## The Annals of Statistics

- Ann. Statist.
- Volume 41, Number 4 (2013), 1742-1779.

### Reversible MCMC on Markov equivalence classes of sparse directed acyclic graphs

Yangbo He, Jinzhu Jia, and Bin Yu

#### 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.

#### Article information

**Source**

Ann. Statist., Volume 41, Number 4 (2013), 1742-1779.

**Dates**

First available in Project Euclid: 5 September 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1378386238

**Digital Object Identifier**

doi:10.1214/13-AOS1125

**Mathematical Reviews number (MathSciNet)**

MR3127848

**Zentralblatt MATH identifier**

1360.62369

**Subjects**

Primary: 62H05: Characterization and structure theory 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 05C81: Random walks on graphs

**Keywords**

Sparse graphical model reversible Markov chain Markov equivalence class Causal inference

#### Citation

He, Yangbo; Jia, Jinzhu; Yu, Bin. Reversible MCMC on Markov equivalence classes of sparse directed acyclic graphs. Ann. Statist. 41 (2013), no. 4, 1742--1779. doi:10.1214/13-AOS1125. https://projecteuclid.org/euclid.aos/1378386238

#### Supplemental materials

- Supplementary material: Supplement to “Reversible MCMC on Markov equivalence classes of sparse directed acyclic graphs”. In this supplementary note, we give some algorithms, examples, an experiment and the proofs of the results in this paper.Digital Object Identifier: doi:10.1214/13-AOS1125SUPP