## Bayesian Analysis

### Marginal Pseudo-Likelihood Learning of Discrete Markov Network Structures

#### Abstract

Markov networks are a popular tool for modeling multivariate distributions over a set of discrete variables. The core of the Markov network representation is an undirected graph which elegantly captures the dependence structure over the variables. Traditionally, the Bayesian approach of learning the graph structure from data has been done under the assumption of chordality since non-chordal graphs are difficult to evaluate for likelihood-based scores. Recently, there has been a surge of interest towards the use of regularized pseudo-likelihood methods as such approaches can avoid the assumption of chordality. Many of the currently available methods necessitate the use of a tuning parameter to adapt the level of regularization for a particular dataset. Here we introduce the marginal pseudo-likelihood which has a built-in regularization through marginalization over the graph-specific nuisance parameters. We prove consistency of the resulting graph estimator via comparison with the pseudo-Bayesian information criterion. To identify high-scoring graph structures in a high-dimensional setting we design a two-step algorithm that exploits the decomposable structure of the score. Using synthetic and existing benchmark networks, the marginal pseudo-likelihood method is shown to perform favorably against recent popular structure learning methods.

#### Article information

Source
Bayesian Anal., Volume 12, Number 4 (2017), 1195-1215.

Dates
First available in Project Euclid: 31 October 2016

https://projecteuclid.org/euclid.ba/1477918728

Digital Object Identifier
doi:10.1214/16-BA1032

Mathematical Reviews number (MathSciNet)
MR3724983

Zentralblatt MATH identifier
1384.62178

#### Citation

Pensar, Johan; Nyman, Henrik; Niiranen, Juha; Corander, Jukka. Marginal Pseudo-Likelihood Learning of Discrete Markov Network Structures. Bayesian Anal. 12 (2017), no. 4, 1195--1215. doi:10.1214/16-BA1032. https://projecteuclid.org/euclid.ba/1477918728

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#### Supplemental materials

• Appendix: Supplementary Appendix to “Marginal Pseudo-Likelihood Learning of Discrete Markov Network Structures”. The appendix contains a proof of the consistency theorem, pseudocode of the search algorithms, and detailed results from the numerical experiments.