Objective Bayes factors for Gaussian directed acyclic graphical models.

*(English)*Zbl 1253.62015Summary: 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 factors (BFs), 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 BFs. 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.

##### MSC:

62F15 | Bayesian inference |

05C20 | Directed graphs (digraphs), tournaments |

62H10 | Multivariate distribution of statistics |

05C90 | Applications of graph theory |

##### Keywords:

Bayesian model selection; exponential family; fractional Bayes factor; Gaussian graphical model; standard conjugate prior; structural learning
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\textit{G. Consonni} and \textit{L. La Rocca}, Scand. J. Stat. 39, No. 4, 743--756 (2012; Zbl 1253.62015)

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