational_loc.numpy()
plt.figure(figsize=(15, 5))
plt.scatter(inducing_index_points_[..., 0], variational_loc_,
marker='x', s=50, color='k', zorder=10)
plt.scatter(x_train_[..., 0], y_train_, color='#00ff00', alpha=.1, zorder=9)
plt.plot(np.tile(index_points_, num_samples),
samples_.T, color='r', alpha=.1)
plt.plot(index_points_, mean_, color='k')
plt.plot(index_points_, f(index_points_), color='b')
References
[1]: Titsias, M. "Variational Model Selection for Sparse Gaussian Process Regression", 2009. http://proceedings.mlr.press/v5/titsias09a/titsias09a.pdf [2]: Hensman, J., Lawrence, N. "Gaussian Processes for Big Data", 2013 https://arxiv.org/abs/1309.6835 [3]: Carl Rasmussen, Chris Williams. Gaussian Processes For Machine Learning, 2006. http://www.gaussianprocess.org/gpml/
Args  

kernel

PositiveSemidefiniteKernel like instance representing the
GP's covariance function.

index_points

float Tensor representing finite (batch of) vector(s) of
points in the index set over which the VGP is defined. Shape has the
form [b1, ..., bB, e1, f1, ..., fF] where F is the number of feature
dimensions and must equal kernel.feature_ndims and e1 is the number
(size) of index points in each batch (we denote it e1 to distinguish
it from the numer of inducing index points, denoted e2 below).
Ultimately the VariationalGaussianProcess distribution corresponds to an
e1 dimensional multivariate normal. The batch shape must be
broadcastable with kernel.batch_shape , the batch shape of
inducing_index_points , and any batch dims yielded by mean_fn .

inducing_index_points

float Tensor of locations of inducing points in
the index set. Shape has the form [b1, ..., bB, e2, f1, ..., fF] , just
like index_points . The batch shape components needn't be identical to
those of index_points , but must be broadcast compatible with them.

variational_inducing_observations_loc

float Tensor ; the mean of the
(fullrank Gaussian) variational posterior over function values at the
inducing points, conditional on observed data. Shape has the form [b1,
..., bB, e2] , where b1, ..., bB is broadcast compatible with other
parameters' batch shapes, and e2 is the number of inducing points.

variational_inducing_observations_scale

float Tensor ; the scale
matrix of the (fullrank Gaussian) variational posterior over function
values at the inducing points, conditional on observed data. Shape has
the form [b1, ..., bB, e2, e2] , where b1, ..., bB is broadcast
compatible with other parameters and e2 is the number of inducing
points.

mean_fn

Python callable that acts on index points to produce a (batch
of) vector(s) of mean values at those 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.

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 additional
variance in the posterior predictive model. If None , we simply reuse
observation_noise_variance for the posterior predictive noise. If set
explicitly, however, we use the given 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.

jitter

float scalar Tensor added to the diagonal of the covariance
matrix to ensure positive definiteness of the covariance matrix.
Default value: 1e6 .

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: "VariationalGaussianProcess".

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, nonidentical parameterizations of this distribution. 
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. 
index_points


inducing_index_points


jitter


kernel


mean_fn


name

Name prepended to all ops created by this Distribution .

observation_noise_variance


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

trainable_variables


validate_args

Python bool indicating possibly expensive checks are enabled.

variables


variational_inducing_observations_loc


variational_inducing_observations_scale

Methods
batch_shape_tensor
batch_shape_tensor(
name='batch_shape_tensor'
)
Shape of a single sample from a single event index as a 1D Tensor
.
The batch dimensions are indexes into independent, nonidentical parameterizations of this distribution.
Args  

name

name to give to the op 
Returns  

batch_shape

Tensor .

cdf
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
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', **kwargs
)
Covariance.
Covariance is (possibly) defined only for nonscalarevent distributions.
For example, for a lengthk
, vectorvalued 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 nonvector, multivariate distributions (e.g.,
matrixvalued, 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
lengthk'
vector.
Args  

name

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

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

covariance

Floatingpoint 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
.
other
types with builtin 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
entropy(
name='entropy', **kwargs
)
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 1D int32 Tensor
.
Args  

name

name to give to the op 
Returns  

event_shape

Tensor .

experimental_default_event_space_bijector
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 * (k1) // 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 .

get_marginal_distribution
get_marginal_distribution(
index_points=None
)
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
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 KullbackLeibler 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 builtin 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 KullbackLeibler
divergence.

