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Bayesian Model Selection of Gaussian Directed Acyclic Graph Structures

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  • Federico Castelletti

Abstract

During the last years, graphical models have become a popular tool to represent dependencies among variables in many scientific areas. Typically, the objective is to discover dependence relationships that can be represented through a directed acyclic graph (DAG). The set of all conditional independencies encoded by a DAG determines its Markov property. In general, DAGs encoding the same conditional independencies are not distinguishable from observational data and can be collected into equivalence classes, each one represented by a chain graph called essential graph (EG). However, both the DAG and EG space grow super exponentially in the number of variables, and so, graph structural learning requires the adoption of Markov chain Monte Carlo (MCMC) techniques. In this paper, we review some recent results on Bayesian model selection of Gaussian DAG models under a unified framework. These results are based on closed‐form expressions for the marginal likelihood of a DAG and EG structure, which is obtained from a few suitable assumptions on the prior for model parameters. We then introduce a general MCMC scheme that can be adopted both for model selection of DAGs and EGs together with a couple of applications on real data sets.

Suggested Citation

  • Federico Castelletti, 2020. "Bayesian Model Selection of Gaussian Directed Acyclic Graph Structures," International Statistical Review, International Statistical Institute, vol. 88(3), pages 752-775, December.
    • Handle: RePEc:bla:istatr:v:88:y:2020:i:3:p:752-775
    DOI: 10.1111/insr.12379
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    References listed on IDEAS

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    1. Federico Castelletti & Guido Consonni, 2020. "Discovering causal structures in Bayesian Gaussian directed acyclic graph models," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 183(4), pages 1727-1745, October.
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    4. Guido Consonni & Luca La Rocca & Stefano Peluso, 2017. "Objective Bayes Covariate-Adjusted Sparse Graphical Model Selection," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 44(3), pages 741-764, September.
    5. Guido Consonni & Luca La Rocca, 2012. "Objective Bayes Factors for Gaussian Directed Acyclic Graphical Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 39(4), pages 743-756, December.
    6. Alberto Roverato, 2005. "A Unified Approach to the Characterization of Equivalence Classes of DAGs, Chain Graphs with no Flags and Chain Graphs," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(2), pages 295-312, June.
    7. Anindya Bhadra & Bani K. Mallick, 2013. "Joint High-Dimensional Bayesian Variable and Covariance Selection with an Application to eQTL Analysis," Biometrics, The International Biometric Society, vol. 69(2), pages 447-457, June.
    8. J. Peters & P. Bühlmann, 2014. "Identifiability of Gaussian structural equation models with equal error variances," Biometrika, Biometrika Trust, vol. 101(1), pages 219-228.
    9. Steen A. Andersson & David Madigan & Michael D. Perlman, 2001. "Alternative Markov Properties for Chain Graphs," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 28(1), pages 33-85, March.
    10. Christine Peterson & Francesco C. Stingo & Marina Vannucci, 2015. "Bayesian Inference of Multiple Gaussian Graphical Models," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 110(509), pages 159-174, March.
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