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 = k(t, x) @ inv(k(x, x) + v * I) @ (y - loc)
  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 as tf
import tensorflow_probability as tfp

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 transform the trainable
# variables to apply a positivity constraint.
amplitude = tf.exp(tf.Variable(np.float64(0)), name='amplitude')
length_scale = tf.exp(tf.Variable(np.float64(0)), name='length_scale')
kernel = psd_kernels.ExponentiatedQuadratic(amplitude, length_scale)

observation_noise_variance = tf.exp(
    tf.Variable(np.float64(-5)), 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)
neg_log_likelihood = -gp.log_prob(observations)

optimizer = tf.train.AdamOptimizer(learning_rate=.05, beta1=.5, beta2=.99)
optimize = optimizer.minimize(neg_log_likelihood)

# 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)

samples = gprm.sample(10)
# ==> 10 independently drawn, joint samples at `index_points`.

# Now execute the above ops in a Session, first training the model
# parameters, then drawing and plotting posterior samples.
with tf.Session() as sess:
  sess.run(tf.global_variables_initializer())

  for i in range(1000):
    _, neg_log_likelihood_ = sess.run([optimize, neg_log_likelihood])
    if i % 100 == 0:
      print("Step {}: NLL = {}".format(i, neg_log_likelihood_))

  print("Final NLL = {}".format(neg_log_likelihood_))
  samples_ = sess.run(samples)

  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)

def joint_log_prob(
    index_points, observations, amplitude, length_scale, noise_variance):

  # Hyperparameter Distributions.
  rv_amplitude = tfd.LogNormal(np.float64(0.), np.float64(1))
  rv_length_scale = tfd.LogNormal(np.float64(0.), np.float64(1))
  rv_noise_variance = tfd.LogNormal(np.float64(0.), np.float64(1))

  gp = tfd.GaussianProcess(
      kernel=psd_kernels.ExponentiatedQuadratic(amplitude, length_scale),
      index_points=index_points,
      observation_noise_variance=noise_variance)

  return (
      rv_amplitude.log_prob(amplitude) +
      rv_length_scale.log_prob(length_scale) +
      rv_noise_variance.log_prob(noise_variance) +
      gp.log_prob(observations)
  )

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(amplitude, length_scale, noise_variance):
  return joint_log_prob(
      observation_index_points, observations, amplitude, length_scale,
      noise_variance)

num_results = 200
[
    amplitudes,
    length_scales,
    observation_noise_variances
], kernel_results = 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))

# 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()

with tf.Session() as sess:
  kernel_results_, samples_ = sess.run([kernel_results, samples])

  print("Acceptance rate: {}".format(
      np.mean(kernel_results_.inner_results.is_accepted)))

  # Plot posterior samples and their mean, target function, and observations.
  plt.plot(np.stack([index_points[:, 0]]*num_results).T,
           samples_.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'.

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 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.
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
assert list(a.submodules) == [b, c]
assert list(b.submodules) == [c]
assert list(c.submodules) == []

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.

Raises:

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

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:

Attributes:

Raises:

Methods

__getitem__

Args:

Returns:

__iter__

batch_shape_tensor

Args:

Returns:

cdf

Args:

Returns:

copy

Args:

Returns:

covariance

Args:

Returns:

cross_entropy

Args:

Returns:

entropy

event_shape_tensor

Args:

Returns:

get_marginal_distribution

Args:

Returns:

is_scalar_batch

Args:

Returns:

is_scalar_event

Args:

Returns:

kl_divergence

Args:

Returns:

log_cdf

Args:

Returns:

log_prob

Args:

Returns:

log_survival_function

Args:

Returns:

mean

mode

param_shapes

Args:

Returns:

param_static_shapes

Args:

Returns:

Raises:

prob

Args:

Returns:

quantile

Args:

Returns:

sample

Args:

Returns:

stddev

Args:

Returns:

survival_function

Args:

Returns:

variance

`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`