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

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', **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
mean(
name='mean', **kwargs
)
Mean.
mode
mode(
name='mode', **kwargs
)
Mode.
optimal_variational_posterior
@staticmethod
optimal_variational_posterior( kernel, inducing_index_points, observation_index_points, observations, observation_noise_variance, mean_fn=None, jitter=1e06, name=None )
Model selection for optimal variational hyperparameters.
Given the full training set (parameterized by observations
and
observation_index_points
), compute the optimal variational
location and scale for the VGP. This is based of the method suggested
in [Titsias, 2009][1].
Args  

kernel

PositiveSemidefiniteKernel like instance representing the
GP's covariance function.

inducing_index_points

float Tensor of locations of inducing points in
the index set. Shape has the form [b1, ..., bB, e2, f1, ..., fF] , just
like observation_index_points . The batch shape components needn't be
identical to those of observation_index_points , but must be broadcast
compatible with them.

observation_index_points

float Tensor representing finite (batch of)
vector(s) of points where observations are defined. Shape has the
form [b1, ..., bB, e1, f1, ..., fF] where F is the number of feature
dimensions and must equal kernel.feature_ndims and e1 is the number
(size) of index points in each batch (we denote it e1 to distinguish
it from the numer of inducing index points, denoted e2 below).

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 , observation_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.

mean_fn

Python callable that acts on index points to produce a (batch
of) vector(s) of mean values at those 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: 1e6 .

name

Python str name prefixed to Ops created by this class.
Default value: "optimal_variational_posterior".

Returns  

loc, scale: Tuple representing the variational location and scale. 
Raises  

ValueError

if mean_fn is not None and is not callable.

References
[1]: Titsias, M. "Variational Model Selection for Sparse Gaussian Process Regression", 2009. http://proceedings.mlr.press/v5/titsias09a/titsias09a.pdf
param_shapes
@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
@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
constantvalued 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
@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.
Args  

dtype

Optional float dtype to assume for continuousvalued 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 categoricallike distributions.
Otherwise ignored.

Returns  

parameter_properties

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

prob
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
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
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

Python integer or tfp.util.SeedStream instance, for seeding PRNG.

name

name to give to the op. 
**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

samples

a Tensor with prepended dimensions sample_shape .

stddev
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

Floatingpoint Tensor with shape identical to
batch_shape + event_shape , i.e., the same shape as self.mean() .

surrogate_posterior_expected_log_likelihood
surrogate_posterior_expected_log_likelihood(
observations, observation_index_points=None, log_likelihood_fn=None,
quadrature_size=10, name=None
)
Compute the expected log likelihood term in the ELBO, using quadrature.
In variational inference, we're interested in optimizing the ELBO, which looks like
ELBO = E_{q(z)} log p(x  z) + KL(q(z)  p(z))
where q(z)
is the variational, or "surrogate", posterior over latents z
,
p(x  z)
is the likelihood of some data x
conditional on latents z
,
and p(z)
is the prior over z
.
In the specific case of the VariationalGaussianProcess model, the
surrograte posterior q(z)
is such that the above expectation factorizes
into a sum over 1dimensional integrals of the log likelihood times a
certain Gaussian distribution (a 1dimensional marginal of the full
variational GP). This means we can get a really good estimate of the
likelihood term using GaussHermite quadrature, which is what this method
does. In the particular case of a Gaussian likelihood, we can actually get
an exact answer with 3 quadrature points (we could also work it out
analytically, but it's still exact and a bit simpler to just have one
implementation for all likelihoods).
The observation_index_points
arguments are optional and if omitted default
to the index_points
of this class (ie, the predictive locations).
Example: binary classification
def log_prob(observations, f):
# Parameterize a collection of independent Bernoulli random variables
# with logits given by the passedin function values `f`. Return the
# joint log probability of the (binary) `observations` under that
# model.
berns = tfd.Independent(tfd.Bernoulli(logits=f),
reinterpreted_batch_ndims=1)
return berns.log_prob(observations)
# Compute the expected log likelihood using GaussHermite quadrature.
recon = vgp.surrogate_posterior_expected_log_likelihood(
observations,
observation_index_points,
log_likelihood_fn=log_prob,
quadrature_size=20)
elbo = recon + vgp.surrogate_posterior_kl_divergence_prior()
Args  

