Learning Markov Equivalence Classes of Directed Acyclic Graphs: An Objective Bayes Approach

Abstract

A Markov equivalence class contains all the Directed Acyclic Graphs (DAGs) encoding the same conditional independencies, and is represented by a Completed Partially Directed Acyclic Graph (CPDAG), also named Essential Graph (EG). We approach the problem of model selection among noncausal sparse Gaussian DAGs by directly scoring EGs, using an objective Bayes method. Specifically, we construct objective priors for model selection based on the Fractional Bayes Factor, leading to a closed form expression for the marginal likelihood of an EG. Next we propose a Markov Chain Monte Carlo (MCMC) strategy to explore the space of EGs using sparsity constraints, and illustrate the performance of our method on simulation studies, as well as on a real dataset. Our method provides a coherent quantification of inferential uncertainty, requires minimal prior specification, and shows to be competitive in learning the structure of the data-generating EG when compared to alternative state-of-the-art algorithms.