tfp.distributions.GaussianProcessRegressionModel

Posterior predictive distribution in a conjugate GP regression model.

Inherits From: GaussianProcess

tfp.distributions.GaussianProcessRegressionModel(
    kernel, index_points=None, observation_index_points=None, observations=None,
    observation_noise_variance=0.0, predictive_noise_variance=None, mean_fn=None,
    jitter=1e-06, validate_args=False, allow_nan_stats=False,
    name='GaussianProcessRegressionModel'
)

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 collection x",
m(t) refers to the collection of values of the mean function at the index points in the collection t, and
k(x, t) refers to the matrix whose entries are values of the kernel function k at all pairs of index points from x and t.

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, f(x)) ~ 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
import tensorflow.compat.v2 as tf
import tensorflow_probability as tfp

tf.enable_v2_behavior()

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.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 = [
    1e-1 * tf.ones([], dtype=np.float64, name='init_amplitude'),
    1e-1 * tf.ones([], dtype=np.float64, name='init_length_scale'),
    1e-1 * 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 real-space.
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.

Args
`kernel`	`PositiveSemidefiniteKernel`-like instance representing the GP's covariance function.
`index_points`	`float` `Tensor` representing finite collection, or batch of collections, of points in the index set over which the GP is defined. Shape has the form `[b1, ..., bB, e, f1, ..., fF]` where `F` is the number of feature dimensions and must equal `kernel.feature_ndims` and `e` is the number (size) of index points in each batch. Ultimately this distribution corresponds to an `e`-dimensional multivariate normal. The batch shape must be broadcastable with `kernel.batch_shape` and any batch dims yielded by `mean_fn`.
`observation_index_points`	`float` `Tensor` representing finite collection, or batch of collections, of points in the index set for which some data has been observed. Shape has the form `[b1, ..., bB, e, f1, ..., fF]` where `F` is the number of feature dimensions and must equal `kernel.feature_ndims`, and `e` is the number (size) of index points in each batch. `[b1, ..., bB, e]` must be broadcastable with the shape of `observations`, and `[b1, ..., bB]` must be broadcastable with the shapes of all other batched parameters (`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`).
`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 re-use `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 `Tensor` of shape `[b1, ..., bB, f1, ..., fF]` and returns a `Tensor` whose shape is broadcastable with `[b1, ..., bB]`. Default value: `None` implies the constant zero function.
`jitter`	`float` scalar `Tensor` added to the diagonal of the covariance matrix to ensure positive definiteness of the covariance matrix. Default value: `1e-6`.
`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: 'GaussianProcessRegressionModel'.

Raises
`ValueError`	if either only one of `observations` and `observation_index_points` is given, or `mean_fn` is not `None` and not callable.

Raises

ValueError

if either

only one of observations and observation_index_points is given, or
mean_fn is not None and 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.
`dtype`	The `DType` of `Tensor`s handled by this `Distribution`.
`event_shape`	Shape of a single sample from a single batch as a `TensorShape`. May be partially defined or unknown.
`index_points`
`jitter`
`kernel`
`mean_fn`
`name`	Name prepended to all ops created by this `Distribution`.
`name_scope`	Returns a `tf.name_scope` instance for this class.
`observation_index_points`
`observation_noise_variance`
`observations`
`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.

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

Args
`sample_shape`	0D or 1D `int32` `Tensor`. Shape of the generated samples.
`seed`	Python integer or `tfp.util.SeedStream` instance, for seeding PRNG.
`name`	name to give to the op.
`**kwargs`	Named arguments forwarded to subclass implementation.

tfp.distributions.GaussianProcessRegressionModel

Mathematical Details

Examples

Draw joint samples from the posterior predictive distribution in a GP

Optimize model parameters via maximum marginal likelihood

Marginalization of model hyperparameters

References

Args

Raises

Attributes

Methods

batch_shape_tensor

cdf

copy

covariance

cross_entropy

entropy

event_shape_tensor

get_marginal_distribution

is_scalar_batch

is_scalar_event

kl_divergence

log_cdf

log_prob

log_survival_function

mean

mode

param_shapes

param_static_shapes

prob

quantile

sample

stddev

survival_function

variance

with_name_scope

__getitem__

__iter__

`batch_shape_tensor`

`cdf`

`copy`

`covariance`

`cross_entropy`

`entropy`

`event_shape_tensor`

`get_marginal_distribution`

`is_scalar_batch`

`is_scalar_event`

`kl_divergence`

`log_cdf`

`log_prob`

`log_survival_function`

`mean`

`mode`

`param_shapes`

`param_static_shapes`

`prob`

`quantile`

`sample`

`stddev`

`survival_function`

`variance`

`with_name_scope`

`getitem`

`iter`