observations

observed data at the given observation_index_points ; must
be acceptable inputs to the given log_likelihood_fn callable.

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 Fis the number of feature dimensions and must equal self.kernel.feature_ndims, and eis 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 of this VariationalGaussianProcessinstance ( kernel.batch_shape, index_points, etc).
</td>
</tr><tr>
<td> log_likelihood_fn</td>
<td>
A callable, which takes a set of observed data and
function values (ie, events under this GP model at the
observation_index_points) and returns the log likelihood of those data
conditioned on those function values. Default value is None, which
implies a Normallikelihood and 3 qudrature points.
</td>
</tr><tr>
<td> quadrature_size</td>
<td>
number of grid points to use in GaussHermite quadrature
scheme. Default of 10(arbitrarily), or if 3if log_likelihood_fnis None(implying a Gaussian likelihood for which 3points will give
an exact answer.)
</td>
</tr><tr>
<td> name</td>
<td>
Python str` name prefixed to Ops created by this class.
Default value: "surrogate_posterior_expected_log_likelihood".

Returns  

surrogate_posterior_expected_log_likelihood

the value of the expected log likelihood of the given observed data under the surrogate posterior model of latent function values and given likelihood model. 
surrogate_posterior_kl_divergence_prior
surrogate_posterior_kl_divergence_prior(
name=None
)
The KL divergence between the surrograte posterior and GP prior.
Args  

name

Python str name prefixed to Ops created by this class.
Default value: "surrogate_posterior_kl_divergence_prior".

Returns  

kl_divergence

the value of the KL divergence between the surrograte
posterior implied by this VariationalGaussianProcess instance and the
prior, which is an unconditional GP with the same kernel and prior
mean_fn

survival_function
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 .

variance
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

Floatingpoint Tensor with shape identical to
batch_shape + event_shape , i.e., the same shape as self.mean() .

variational_loss
variational_loss(
observations, observation_index_points=None, log_likelihood_fn=None,
quadrature_size=3, kl_weight=1.0, name='variational_loss'
)
Variational loss for the VGP.
Given observations
and observation_index_points
, compute the
negative variational lower bound as specified in [Hensman, 2013][1].
Args  

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 , observation_index_points , etc.).

observation_index_points

float Tensor representing finite (batch of)
vector(s) of points where observations are defined. Shape has the
form [b1, ..., bB, e1, f1, ..., fF] where F is the number of feature
dimensions and must equal kernel.feature_ndims and e1 is the number
(size) of index points in each batch (we denote it e1 to distinguish
it from the numer of inducing index points, denoted e2 below). If
set to None uses index_points as the origin for observations.
Default value: None.

log_likelihood_fn

log likelihood function. 
quadrature_size

num quadrature grid points. 
kl_weight

Amount by which to scale the KL divergence loss between prior and posterior. Default value: 1. 
name

Python str name prefixed to Ops created by this class.
Default value: 'variational_loss'.

Returns  

loss

Scalar tensor representing the negative variational lower bound.
Can be directly used in a tf.Optimizer .

References
[1]: Hensman, J., Lawrence, N. "Gaussian Processes for Big Data", 2013 https://arxiv.org/abs/1309.6835
__getitem__
__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__
__iter__()