tfp.distributions.VariationalGaussianProcess

Posterior predictive of a variational Gaussian process.

Inherits From: GaussianProcess, AutoCompositeTensorDistribution, Distribution, AutoCompositeTensor

tfp.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,
    cholesky_fn=None,
    use_whitening_transform=False,
    jitter=1e-06,
    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 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:

function values at non-inducing 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) 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
import tensorflow.compat.v2 as tf
import tensorflow_probability as tfp

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')
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 numpy-based 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(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')
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 numpy-based 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/ [4]: Hensman, J., Matthews, A. G., Filippone M., Ghahramani Z. "MCMC for Variationally Sparse Gaussian Processes" https://arxiv.org/abs/1506.04000

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 (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, e, f1, ..., fF]` and returns a `Tensor` whose shape is (broadcastable with) `[b1, ..., bB, e]`. 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.
`cholesky_fn`	Callable which takes a single (batch) matrix argument and returns a Cholesky-like lower triangular factor. Default value: `None`, in which case `make_cholesky_with_jitter_fn` is used with the `jitter` parameter.
`use_whitening_transform`	Python `bool`. Whether to reparameterize the variational scale as `m = chol(K_zz)m', s = chol(K_zz)s'`, where `chol(K_zz)` is the cholesky factor of the kernel at the `inducing_index_points`, `m` is the variational location, `s` is the variational scale, and `m', s'` is the reparameterization when this parameter is True. Default value: `False`.
`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".

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, non-identical parameterizations of this distribution.
`cholesky_fn`
`dtype`	The `DType` of `Tensor`s handled by this `Distribution`.
`event_shape`	Shape of a single sample from a single batch as a `TensorShape`. May be partially defined or unknown.
`experimental_shard_axis_names`	The list or structure of lists of active shard axis names.
`index_points`
`inducing_index_points`
`jitter`	DEPRECATED FUNCTION Deprecated: THIS FUNCTION IS DEPRECATED. It will be removed after 2022-02-04. Instructions for updating: the `jitter` property of `tfd.GaussianProcess` is deprecated; use the `marginal_fn` property instead.
`kernel`
`marginal_fn`
`mean_fn`
`name`	Name prepended to all ops created by this `Distribution`.
`name_scope`	Returns a `tf.name_scope` instance for this class.
`non_trainable_variables`	Sequence of non-trainable variables owned by this module and its submodules. Note: this method uses reflection to find variables on the current instance and submodules. For performance reasons you may wish to cache the result of calling this method if you don't expect the return value to change.
`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. Note: this method uses reflection to find variables on the current instance and submodules. For performance reasons you may wish to cache the result of calling this method if you don't expect the return value to change.
`validate_args`	Python `bool` indicating possibly expensive checks are enabled.
`variables`	Sequence of variables owned by this module and its submodules. Note: this method uses reflection to find variables on the current instance and submodules. For performance reasons you may wish to cache the result of calling this method if you don't expect the return value to change.
`variational_inducing_observations_loc`
`variational_inducing_observations_scale`

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.

Args
`other`	`tfp.distributions.Distribution` instance.
`name`	Python `str` prepended to names of ops created by this function.

Args
`*args`	Passed to implementation `_default_event_space_bijector`.
`**kwargs`	Passed to implementation `_default_event_space_bijector`.

Args
`value`	a `Tensor` valid sample from this distribution family.
`sample_ndims`	Positive `int` Tensor number of leftmost dimensions of `value` that index i.i.d. samples. Default value: `1`.
`validate_args`	Python `bool`, default `False`. When `True`, distribution parameters are checked for validity despite possibly degrading runtime performance. When `False`, invalid inputs may silently render incorrect outputs. Default value: `False`.
`**init_kwargs`	Additional keyword arguments passed through to `cls.__init__`. These take precedence in case of collision with the fitted parameters; for example, `tfd.Normal.experimental_fit([1., 1.], scale=20.)` returns a Normal distribution with `scale=20.` rather than the maximum likelihood parameter `scale=0.`.

Args
`value`	`float` or `double` `Tensor`.
`backward_compat`	`bool` specifying whether to fall back to returning `FullSpace` as the tangent space, and representing R^n with the standard basis.
`**kwargs`	Named arguments forwarded to subclass implementation.

