On a wider class of prior distributions for graphical models

TL;DR

This paper introduces a new prior for Gaussian graphical models based on cycle space subspaces, enabling efficient Bayesian inference by restricting the graph search space and simplifying MCMC implementation.

Contribution

It proposes a novel prior on cycle space subspaces, providing theoretical insights and practical algorithms for Bayesian graph inference in Gaussian models.

Findings

Posterior edge inclusion estimates are comparable to standard methods.

The vector space approach simplifies MCMC implementation.

The new prior effectively restricts the graph search space.

Abstract

Gaussian graphical models are useful tools for conditional independence structure inference of multivariate random variables. Unfortunately, Bayesian inference of latent graph structures is challenging due to exponential growth of $\mathcal{G}_n$, the set of all graphs in $n$ vertices. One approach that has been proposed to tackle this problem is to limit search to subsets of $\mathcal{G}_n$. In this paper, we study subsets that are vector subspaces with the cycle space $\mathcal{C}_n$ as main example. We propose a novel prior on $\mathcal{C}_n$ based on linear combinations of cycle basis elements and present its theoretical properties. Using this prior, we implement a Markov chain Monte Carlo algorithm, and show that (i) posterior edge inclusion estimates computed with our technique are comparable to estimates from the standard technique despite searching a smaller graph space, and (ii) the vector space perspective enables straightforward implementation of MCMC algorithms.