Academic

Abstract

Gaussian graphical models are a powerful tool to learn the conditional independence structure in multivariate data with inference often focused on estimating individual edges in the latent graph. On the other hand, there is increasing interest in inferring more complex structures, such as communities, for multiple reasons including interpretability. Bayesian modelling provides an attractive framework for such inference. However, posterior computation under the conjugate G-Wishart prior distribution on the precision matrix is expensive for general non-decomposable graphs. Firstly, we therefore propose a new Markov chain Monte Carlo (MCMC) method named the G-Wishart weighted proposal algorithm (WWA). WWA’s distinctive features include delayed acceptance MCMC, Gibbs updates for the precision matrix and an informed proposal distribution on the graph space that enables embarrassingly parallel computations. They result in faster MCMC convergence, improved MCMC mixing and reduced computing time. Secondly, stochastic blockmodels offer a powerful tool to detect structure in a network and we propose to exploit such advances from random graph theory within the graphical models framework. This results in the propagation of the uncertainty in graph estimation to the large-scale structure learning. We consider Bayesian nonparametric stochastic blockmodels as priors on the graph which we extend to clique-based blocks and to the multiple graph scenario using a novel dependent Dirichlet process.