R/distributions.R
tfd_gaussian_process.Rd
A Gaussian process (GP) is an indexed collection of random variables, any finite collection of which are jointly Gaussian. While this definition applies to finite index sets, it is typically implicit that the index set is infinite; in applications, it is often some finite dimensional real or complex vector space. In such cases, the GP may be thought of as a distribution over (real or complexvalued) functions defined over the index set.
tfd_gaussian_process( kernel, index_points, mean_fn = NULL, observation_noise_variance = 0, jitter = 1e06, validate_args = FALSE, allow_nan_stats = FALSE, name = "GaussianProcess" )
kernel 


index_points 

mean_fn  function that acts on index points to produce a (batch
of) vector(s) of mean values at those index points. Takes a 
observation_noise_variance 

jitter 

validate_args  Logical, 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  Logical, 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. 
name  name prefixed to Ops created by this class. 
a distribution instance.
Just as Gaussian distributions are fully specified by their first and second
moments, a Gaussian process can be completely specified by a mean and
covariance function.
Let S
denote the index set and K
the space in which
each indexed random variable takes its values (again, often R or C). The mean
function is then a map m: S > K
, and the covariance function, or kernel, is
a positivedefinite function k: (S x S) > K
. The properties of functions
drawn from a GP are entirely dictated (up to translation) by the form of the
kernel function.
This Distribution
represents the marginal joint distribution over function
values at a given finite collection of points [x[1], ..., x[N]]
from the
index set S
. By definition, this marginal distribution is just a
multivariate normal distribution, whose mean is given by the vector
[ m(x[1]), ..., m(x[N]) ]
and whose covariance matrix is constructed from
pairwise applications of the kernel function to the given inputs:
 k(x[1], x[1]) k(x[1], x[2]) ... k(x[1], x[N])   k(x[2], x[1]) k(x[2], x[2]) ... k(x[2], x[N])   ... ... ...   k(x[N], x[1]) k(x[N], x[2]) ... k(x[N], x[N]) 
For this to be a valid covariance matrix, it must be symmetric and positive
definite; hence the requirement that k
be a positive definite function
(which, by definition, says that the above procedure will yield PD matrices).
We also support the inclusion of zeromean Gaussian noise in the model, via
the observation_noise_variance
parameter. This augments the generative model
to
f ~ GP(m, k) (y[i]  f, x[i]) ~ Normal(f(x[i]), s)
where
m
is the mean function
k
is the covariance kernel function
f
is the function drawn from the GP
x[i]
are the index points at which the function is observed
y[i]
are the observed values at the index points
s
is the scale of the observation noise.
Note that this class represents an unconditional Gaussian process; it does not implement posterior inference conditional on observed function evaluations. This class is useful, for example, if one wishes to combine a GP prior with a nonconjugate likelihood using MCMC to sample from the posterior.
Mathematical Details
The probability density function (pdf) is a multivariate normal whose parameters are derived from the GP's properties:
pdf(x; index_points, mean_fn, kernel) = exp(0.5 * y) / Z K = (kernel.matrix(index_points, index_points) + (observation_noise_variance + jitter) * eye(N)) y = (x  mean_fn(index_points))^T @ K @ (x  mean_fn(index_points)) Z = (2 * pi)**(.5 * N) det(K)**(.5)
where:
index_points
are points in the index set over which the GP is defined,
mean_fn
is a callable mapping the index set to the GP's mean values,
kernel
is PositiveSemidefiniteKernel
like and represents the covariance
function of the GP,
observation_noise_variance
represents (optional) observation noise.
jitter
is added to the diagonal to ensure positive definiteness up to
machine precision (otherwise Choleskydecomposition is prone to failure),
eye(N)
is an NbyN identity matrix.
For usage examples see e.g. tfd_sample()
, tfd_log_prob()
, tfd_mean()
.
Other distributions:
tfd_autoregressive()
,
tfd_batch_reshape()
,
tfd_bates()
,
tfd_bernoulli()
,
tfd_beta_binomial()
,
tfd_beta()
,
tfd_binomial()
,
tfd_categorical()
,
tfd_cauchy()
,
tfd_chi2()
,
tfd_chi()
,
tfd_cholesky_lkj()
,
tfd_continuous_bernoulli()
,
tfd_deterministic()
,
tfd_dirichlet_multinomial()
,
tfd_dirichlet()
,
tfd_empirical()
,
tfd_exp_gamma()
,
tfd_exp_inverse_gamma()
,
tfd_exponential()
,
tfd_gamma_gamma()
,
tfd_gamma()
,
tfd_gaussian_process_regression_model()
,
tfd_generalized_normal()
,
tfd_geometric()
,
tfd_gumbel()
,
tfd_half_cauchy()
,
tfd_half_normal()
,
tfd_hidden_markov_model()
,
tfd_horseshoe()
,
tfd_independent()
,
tfd_inverse_gamma()
,
tfd_inverse_gaussian()
,
tfd_johnson_s_u()
,
tfd_joint_distribution_named_auto_batched()
,
tfd_joint_distribution_named()
,
tfd_joint_distribution_sequential_auto_batched()
,
tfd_joint_distribution_sequential()
,
tfd_kumaraswamy()
,
tfd_laplace()
,
tfd_linear_gaussian_state_space_model()
,
tfd_lkj()
,
tfd_log_logistic()
,
tfd_log_normal()
,
tfd_logistic()
,
tfd_mixture_same_family()
,
tfd_mixture()
,
tfd_multinomial()
,
tfd_multivariate_normal_diag_plus_low_rank()
,
tfd_multivariate_normal_diag()
,
tfd_multivariate_normal_full_covariance()
,
tfd_multivariate_normal_linear_operator()
,
tfd_multivariate_normal_tri_l()
,
tfd_multivariate_student_t_linear_operator()
,
tfd_negative_binomial()
,
tfd_normal()
,
tfd_one_hot_categorical()
,
tfd_pareto()
,
tfd_pixel_cnn()
,
tfd_poisson_log_normal_quadrature_compound()
,
tfd_poisson()
,
tfd_power_spherical()
,
tfd_probit_bernoulli()
,
tfd_quantized()
,
tfd_relaxed_bernoulli()
,
tfd_relaxed_one_hot_categorical()
,
tfd_sample_distribution()
,
tfd_sinh_arcsinh()
,
tfd_skellam()
,
tfd_spherical_uniform()
,
tfd_student_t_process()
,
tfd_student_t()
,
tfd_transformed_distribution()
,
tfd_triangular()
,
tfd_truncated_cauchy()
,
tfd_truncated_normal()
,
tfd_uniform()
,
tfd_variational_gaussian_process()
,
tfd_vector_diffeomixture()
,
tfd_vector_exponential_diag()
,
tfd_vector_exponential_linear_operator()
,
tfd_vector_laplace_diag()
,
tfd_vector_laplace_linear_operator()
,
tfd_vector_sinh_arcsinh_diag()
,
tfd_von_mises_fisher()
,
tfd_von_mises()
,
tfd_weibull()
,
tfd_wishart_linear_operator()
,
tfd_wishart_tri_l()
,
tfd_wishart()
,
tfd_zipf()