Learning Block Structured Graphs in Gaussian Graphical Models

A prior distribution for the underlying graph is introduced in the framework of Gaussian graphical models. Such a prior distribution induces a block structure in the graph’s adjacency matrix, allowing learning relationships between fixed groups of variables. A novel sampling strategy named Double Reversible Jumps Markov chain Monte Carlo is developed for learning block structured graphs under the conjugate G-Wishart prior. The algorithm proposes moves that add or remove not just a single edge of the graph but an entire group of edges. The method is then applied to smooth functional data. The classical smoothing procedure is improved by placing a graphical model on the basis expansion coefficients, providing an estimate of their conditional dependence structure. Since the elements of a B-Spline basis have compact support, the conditional dependence structure is reflected on well-defined portions of the domain. A known partition of the functional domain is exploited to investigate relationships among portions of the domain and improve the interpretability of the results. Supplementary materials for this article are available online.