Bayesian Inference in Decomposable Graphical Models Using Sequential Monte Carlo Methods

In this study we present a sequential sampling methodology for Bayesian inference in decomposable graphical models. We recast the problem of graph estimation, which in general lacks natural sequential interpretation, into a sequential setting. Specifically, we propose a recursive Feynman-Kac model which generates a flow of junction tree distributions over a space of increasing dimensions and develop an efficient sequential Monte Carlo sampler. As a key ingredient of the proposal kernel in our sampler we use the Christmas tree algorithm developed in the companion paper Olsson et al. [2017]. We focus on particle MCMC methods, in particular particle Gibbs (PG) as it allows for generating MCMC chains with global moves on an underlying space of decomposable graphs. To further improve the algorithm mixing properties of this PG, we incorporate a systematic refreshment step implemented through direct sampling from a backward kernel. The theoretical properties of the algorithm are investigated, showing in particular that the refreshment step improves the algorithm performance in terms of asymptotic variance of the estimated distribution. Performance accuracy of the graph estimators are illustrated through a collection of numerical examples demonstrating the feasibility of the suggested approach in both discrete and continuous graphical models.