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Consonni, Guido
La Rocca, Luca
Year of Publication: 
Series/Report no.: 
Quaderni di Dipartimento 141
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.
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
Document Type: 
Working Paper

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