  We develop a Bayesian graphical modeling framework for functional data for
correlated multivariate random variables observed over a continuous domain. Our
method leads to graphical Markov models for functional data which allows the
graphs to vary over the functional domain. The model involves estimation of
graphical models that evolve functionally in a nonparametric fashion while
accounting for within-functional correlations and borrowing strength across
functional positions so contiguous locations are encouraged but not forced to
have similar graph structure and edge strength. We utilize a strategy that
combines nonparametric basis function modeling with modified Bayesian graphical
regularization techniques, which induces a new class of hypoexponential normal
scale mixture distributions that not only leads to adaptively shrunken
estimators of the conditional cross-covariance but also facilitates a thorough
theoretical investigation of the shrinkage properties. Our approach scales up
to large functional datasets collected on a fine grid. We show through
simulations and real data analysis that the Bayesian functional graphical model
can efficiently reconstruct the functionally-evolving graphical models by
accounting for within-function correlations.
