Decomposable graphical Gaussian model determination

Abstract

We propose a methodology for Bayesian model determination in decomposable graphical gaussian models. To achieve this aim we consider a hyper inverse Wishart prior distribution on the concentration matrix for each given graph. To ensure compatibility across models, such prior distributions are obtained by marginalisation from the prior conditional on the complete graph. We explore alternative structures for the hyperparameters of the latter, and their consequences for the model. Model determination is carried out by implementing a reversible jump MCMC sampler. In particular, the dimension-changing move we propose involves adding or dropping an edge from the graph. We characterise the set of moves which preserve the decomposability of the graph, giving a fast algorithm for maintaining the junction tree representation of the graph at each sweep. As state variable, we propose to use the incomplete variance-covariance matrix, containing only the elements for which the corresponding element of the inverse is nonzero. This allows all computations to be performed locally, at the clique level, which is a clear advantage for the analysis of large and complex data-sets. Finally, the statistical and computational performance of the procedure is illustrated by means of both arti cial and real data-sets.

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“…If the aim of the approximate inference is heavily focused on quantitative learning aspects we suggest considering, as a more powerful alternative, the reversible jump approach suggested in Giudici and Green (1999) and Dellaportas and Forster (1999) which does MCMC model determination over both the model and the parameter space.…”

“…Furthermore, in order to alleviate possible sensitivity of the results to the prior hyperparameters, a hierarchical prior distribution may be considered, as in Giudici and Green (1999).…”

“…Legal additions of edges for chordal graphs that determine UDG models were characterized by Giudici and Green (1999), while legal removals were characterized by Frydenberg and Lauritzen (1989). A chordal graph has a representation as a junction forest.…”

“…We remark that, while MC 3 addresses the structural learning problem, the reversible jump algorithm, applied to graphical models in Giudici and Green (1999) and Dellaportas and Forster (1999), although certainly more difficult to implement and test, is indeed more suited for problems in which also quantitative learning requires the development of approximate inferences (as occurs, for instance, with incomplete data).…”

Stratified Gaussian graphical models

“…Since their original introduction in the 1970's, Gaussian graphical models (GGMs) are by now ubiquitous in statistical analysis of multivariate systems, given their beneficial characteristics regarding modularity and tractability of statistical inference, see Dempster (1972), Whittaker (1990), Lauritzen (1996), Giudici and Green (1999), Wong et al (2003), Atay-Kayis and Massam (2005), Jones and West (2005), Li and Gui (2006), Yuan and Lin (2007), Carvalho and Scott (2009), Sun and Li (2012). However, unlike their discrete counterparts, log-linear graphical models, GGMs do not allow for very flexible representation of marginal and conditional dependence between variables, since their characteristics are determined by the properties of the multivariate normal distribution.…”