Bayesian learning of weakly structural Markov graph laws using sequential Monte Carlo methods

TL;DR

This paper introduces a sequential Monte Carlo-based Bayesian method for learning weakly structural Markov graph laws, enabling efficient graph estimation in decomposable graphical models with improved sampling techniques.

Contribution

It recasts graph estimation into a sequential framework using a recursive Feynman-Kac model and enhances particle Gibbs sampling with a systematic refreshment step for better performance.

Findings

Demonstrates high accuracy in Bayesian graph structure learning

Shows improved mixing properties with the refreshment step

Validates the methodology through numerical examples

Abstract

We present a sequential sampling methodology for weakly structural Markov laws, arising naturally in a Bayesian structure learning context for decomposable graphical models. As a key component of our suggested approach, we show that the problem of graph estimation, which in general lacks natural sequential interpretation, can be recast into a sequential setting by proposing a recursive Feynman-Kac model that generates a flow of junction tree distributions over a space of increasing dimensions. We focus on particle McMC methods to provide samples on this space, in particular on particle Gibbs (PG), as it allows for generating McMC chains with global moves on an underlying space of decomposable graphs. To further improve the PG mixing properties, we incorporate a systematic refreshment step implemented through direct sampling from a backward kernel. The theoretical properties of the algorithm are investigated, showing that the proposed refreshment step improves the performance in terms of asymptotic variance of the estimated distribution. The suggested sampling methodology is illustrated through a collection of numerical examples demonstrating high accuracy in Bayesian graph structure learning in both discrete and continuous graphical models.