Objective Bayes Factors for Gaussian Directed Acyclic Graphical Models

Guido Consonni and Luca La Rocca
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Guido Consonni: Department of Economics and Quantitative Methods, University of Pavia
Luca La Rocca: Dipartimento di Comunicazione e Economia, University of Modena and Reggio Emilia

No 141, Quaderni di Dipartimento from University of Pavia, Department of Economics and Quantitative Methods

Abstract: We propose an objective Bayesian method for the comparison of all Gaussian directed acyclic graphical models defined on a given set of variables. The method, which is based on the notion of fractional Bayes factor, requires a single default (typically improper) prior on the space of unconstrained covariance matrices, together with a prior sample size hyper-parameter, which can be set to its minimal value. We show that our approach produces genuine Bayes factors. The implied prior on the concentration matrix of any complete graph is a data-dependent Wishart distribution, and this in turn guarantees that Markov equivalent graphs are scored with the same marginal likelihood. We specialize our results to the smaller class of Gaussian decomposable undirected graphical models, and show that in this case they coincide with those recently obtained using limiting versions of hyper-inverse Wishart distributions as priors on the graph-constrained covariance matrices.

Keywords: Bayes factor; Bayesian model selection; Directed acyclic graph; Exponential family; Fractional Bayes factor; Gaussian graphical model; Objective Bayes; Standard conjugate prior; Structural learning. network; Stochastic search; Structural learning. (search for similar items in EconPapers)
Pages: 25 pages
Date: 2011-03
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