Objective Bayes Factors for Gaussian Directed Acyclic Graphical Models

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.

On the inconsistency of ℓ 1-penalised sparse precision matrix estimation

“…The FMPL method aims at circumventing this problem by replacing the true likelihood in the marginal likelihood with pseudo-likelihood. This leads to a convenient factorisation of marginal likelihood over variables and the resulting expression can be evaluated in closed form using previous results regarding objective comparison of Gaussian directed acyclic graphs [19, 20]. In practice, the factorisation allows the method to identify optimal Markov blankets independently for each of the variables using a greedy hill-climbing algorithm.…”

Partition MCMC for Inference on Acyclic Digraphs

“…We work with the BGe score (Geiger and Heckerman, 1994;Heckerman and Geiger, 1995;Geiger and Heckerman, 2002), corrected as in Consonni and Rocca (2012); Kuipers et al (2014) and sped up following Kuipers et al (2014). Since this score is modular, each node is scored separately from the others just depending on its parent set.…”

“…In our examples we will focus on continuous variables with a multivariate Gaussian distribution (Geiger and Heckerman, 2002; see also the correction in Consonni and Rocca, 2012;Kuipers et al, 2014).…”

“…Here we also use the same dataset to compare the performance of the different algorithms we consider. We cannot compare directly to the results in Friedman and Koller (2003); Grzegorczyk and Husmeier (2008) since their implementation is not available and moreover they are most likely based on one of the incorrect versions of the BGe score (Geiger and Heckerman, 1994;Heckerman and Geiger, 1995;Geiger and Heckerman, 2002), which has only recently been corrected (Consonni and Rocca, 2012;Kuipers et al, 2014).…”

Stratified Gaussian graphical models

“…Despite of our stratification of the outcome space, the resulting models are not typical mixture-type latent class models, for which inference is notoriously challenging, especially in the multivariate setting considered here. An interesting further generalization of the SGGM class would be to consider an adaptation to directed Gaussian graphical models, for which Bayesian learning has been recently considered in Consonni and Rocca (2012). A potential solution to obtaining such a generalization would be to employ the concept of labeled directed acyclic graphs, introduced for discrete-valued systems by Pensar et al (2014).…”