Posterior predictive distribution in a conjugate GP regression model.
Inherits From: GaussianProcess
, AutoCompositeTensorDistribution
, Distribution
oryx.distributions.GaussianProcessRegressionModel(
kernel,
index_points=None,
observation_index_points=None,
observations=None,
observation_noise_variance=0.0,
predictive_noise_variance=None,
mean_fn=None,
cholesky_fn=None,
jitter=1e06,
validate_args=False,
allow_nan_stats=False,
name='GaussianProcessRegressionModel',
_conditional_kernel=None,
_conditional_mean_fn=None
)
This class represents the distribution over function values at a set of points
in some index set, conditioned on noisy observations at some other set of
points. More specifically, we assume a Gaussian process prior, f ~ GP(m, k)
with IID normal noise on observations of function values. In this model
posterior inference can be done analytically. This Distribution
is
parameterized by
 the mean and covariance functions of the GP prior,
 the set of (noisy) observations and index points to which they correspond,
 the set of index points at which the resulting posterior predictive distribution over function values is defined,
 the observation noise variance,
 jitter, to compensate for numerical instability of Cholesky decomposition,
in addition to the usual params like validate_args
and allow_nan_stats
.
Mathematical Details
Gaussian process regression (GPR) assumes a Gaussian process (GP) prior and a
normal likelihood as a generative model for data. Given GP mean function m
,
covariance kernel k
, and observation noise variance v
, we have
f ~ GP(m, k)
iid
(y[i]  f, x[i]) ~ Normal(f(x[i]), v), i = 1, ... , N
where y[i]
are the noisy observations of function values at points x[i]
.
In practice, f
is an infinite object (eg, a function over R^n
) which can't
be realized on a finite machine, but fortunately the marginal distribution
over function values at a finite set of points is just a multivariate normal
with mean and covariance given by the mean and covariance functions applied at
our finite set of points (see [Rasmussen and Williams, 2006][1] for a more
extensive discussion of these facts).
We spell out the generative model in detail below, but first, a digression on
notation. In what follows we drop the indices on vectorial objects such as
x[i]
, it being implied that we are generally considering finite collections
of index points and corresponding function values and noisy observations
thereof. Thus x
should be considered to stand for a collection of index
points (indeed, themselves often vectorial). Furthermore:
f(x)
refers to the collection of function values at the index points in the collectionx
",m(t)
refers to the collection of values of the mean function at the index points in the collectiont
, andk(x, t)
refers to the matrix whose entries are values of the kernel functionk
at all pairs of index points fromx
andt
.
With these conventions in place, we may write
(f(x)  x) ~ MVN(m(x), k(x, x))
(y  f(x), x) ~ Normal(f(x), v)
When we condition on observed data y
at the points x
, we can derive the
posterior distribution over function values f(x)
at those points. We can
then compute the posterior predictive distribution over function values f(t)
at a new set of points t
, conditional on those observed data.
(f(t)  t, x, y) ~ MVN(loc, cov)
where
loc = m(t) + k(t, x) @ inv(k(x, x) + v * I) @ (y  m(x))
cov = k(t, t)  k(t, x) @ inv(k(x, x) + v * I) @ k(x, t)
where I
is the identity matrix of appropriate dimension. Finally, the
distribution over noisy observations at the new set of points t
is obtained
by adding IID noise from Normal(0., observation_noise_variance)
.
Examples
Draw joint samples from the posterior predictive distribution in a GP
regression model
import numpy as np
from tensorflow_probability.python.internal.backend.jax.compat import v2 as tf
from tensorflow_probability.substrates import numpy as tfp
tfb = tfp.bijectors
tfd = tfp.distributions
psd_kernels = tfp.math.psd_kernels
# Generate noisy observations from a known function at some random points.
observation_noise_variance = .5
f = lambda x: np.sin(10*x[..., 0]) * np.exp(x[..., 0]**2)
observation_index_points = np.random.uniform(1., 1., 50)[..., np.newaxis]
observations = (f(observation_index_points) +
np.random.normal(0., np.sqrt(observation_noise_variance)))
index_points = np.linspace(1., 1., 100)[..., np.newaxis]
kernel = psd_kernels.MaternFiveHalves()
gprm = tfd.GaussianProcessRegressionModel(
kernel=kernel,
index_points=index_points,
observation_index_points=observation_index_points,
observations=observations,
observation_noise_variance=observation_noise_variance)
samples = gprm.sample(10)
# ==> 10 independently drawn, joint samples at `index_points`.
Above, we have used the kernel with default parameters, which are unlikely to be good. Instead, we can train the kernel hyperparameters on the data, as in the next example.
Optimize model parameters via maximum marginal likelihood
Here we learn the kernel parameters as well as the observation noise variance using gradient descent on the maximum marginal likelihood.
