# tfp.experimental.substrates.numpy.distributions.VariationalGaussianProcess

Posterior predictive of a variational Gaussian process.

Inherits From: GaussianProcess

This distribution implements the variational Gaussian process (VGP), as described in [Titsias, 2009][1] and [Hensman, 2013][2]. The VGP is an inducing point-based 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 (full-rank, 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 points
• t[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 asK_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 finite-dimensional, 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 mean-zero 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 point-based approximation of an exact GP posterior, where two approximating assumptions have been made:

1. function values at non-inducing points are mutually independent conditioned on function values at the inducing points,
2. the (expensive) posterior over function values at inducing points conditional on observations is replaced with an arbitrary (learnable) full-rank Gaussian distribution,
q(f(Z)) = MVN(loc=m, scale=S),
where `m` and `S` 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.numpy.compat import v2 as tf
import tensorflow_probability as tfp; tfp = tfp.experimental.substrates.numpy

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(1e-1) ** 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')
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 numpy-based minibatching.
batch_size = 64

@tf.function
def optimize(x_train_batch, y_train_batch):
# 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_))
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')

# 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(1e-1) ** 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')
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 numpy-based minibatching.
batch_size = 64

@tf.function
def optimize(x_train_batch, y_train_batch):
# 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_))
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/

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 (full-rank 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 (full-rank 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 re-use 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: 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: "VariationalGaussianProcess".

ValueError if mean_fn is not None and is 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.

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.

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 tfd.FULLY_REPARAMETERIZED or tfd.NOT_REPARAMETERIZED.

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

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.

### 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.

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

View source

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.

View source

Mean.

View source

Mode.

### optimal_variational_posterior

View source

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: 1e-6.
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

View source

Shapes of parameters given the desired shape of a call to sample().

This is a class method that describes what key/value arguments are required to instantiate the given Distribution so that a particular shape is returned for that instance's call to sample().

Subclasses should override class method _param_shapes.

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

Returns
dict of parameter name to Tensor shapes.

### param_static_shapes

View source

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

View source

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

View source

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

View source

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

View source

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().

### surrogate_posterior_expected_log_likelihood

View source

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 1-dimensional integrals of the log likelihood times a certain Gaussian distribution (a 1-dimensional marginal of the full variational GP). This means we can get a really good estimate of the likelihood term using Gauss-Hermite 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 passed-in 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 Gauss-Hermite quadrature.
recon = vgp.surrogate_posterior_expected_log_likelihood(
observations,
observation_index_points,
log_likelihood_fn=log_prob,

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]' whereFis the number of feature dimensions and must equalself.kernel.feature_ndims, andeis the number (size) of index points in each batch.[b1, ..., bB, e]must be broadcastable with the shape ofobservations, and[b1, ..., bB]must be broadcastable with the shapes of all other batched parameters of thisVariationalGaussianProcessinstance (kernel.batch_shape,index_points, etc). </td> </tr><tr> <td>log_likelihood_fn</td> <td> Acallable, 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 isNone, which implies aNormallikelihood and 3 qudrature points. </td> </tr><tr> <td>quadrature_size</td> <td> number of grid points to use in Gauss-Hermite quadrature scheme. Default of10(arbitrarily), or if3iflog_likelihood_fnisNone(implying a Gaussian likelihood for which3points will give an exact answer.) </td> </tr><tr> <td>name</td> <td> Pythonstr` 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

View source

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

View source

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

View source

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().

### variational_loss

View source

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.
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__

View source

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.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.

View source