# tfp.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] and [Hensman, 2013]. 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] and [Hensman, 2013] 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] 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, ..., x[N]`: observation index points -- locations of our observed data.
• `z, ..., z[M]`: inducing index points -- locations of the "summarizing" inducing points
• `t, ..., 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, z)    k(t, z)  ...  k(t, z[M]) |
K_tz = | k(t, z)    k(t, z)  ...  k(t, z[M]) |
|      ...              ...                 ...      |
| k(t[P], z)    k(t[P], z)  ...  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] 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] and [Hensman, 2013] 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], 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
import tensorflow.compat.v2 as tf
import tensorflow_probability as tfp

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')
``````

: Titsias, M. "Variational Model Selection for Sparse Gaussian Process Regression", 2009. http://proceedings.mlr.press/v5/titsias09a/titsias09a.pdf : Hensman, J., Lawrence, N. "Gaussian Processes for Big Data", 2013 https://arxiv.org/abs/1309.6835 : 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 `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`.
`name_scope` Returns a `tf.name_scope` instance for this class.
`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`.

`submodules` Sequence of all sub-modules.

Submodules are modules which are properties of this module, or found as properties of modules which are properties of this module (and so on).

````a = tf.Module()`
`b = tf.Module()`
`c = tf.Module()`
`a.b = b`
`b.c = c`
`list(a.submodules) == [b, c]`
`True`
`list(b.submodules) == [c]`
`True`
`list(c.submodules) == []`
`True`
```

`trainable_variables` Sequence of trainable variables owned by this module and its submodules.

`validate_args` Python `bool` indicating possibly expensive checks are enabled.
`variables` Sequence of variables owned by this module and its submodules.
`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`

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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`

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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`.

View source

Mean.

View source

Mode.

### `optimal_variational_posterior`

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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].

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

: Titsias, M. "Variational Model Selection for Sparse Gaussian Process Regression", 2009. http://proceedings.mlr.press/v5/titsias09a/titsias09a.pdf

### `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.

### `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`

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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`

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`

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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]' where```F`is the number of feature dimensions and must equal`self.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 of this```VariationalGaussianProcess`instance (`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```Normal```likelihood and 3 qudrature points. </td> </tr><tr> <td>```quadrature_size```</td> <td> number of grid points to use in Gauss-Hermite quadrature scheme. Default of```10`(arbitrarily), or if`3`if`log_likelihood_fn`is`None`(implying a Gaussian likelihood for which`3```points 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`

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].

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

: Hensman, J., Lawrence, N. "Gaussian Processes for Big Data", 2013 https://arxiv.org/abs/1309.6835

### `with_name_scope`

Decorator to automatically enter the module name scope.

````class MyModule(tf.Module):`
`  @tf.Module.with_name_scope`
`  def __call__(self, x):`
`    if not hasattr(self, 'w'):`
`      self.w = tf.Variable(tf.random.normal([x.shape, 3]))`
`    return tf.matmul(x, self.w)`
```

Using the above module would produce `tf.Variable`s and `tf.Tensor`s whose names included the module name:

````mod = MyModule()`
`mod(tf.ones([1, 2]))`
`<tf.Tensor: shape=(1, 3), dtype=float32, numpy=..., dtype=float32)>`
`mod.w`
`<tf.Variable 'my_module/Variable:0' shape=(2, 3) dtype=float32,`
`numpy=..., dtype=float32)>`
```

Args
`method` The method to wrap.

Returns
The original method wrapped such that it enters the module's name scope.

### `__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.normal([5, 3, 2, 2])
cov = tf.matmul(x, x, transpose_b=True)
chol = tf.cholesky(cov)
loc = tf.random.normal([4, 1, 3, 1])
mvn = tfd.MultivariateNormalTriL(loc, chol)
mvn.batch_shape  # => [4, 5, 3]
mvn.event_shape  # => 
mvn2 = mvn[:, 3:, ..., ::-1, tf.newaxis]
mvn2.batch_shape  # => [4, 2, 3, 1]
mvn2.event_shape  # => 
``````

Args
`slices` slices from the [] operator

Returns
`dist` A new `tfd.Distribution` instance with sliced parameters.

View source