# Suppose we have some data from a known function. Note the index points in
# general have shape `[b1, ..., bB, f1, ..., fF]` (here we assume `F == 1`),
# so we need to explicitly consume the feature dimensions (just the last one
# here).
f = lambda x: np.sin(10*x[..., 0]) * np.exp(x[..., 0]**2)
observation_index_points = np.random.uniform(1., 1., 50)[..., np.newaxis]
observations = f(observation_index_points) + np.random.normal(0., .05, 50)
# Define a kernel with trainable parameters. Note we use TransformedVariable
# to apply a positivity constraint.
amplitude = tfp.util.TransformedVariable(
1., tfb.Exp(), dtype=tf.float64, name='amplitude')
length_scale = tfp.util.TransformedVariable(
1., tfb.Exp(), dtype=tf.float64, name='length_scale')
kernel = psd_kernels.ExponentiatedQuadratic(amplitude, length_scale)
observation_noise_variance = tfp.util.TransformedVariable(
np.exp(5), tfb.Exp(), name='observation_noise_variance')
# We'll use an unconditioned GP to train the kernel parameters.
gp = tfd.GaussianProcess(
kernel=kernel,
index_points=observation_index_points,
observation_noise_variance=observation_noise_variance)
optimizer = tf_keras.optimizers.Adam(learning_rate=.05, beta_1=.5, beta_2=.99)
@tf.function
def optimize():
with tf.GradientTape() as tape:
loss = gp.log_prob(observations)
grads = tape.gradient(loss, gp.trainable_variables)
optimizer.apply_gradients(zip(grads, gp.trainable_variables))
return loss
# We can construct the posterior at a new set of `index_points` using the same
# kernel (with the same parameters, which we'll optimize below).
index_points = np.linspace(1., 1., 100)[..., np.newaxis]
gprm = tfd.GaussianProcessRegressionModel(
kernel=kernel,
index_points=index_points,
observation_index_points=observation_index_points,
observations=observations,
observation_noise_variance=observation_noise_variance)
# First train the model, then draw and plot posterior samples.
for i in range(1000):
neg_log_likelihood_ = optimize()
if i % 100 == 0:
print("Step {}: NLL = {}".format(i, neg_log_likelihood_))
print("Final NLL = {}".format(neg_log_likelihood_))
samples = gprm.sample(10).numpy()
# ==> 10 independently drawn, joint samples at `index_points`.
import matplotlib.pyplot as plt
plt.scatter(np.squeeze(observation_index_points), observations)
plt.plot(np.stack([index_points[:, 0]]*10).T, samples.T, c='r', alpha=.2)
Marginalization of model hyperparameters
Here we use TensorFlow Probability's MCMC functionality to perform marginalization of the model hyperparameters: kernel params as well as observation noise variance.
f = lambda x: np.sin(10*x[..., 0]) * np.exp(x[..., 0]**2)
observation_index_points = np.random.uniform(1., 1., 25)[..., np.newaxis]
observations = np.random.normal(f(observation_index_points), .05)
gaussian_process_model = tfd.JointDistributionSequential([
tfd.LogNormal(np.float64(0.), np.float64(1.)),
tfd.LogNormal(np.float64(0.), np.float64(1.)),
tfd.LogNormal(np.float64(0.), np.float64(1.)),
lambda noise_variance, length_scale, amplitude: tfd.GaussianProcess(
kernel=psd_kernels.ExponentiatedQuadratic(amplitude, length_scale),
index_points=observation_index_points,
observation_noise_variance=noise_variance)
])
initial_chain_states = [
1e1 * tf.ones([], dtype=np.float64, name='init_amplitude'),
1e1 * tf.ones([], dtype=np.float64, name='init_length_scale'),
1e1 * tf.ones([], dtype=np.float64, name='init_obs_noise_variance')
]
# Since HMC operates over unconstrained space, we need to transform the
# samples so they live in realspace.
unconstraining_bijectors = [
tfp.bijectors.Softplus(),
tfp.bijectors.Softplus(),
tfp.bijectors.Softplus(),
]
def unnormalized_log_posterior(*args):
return gaussian_process_model.log_prob(*args, x=observations)
num_results = 200
@tf.function
def run_mcmc():
return tfp.mcmc.sample_chain(
num_results=num_results,
num_burnin_steps=500,
num_steps_between_results=3,
current_state=initial_chain_states,
kernel=tfp.mcmc.TransformedTransitionKernel(
inner_kernel = tfp.mcmc.HamiltonianMonteCarlo(
target_log_prob_fn=unnormalized_log_posterior,
step_size=[np.float64(.15)],
num_leapfrog_steps=3),
bijector=unconstraining_bijectors),
trace_fn=lambda _, pkr: pkr.inner_results.is_accepted)
[
amplitudes,
length_scales,
observation_noise_variances
], is_accepted = run_mcmc()
print("Acceptance rate: {}".format(np.mean(is_accepted)))
# Now we can sample from the posterior predictive distribution at a new set
# of index points.
index_points = np.linspace(1., 1., 200)[..., np.newaxis]
gprm = tfd.GaussianProcessRegressionModel(
# Batch of `num_results` kernels parameterized by the MCMC samples.
kernel=psd_kernels.ExponentiatedQuadratic(amplitudes, length_scales),
index_points=index_points,
observation_index_points=observation_index_points,
observations=observations,
observation_noise_variance=observation_noise_variances)
samples = gprm.sample()
# Plot posterior samples and their mean, target function, and observations.
plt.plot(np.stack([index_points[:, 0]]*num_results).T,
samples.numpy().T,
c='r',
alpha=.01)
plt.plot(index_points[:, 0], np.mean(samples, axis=0), c='k')
plt.plot(index_points[:, 0], f(index_points))
plt.scatter(observation_index_points[:, 0], observations)
References
[1]: Carl Rasmussen, Chris Williams. Gaussian Processes For Machine Learning, 2006.
Raises  

