Reversible MCMC on Markov equivalence classes of sparse directed acyclic graphs

TL;DR

This paper develops reversible Markov chains for sampling Markov equivalence classes of sparse DAGs, enabling efficient exploration of large graphical model spaces with thousands of vertices.

Contribution

It introduces a perfect set of operators for constructing reversible Markov chains on sparse equivalence classes, allowing scalable sampling and analysis.

Findings

Most edges in Markov equivalence classes are directed.

Undirected subgraphs tend to be small.

Number of undirected subgraphs grows linearly with vertices.

Abstract

Graphical models are popular statistical tools which are used to represent dependent or causal complex systems. Statistically equivalent causal or directed graphical models are said to belong to a Markov equivalent class. It is of great interest to describe and understand the space of such classes. However, with currently known algorithms, sampling over such classes is only feasible for graphs with fewer than approximately 20 vertices. In this paper, we design reversible irreducible Markov chains on the space of Markov equivalent classes by proposing a perfect set of operators that determine the transitions of the Markov chain. The stationary distribution of a proposed Markov chain has a closed form and can be computed easily. Specifically, we construct a concrete perfect set of operators on sparse Markov equivalence classes by introducing appropriate conditions on each possible operator. Algorithms and their accelerated versions are provided to efficiently generate Markov chains and to explore properties of Markov equivalence classes of sparse directed acyclic graphs (DAGs) with thousands of vertices. We find experimentally that in most Markov equivalence classes of sparse DAGs, (1) most edges are directed, (2) most undirected subgraphs are small and (3) the number of these undirected subgraphs grows approximately linearly with the number of vertices. The article contains supplement arXiv:1303.0632, http://dx.doi.org/10.1214/13-AOS1125SUPP