Learning Block Structured Graphs in Gaussian Graphical Models

Abstract

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.

KEYWORDS:

• Bayesian statistics
• Conditional independence
• Functional data analysis
• G-Wishart prior
• Reversible jump MCMC

Supplementary Materials

AppendixThe appendix provides the details on how to derive the acceptance-rejection probability of the BDRJ algorithm and its schematic description (appendix.pdf)

R-package BGSL:R-package BGSL containing code implementing the BDRJ algorithm described in the article. The package also contains all datasets used as examples in the article. (BGSL-main.zip)

Acknowledgments

We would like to thank the Editor, the Associate Editor and two Referees for the insightful and constructive comments. The author thank Alessandra Guglielmi for valuable suggestions. Moreover, the author thank Matteo Gianella and Laura Codazzi for the useful discussion about the real data analysis.

Disclosure Statement

The authors report there are no competing interests to declare.

Additional information

Funding

The research of the third and fourth author has been partially supported by a grant from Università Cattolica del Sacro Cuore, Italy (track D1).