ValueError

if either

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

name

name to give to the op 
Returns  

batch_shape

Tensor .

cdf
cdf(
value, name='cdf', **kwargs
)
Cumulative distribution function.
Given random variable X
, the cumulative distribution function cdf
is:
cdf(x) := P[X <= x]
Args  

value

float or double Tensor .

name

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

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

cdf

a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .

copy
copy(
**override_parameters_kwargs
)
Creates a deep copy of the distribution.
Args  

**override_parameters_kwargs

String/value dictionary of initialization arguments to override with new values. 
Returns  

distribution

A new instance of type(self) initialized from the union
of self.parameters and override_parameters_kwargs, i.e.,
dict(self.parameters, **override_parameters_kwargs) .

covariance
covariance(
name='covariance', **kwargs
)
Covariance.
Covariance is (possibly) defined only for nonscalarevent distributions.
For example, for a lengthk
, vectorvalued distribution, it is calculated
as,
Cov[i, j] = Covariance(X_i, X_j) = E[(X_i  E[X_i]) (X_j  E[X_j])]
where Cov
is a (batch of) k x k
matrix, 0 <= (i, j) < k
, and E
denotes expectation.
Alternatively, for nonvector, multivariate distributions (e.g.,
matrixvalued, Wishart), Covariance
shall return a (batch of) matrices
under some vectorization of the events, i.e.,
Cov[i, j] = Covariance(Vec(X)_i, Vec(X)_j) = [as above]
where Cov
is a (batch of) k' x k'
matrices,
0 <= (i, j) < k' = reduce_prod(event_shape)
, and Vec
is some function
mapping indices of this distribution's event dimensions to indices of a
lengthk'
vector.
Args  

name

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

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

covariance

Floatingpoint Tensor with shape [B1, ..., Bn, k', k']
where the first n dimensions are batch coordinates and
k' = reduce_prod(self.event_shape) .

cross_entropy
cross_entropy(
other, name='cross_entropy'
)
Computes the (Shannon) cross entropy.
Denote this distribution (self
) by P
and the other
distribution by
Q
. Assuming P, Q
are absolutely continuous with respect to
one another and permit densities p(x) dr(x)
and q(x) dr(x)
, (Shannon)
cross entropy is defined as:
H[P, Q] = E_p[log q(X)] = int_F p(x) log q(x) dr(x)
where F
denotes the support of the random variable X ~ P
.
other
types with builtin registrations: MultivariateNormalDiag
, MultivariateNormalDiagPlusLowRank
, MultivariateNormalFullCovariance
, MultivariateNormalLinearOperator
, MultivariateNormalTriL
Args  

other

tfp.distributions.Distribution instance.

