  Graphical models express conditional independence relationships among
variables. Although methods for vector-valued data are well established,
functional data graphical models remain underdeveloped. We introduce a notion
of conditional independence between random functions, and construct a framework
for Bayesian inference of undirected, decomposable graphs in the multivariate
functional data context. This framework is based on extending Markov
distributions and hyper Markov laws from random variables to random processes,
providing a principled alternative to naive application of multivariate methods
to discretized functional data. Markov properties facilitate the composition of
likelihoods and priors according to the decomposition of a graph. Our focus is
on Gaussian process graphical models using orthogonal basis expansions. We
propose a hyper-inverse-Wishart-process prior for the covariance kernels of the
infinite coefficient sequences of the basis expansion, establish existence,
uniqueness, strong hyper Markov property, and conjugacy. Stochastic search
Markov chain Monte Carlo algorithms are developed for posterior inference,
assessed through simulations, and applied to a study of brain activity and
alcoholism.
