  Functional data analysis, which models data as realizations of random
functions over a continuum, has emerged as a useful tool for time series data.
Often, the goal is to infer the dynamic connections (or time-varying
conditional dependencies) among multiple functions or time series. For this
task, we propose a dynamic and Bayesian functional graphical model. Our
modeling approach prioritizes the careful definition of an appropriate graph to
identify both time-invariant and time-varying connectivity patterns. We
introduce a novel block-structured sparsity prior paired with a finite basis
expansion, which together yield effective shrinkage and graph selection with
efficient computations via a Gibbs sampling algorithm. Crucially, the model
includes (one or more) graph changepoints, which are learned jointly with all
model parameters and incorporate graph dynamics. Simulation studies demonstrate
excellent graph selection capabilities, with significant improvements over
competing methods. We apply the proposed approach to study of dynamic
connectivity patterns of sea surface temperatures in the Pacific Ocean and
discovers meaningful edges.