name

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

Returns  

cross_entropy

self.dtype Tensor with shape [B1, ..., Bn]
representing n different calculations of (Shannon) cross entropy.

entropy
entropy(
name='entropy', **kwargs
)
Shannon entropy in nats.
event_shape_tensor
event_shape_tensor(
name='event_shape_tensor'
)
Shape of a single sample from a single batch as a 1D int32 Tensor
.
Args  

name

name to give to the op 
Returns  

event_shape

Tensor .

experimental_default_event_space_bijector
experimental_default_event_space_bijector(
*args, **kwargs
)
Bijector mapping the reals (R**n) to the event space of the distribution.
Distributions with continuous support may implement
_default_event_space_bijector
which returns a subclass of
tfp.bijectors.Bijector
that maps R**n to the distribution's event space.
For example, the default bijector for the Beta
distribution
is tfp.bijectors.Sigmoid()
, which maps the real line to [0, 1]
, the
support of the Beta
distribution. The default bijector for the
CholeskyLKJ
distribution is tfp.bijectors.CorrelationCholesky
, which
maps R^(k * (k1) // 2) to the submanifold of k x k lower triangular
matrices with ones along the diagonal.
The purpose of experimental_default_event_space_bijector
is
to enable gradient descent in an unconstrained space for Variational
Inference and Hamiltonian Monte Carlo methods. Some effort has been made to
choose bijectors such that the tails of the distribution in the
unconstrained space are between Gaussian and Exponential.
For distributions with discrete event space, or for which TFP currently
lacks a suitable bijector, this function returns None
.
Args  

*args

Passed to implementation _default_event_space_bijector .

**kwargs

Passed to implementation _default_event_space_bijector .

Returns  

event_space_bijector

Bijector instance or None .

experimental_fit
@classmethod
experimental_fit( value, sample_ndims=1, validate_args=False, **init_kwargs )
Instantiates a distribution that maximizes the likelihood of x
.
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. .

Returns  

maximum_likelihood_instance

instance of cls with parameters that
maximize the likelihood of value .

experimental_local_measure
experimental_local_measure(
value, backward_compat=False, **kwargs
)
Returns a log probability density together with a TangentSpace
.
A TangentSpace
allows us to calculate the correct pushforward
density when we apply a transformation to a Distribution
on
a strict submanifold of R^n (typically via a Bijector
in the
TransformedDistribution
subclass). The density correction uses
the basis of the tangent space.
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 .

Raises  

UnspecifiedTangentSpaceError if backward_compat is False and
the _experimental_tangent_space attribute has not been defined.

experimental_sample_and_log_prob
experimental_sample_and_log_prob(
sample_shape=(), seed=None, name='sample_and_log_prob', **kwargs
)
Samples from this distribution and returns the log density of the sample.
The default implementation simply calls sample
and log_prob
:
def _sample_and_log_prob(self, sample_shape, seed, **kwargs):
x = self.sample(sample_shape=sample_shape, seed=seed, **kwargs)
return x, self.log_prob(x, **kwargs)
However, some subclasses may provide more efficient and/or numerically stable implementations.
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 .

get_marginal_distribution
get_marginal_distribution(
index_points=None
)
Compute the marginal of this GP over function values at index_points
.
Args  

index_points

(nested) Tensor representing finite (batch of) vector(s)
of points in the index set over which the GP is defined. Shape (or
the 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 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 distribution with vector event shape. 
is_scalar_batch
is_scalar_batch(
name='is_scalar_batch'
)
Indicates that batch_shape == []
.
Args  

name

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

Returns  

is_scalar_batch

bool scalar Tensor .

is_scalar_event
is_scalar_event(
name='is_scalar_event'
)
Indicates that event_shape == []
.
Args  

name

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

Returns  

is_scalar_event

bool scalar Tensor .