Returns
`log_prob`	a `Tensor` representing the log probability density, of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`.
`tangent_space`	a `TangentSpace` object (by default `FullSpace`) representing the tangent space to the manifold at `value`.

Args
`sample_shape`	integer `Tensor` desired shape of samples to draw. Default value: `()`.
`seed`	PRNG seed; see `tfp.random.sanitize_seed` for details. Default value: `None`.
`name`	name to give to the op. Default value: `'sample_and_log_prob'`.
`**kwargs`	Named arguments forwarded to subclass implementation.

Returns
`samples`	a `Tensor`, or structure of `Tensor`s, with prepended dimensions `sample_shape`.
`log_prob`	a `Tensor` of shape `sample_shape(x) + self.batch_shape` with values of type `self.dtype`.

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

Args
`dtype`	Optional float `dtype` to assume for continuous-valued parameters. Some constraining bijectors require advance knowledge of the dtype because certain constants (e.g., `tfb.Softplus.low`) must be instantiated with the same dtype as the values to be transformed.
`num_classes`	Optional `int` `Tensor` number of classes to assume when inferring the shape of parameters for categorical-like distributions. Otherwise ignored.

Args
`observations`	`float` `Tensor` representing collection, or batch of collections, of observations corresponding to `self.index_points`. Shape has the form `[b1, ..., bB, e]`, which must be broadcastable with the batch and example shapes of `self.index_points`. The batch shape `[b1, ..., bB]` must be broadcastable with the shapes of all other batched parameters
`predictive_index_points`	(nested) `Tensor` representing finite collection, or batch of collections, of points in the index set over which the GP is defined. Shape (or shape of each nested component) has the form `[b1, ..., bB, e, f1, ..., fF]` where `F` is the number of feature dimensions and must equal `kernel.feature_ndims` (or its corresponding nested component) and `e` is the number (size) of predictive index points in each batch. The batch shape must be broadcastable with this distributions `batch_shape`. Default value: `None`.
`**kwargs`	Any other keyword arguments to pass / override.

Args
`sample_shape`	0D or 1D `int32` `Tensor`. Shape of the generated samples.
`seed`	PRNG seed; see `tfp.random.sanitize_seed` for details.
`name`	name to give to the op.
`**kwargs`	Named arguments forwarded to subclass implementation.

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'.

tfp.distributions.VariationalGaussianProcess

Mathematical Details

Notation

Preliminaries

The VGP

Optimal variational parameters

Usage Examples

Here we use the same data setup, but compute the optimal variational

parameters instead of training them.

References

Args

Raises

Attributes

Methods

batch_shape_tensor

cdf

copy

covariance

cross_entropy

entropy

event_shape_tensor

experimental_default_event_space_bijector

experimental_fit

experimental_local_measure

experimental_sample_and_log_prob

get_marginal_distribution

is_scalar_batch

is_scalar_event

kl_divergence

log_cdf

log_prob

kwargs:

log_survival_function

mean

mode

optimal_variational_posterior

References

param_shapes

param_static_shapes

parameter_properties

posterior_predictive

prob

quantile

sample

stddev

surrogate_posterior_expected_log_likelihood

Example: binary classification

surrogate_posterior_kl_divergence_prior

survival_function

unnormalized_log_prob

variance

variational_loss

References

with_name_scope

__getitem__

__iter__

`batch_shape_tensor`

`cdf`

`copy`

`covariance`

`cross_entropy`

`entropy`

`event_shape_tensor`

`experimental_default_event_space_bijector`

`experimental_fit`

`experimental_local_measure`

`experimental_sample_and_log_prob`

`get_marginal_distribution`

`is_scalar_batch`

`is_scalar_event`

`kl_divergence`

`log_cdf`

`log_prob`

`kwargs`:

`log_survival_function`

`mean`

`mode`

`optimal_variational_posterior`

`param_shapes`

`param_static_shapes`

`parameter_properties`

`posterior_predictive`

`prob`

`quantile`

`sample`

`stddev`

`surrogate_posterior_expected_log_likelihood`

`surrogate_posterior_kl_divergence_prior`

`survival_function`

`unnormalized_log_prob`

`variance`

`variational_loss`

`with_name_scope`

`getitem`

`iter`