Open Access
August 2013 Reversible MCMC on Markov equivalence classes of sparse directed acyclic graphs
Yangbo He, Jinzhu Jia, Bin Yu
Ann. Statist. 41(4): 1742-1779 (August 2013). DOI: 10.1214/13-AOS1125


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.


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Yangbo He. Jinzhu Jia. Bin Yu. "Reversible MCMC on Markov equivalence classes of sparse directed acyclic graphs." Ann. Statist. 41 (4) 1742 - 1779, August 2013.


Published: August 2013
First available in Project Euclid: 5 September 2013

zbMATH: 1360.62369
MathSciNet: MR3127848
Digital Object Identifier: 10.1214/13-AOS1125

Primary: 05C81 , 60J10 , 62H05

Keywords: Causal inference , Markov equivalence class , reversible Markov chain , Sparse graphical model

Rights: Copyright © 2013 Institute of Mathematical Statistics

Vol.41 • No. 4 • August 2013
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