kl_divergence
kl_divergence(
other, name='kl_divergence'
)
Computes the KullbackLeibler divergence.
Denote this distribution (self
) by p
and the other
distribution by
q
. Assuming p, q
are absolutely continuous with respect to reference
measure r
, the KL divergence is defined as:
KL[p, q] = E_p[log(p(X)/q(X))]
= int_F p(x) log q(x) dr(x) + int_F p(x) log p(x) dr(x)
= H[p, q]  H[p]
where F
denotes the support of the random variable X ~ p
, H[., .]
denotes (Shannon) cross entropy, and H[.]
denotes (Shannon) entropy.
other
types with builtin registrations: MultivariateNormalDiag
, MultivariateNormalDiagPlusLowRank
, MultivariateNormalFullCovariance
, MultivariateNormalLinearOperator
, MultivariateNormalTriL
Args  

other

tfp.distributions.Distribution instance.

name

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

Returns  

kl_divergence

self.dtype Tensor with shape [B1, ..., Bn]
representing n different calculations of the KullbackLeibler
divergence.

log_cdf
log_cdf(
value, name='log_cdf', **kwargs
)
Log cumulative distribution function.
Given random variable X
, the cumulative distribution function cdf
is:
log_cdf(x) := Log[ P[X <= x] ]
Often, a numerical approximation can be used for log_cdf(x)
that yields
a more accurate answer than simply taking the logarithm of the cdf
when
x << 1
.
Args  

value

float or double Tensor .

name

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

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

logcdf

a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .

log_prob
log_prob(
value, name='log_prob', **kwargs
)
Log probability density/mass function.
Additional documentation from GaussianProcess
:
kwargs
:
index_points
: optionalfloat
Tensor
representing a finite (batch of) of points in the index set over which this GP is defined. The shape (or shape of each nested component) has the form[b1, ..., bB, e,f1, ..., fF]
whereF
is the number of feature dimensions and must equalself.kernel.feature_ndims
(or its corresponding nested component) ande
is the number of index points in each batch. Ultimately, this distribution corresponds to ane
dimensional multivariate normal. The batch shape must be broadcastable withkernel.batch_shape
and any batch dims yieldedbymean_fn
. If not specified,self.index_points
is used. Default value:None
.is_missing
: optionalbool
Tensor
of shape[..., e]
, wheree
is the number of index points in each batch. Represents a batch of Boolean masks. Whenis_missing
is notNone
, the returned logprob is for the marginal distribution, in which all dimensions for whichis_missing
isTrue
have been marginalized out. The batch dimensions ofis_missing
must broadcast with the sample and batch dimensions ofvalue
and of thisDistribution
. Default value:None
.
Args  

value

float or double Tensor .

name

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

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

log_prob

a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .

log_survival_function
log_survival_function(
value, name='log_survival_function', **kwargs
)
Log survival function.
Given random variable X
, the survival function is defined:
log_survival_function(x) = Log[ P[X > x] ]
= Log[ 1  P[X <= x] ]
= Log[ 1  cdf(x) ]
Typically, different numerical approximations can be used for the log
survival function, which are more accurate than 1  cdf(x)
when x >> 1
.
Args  

value

float or double Tensor .

name

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

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

Tensor of shape sample_shape(x) + self.batch_shape with values of type
self.dtype .

mean
mean(
name='mean', **kwargs
)
Mean.
mode
mode(
name='mode', **kwargs
)
Mode.
param_shapes
@classmethod
param_shapes( sample_shape, name='DistributionParamShapes' )
Shapes of parameters given the desired shape of a call to sample()
.
This is a class method that describes what key/value arguments are required
to instantiate the given Distribution
so that a particular shape is
returned for that instance's call to sample()
.
Subclasses should override class method _param_shapes
.
Args  

sample_shape

Tensor or python list/tuple. Desired shape of a call to
sample() .

name

name to prepend ops with. 
Returns  

dict of parameter name to Tensor shapes.

param_static_shapes
@classmethod
param_static_shapes( sample_shape )
param_shapes with static (i.e. TensorShape
) shapes.
This is a class method that describes what key/value arguments are required
to instantiate the given Distribution
so that a particular shape is
returned for that instance's call to sample()
. Assumes that the sample's
shape is known statically.
Subclasses should override class method _param_shapes
to return
constantvalued tensors when constant values are fed.
Args  

sample_shape

TensorShape or python list/tuple. Desired shape of a call
to sample() .

Returns  

dict of parameter name to TensorShape .

