tfp.distributions.VariationalGaussianProcess | TensorFlow Probability (original) (raw)

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 thevariational_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 aMultivariateNormalLinearOperator 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 theVariationalGaussianProcess 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. Theoptimal_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", 2013https://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 ane1-dimensional multivariate normal. The batch shape must be broadcastable with kernel.batch_shape, the batch shape ofinducing_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-useobservation_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 jitterparameter.
use_whitening_transform	Python bool. Whether to reparameterize the variational scale as m = chol(K_zz)m', s = chol(K_zz)s', wherechol(K_zz) is the cholesky factor of the kernel at theinducing_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 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.
experimental_shard_axis_names	The list or structure of lists of active shard axis names.
index_points
inducing_index_points
jitter	DEPRECATED FUNCTION
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.
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 instancestfd.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

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 ofvalue 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 tocls.__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 returningFullSpace 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 shapesample_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 Tensors, with prepended dimensionssample_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 tosample().
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 toself.index_points. Shape has the form [b1, ..., bB, e], which must be broadcastable with the batch and example shapes ofself.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	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).
log_likelihood_fn	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.
quadrature_size	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.)
name	Pythonstr` name prefixed to Ops created by this class. Default value: "surrogate_posterior_expected_log_likelihood".

tfp.distributions.VariationalGaussianProcess | TensorFlow Probability (original) (raw)

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

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`