Bayesian Graph Selection Consistency For Decomposable Graphs

Gaussian graphical models are a popular tool to learn the dependence structure in the form of a graph among variables of interest. Bayesian methods have gained in popularity in the last two decades due to their ability to simultaneously learn the covariance and the graph and characterize uncertainty in the selection. For scalability of the Markov chain Monte Carlo algorithms, decomposability is commonly imposed on the graph space. A wide variety of graphical conjugate priors are proposed jointly on the covariance matrix and the graph with improved algorithms to search along the space of decomposable graphs, rendering the methods extremely popular in the context of multivariate dependence modeling. {\it An open problem} in Bayesian decomposable structure learning is whether the posterior distribution is able to select the true graph asymptotically when the dimension of the variables increases with the sample size. In this article, we explore specific conditions on the true precision matrix and the graph which results in an affirmative answer to this question using a commonly used hyper-inverse Wishart prior on the covariance matrix and a suitable complexity prior on the graph space. In absence of structural sparsity assumptions, our strong selection consistency holds in a high dimensional setting where $p = O(n^{\alpha})$ for $\alpha < 1/3$. In addition, we show when the likelihood is misspecified, i.e. when the true graph is non-decomposable, the posterior distribution on the graph concentrates on a set of graphs that are minimal triangulations of the true graph.