Raises  

ValueError

if sample_shape is a TensorShape and is not fully defined.

parameter_properties
@classmethod
parameter_properties( dtype=tf.float32, num_classes=None )
Returns a dict mapping constructor arg names to property annotations.
This dict should include an entry for each of the distribution's
Tensor
valued constructor arguments.
Distribution subclasses are not required to implement
_parameter_properties
, so this method may raise NotImplementedError
.
Providing a _parameter_properties
implementation enables several advanced
features, including:
 Distribution batch slicing (
sliced_distribution = distribution[i:j]
).  Automatic inference of
_batch_shape
and_batch_shape_tensor
, which must otherwise be computed explicitly.  Automatic instantiation of the distribution within TFP's internal property tests.
 Automatic construction of 'trainable' instances of the distribution using appropriate bijectors to avoid violating parameter constraints. This enables the distribution family to be used easily as a surrogate posterior in variational inference.
In the future, parameter property annotations may enable additional
functionality; for example, returning Distribution instances from
tf.vectorized_map
.
Args  

dtype

Optional float dtype to assume for continuousvalued parameters.
Some constraining bijectors require advance knowledge of the dtype
because certain constants (e.g., tfb.Softplus.low ) must be
instantiated with the same dtype as the values to be transformed.

num_classes

Optional int Tensor number of classes to assume when
inferring the shape of parameters for categoricallike distributions.
Otherwise ignored.

Returns  

parameter_properties

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

Raises  

NotImplementedError

if the distribution class does not implement
_parameter_properties .

posterior_predictive
posterior_predictive(
observations, predictive_index_points=None, **kwargs
)
Return the posterior predictive distribution associated with this distribution.
Returns the posterior predictive distribution p(Y'  X, Y, X')
where:
X'
ispredictive_index_points
X
isself.index_points
.Y
isobservations
.
This is equivalent to using the
GaussianProcessRegressionModel.precompute_regression_model
method.
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. 
Returns  

gprm

An instance of Distribution that represents the posterior
predictive.

precompute_regression_model
@staticmethod
precompute_regression_model( kernel, observation_index_points, observations, observations_is_missing=None, index_points=None, observation_noise_variance=0.0, predictive_noise_variance=None, mean_fn=None, cholesky_fn=None, jitter=1e06, validate_args=False, allow_nan_stats=False, name='PrecomputedGaussianProcessRegressionModel', _precomputed_divisor_matrix_cholesky=None, _precomputed_solve_on_observation=None )
Returns a GaussianProcessRegressionModel with precomputed quantities.
This differs from the constructor by precomputing quantities associated with
observations in a nontape safe way. index_points
is the only parameter
that is allowed to vary (i.e. is a Variable
/ changes after
initialization).
Specifically:
 We make
observation_index_points
andobservations
mandatory parameters.  We precompute
kernel(observation_index_points, observation_index_points)
along with any other associated quantities relating to thekernel
,observations
andobservation_index_points
.
A typical usecase would be optimizing kernel hyperparameters for a
GaussianProcess
, and computing the posterior predictive with respect to
those optimized hyperparameters and observation / indexpoints pairs.
Args  

kernel

PositiveSemidefiniteKernel like instance representing the
GP's covariance function.

observation_index_points

(nested) Tensor representing finite
collection, or batch of collections, of points in the index set for
which some data has been observed. 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 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 (kernel.batch_shape , index_points , etc). The
default value is None , which corresponds to the empty set of
observations, and simply results in the prior predictive model (a GP
with noise of variance predictive_noise_variance ).

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 , index_points , etc.). The default value is
None , which corresponds to the empty set of observations, and simply
results in the prior predictive model (a GP with noise of variance
predictive_noise_variance ).

observations_is_missing

bool Tensor of shape [..., e] ,
representing a batch of boolean masks. When observations_is_missing
is not None , the returned distribution is conditioned only on the
observations for which the corresponding elements of
observations_is_missing are True .

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 index points in each
batch. Ultimately this distribution corresponds to an e dimensional
multivariate normal. The batch shape must be broadcastable with
kernel.batch_shape and any batch dims yielded by mean_fn .

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

mean_fn

Python callable that acts on index_points to produce a
collection, or batch of collections, of mean values at index_points .
Takes a (nested) Tensor of shape [b1, ..., bB, e, f1, ..., fF] and
returns a Tensor whose shape is broadcastable with [b1, ..., bB, e] .
Default value: None implies the constant zero function.

cholesky_fn

Callable which takes a single (batch) matrix argument and
returns a Choleskylike lower triangular factor. Default value: None ,
in which case make_cholesky_with_jitter_fn is used with the jitter
parameter.

jitter

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

validate_args

Python bool , default False . When True distribution
parameters are checked for validity despite possibly degrading runtime
performance. When False invalid inputs may silently render incorrect
outputs.
Default value: False .

allow_nan_stats

Python bool , default True . When True ,
statistics (e.g., mean, mode, variance) use the value NaN to
indicate the result is undefined. When False , an exception is raised
if one or more of the statistic's batch members are undefined.
Default value: False .

name

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

_precomputed_divisor_matrix_cholesky

Internal parameter  do not use. 
_precomputed_solve_on_observation

Internal parameter  do not use. 
Returns
An instance of GaussianProcessRegressionModel
with precomputed
quantities associated with observations.
prob
prob(
value, name='prob', **kwargs
)
Probability density/mass function.
Args  

value

float or double Tensor .

name

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

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

prob

a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .

quantile
quantile(
value, name='quantile', **kwargs
)
Quantile function. Aka 'inverse cdf' or 'percent point function'.
Given random variable X
and p in [0, 1]
, the quantile
is:
quantile(p) := x such that P[X <= x] == p
Args  

value

float or double Tensor .

name

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

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

quantile

a Tensor of shape sample_shape(x) + self.batch_shape with
values of type self.dtype .

sample
sample(
sample_shape=(), seed=None, name='sample', **kwargs
)
Generate samples of the specified shape.
Note that a call to sample()
without arguments will generate a single
sample.
Args  

sample_shape

0D or 1D int32 Tensor . Shape of the generated samples.

seed

PRNG seed; see tfp.random.sanitize_seed for details.

name

name to give to the op. 
**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

samples

a Tensor with prepended dimensions sample_shape .

stddev
stddev(
name='stddev', **kwargs
)
Standard deviation.
Standard deviation is defined as,
stddev = E[(X  E[X])**2]**0.5
where X
is the random variable associated with this distribution, E
denotes expectation, and stddev.shape = batch_shape + event_shape
.
Args  

name

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

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

stddev

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

survival_function
survival_function(
value, name='survival_function', **kwargs
)
Survival function.
Given random variable X
, the survival function is defined:
survival_function(x) = P[X > x]
= 1  P[X <= x]
= 1  cdf(x).
Args  

value

float or double Tensor .

name

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

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

Tensor of shape sample_shape(x) + self.batch_shape with values of type
self.dtype .

unnormalized_log_prob
unnormalized_log_prob(
value, name='unnormalized_log_prob', **kwargs
)
Potentially unnormalized log probability density/mass function.
This function is similar to log_prob
, but does not require that the
return value be normalized. (Normalization here refers to the total
integral of probability being one, as it should be by definition for any
probability distribution.) This is useful, for example, for distributions
where the normalization constant is difficult or expensive to compute. By
default, this simply calls log_prob
.
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  

unnormalized_log_prob

a Tensor of shape
sample_shape(x) + self.batch_shape with values of type self.dtype .

variance
variance(
name='variance', **kwargs
)
Variance.
Variance is defined as,
Var = E[(X  E[X])**2]
where X
is the random variable associated with this distribution, E
denotes expectation, and Var.shape = batch_shape + event_shape
.
Args  

name

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

**kwargs

Named arguments forwarded to subclass implementation. 
Returns  

variance

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

__getitem__
__getitem__(
slices
) > 'GaussianProcessRegressionModel'
Slices the batch axes of this distribution, returning a new instance.
b = tfd.Bernoulli(logits=tf.zeros([3, 5, 7, 9]))
b.batch_shape # => [3, 5, 7, 9]
b2 = b[:, tf.newaxis, ..., 2:, 1::2]
b2.batch_shape # => [3, 1, 5, 2, 4]
x = tf.random.stateless_normal([5, 3, 2, 2])
cov = tf.matmul(x, x, transpose_b=True)
chol = tf.linalg.cholesky(cov)
loc = tf.random.stateless_normal([4, 1, 3, 1])
mvn = tfd.MultivariateNormalTriL(loc, chol)
mvn.batch_shape # => [4, 5, 3]
mvn.event_shape # => [2]
mvn2 = mvn[:, 3:, ..., ::1, tf.newaxis]
mvn2.batch_shape # => [4, 2, 3, 1]
mvn2.event_shape # => [2]
Args  

slices

slices from the [] operator 
Returns  

dist

A new tfd.Distribution instance with sliced parameters.

__iter__
__iter__()