Functionally Compatible Local Characteristics for the Local Specification of Priors in Graphical Models

Abstract

The local specification of priors in non-decomposable graphical models does not necessarily yield a proper joint prior for all the parameters of the model. Using results concerning general exponential families with cuts, we derive specific results for the multivariate Gamma distribution (conjugate prior for Poisson counts) and the Wishart distribution (conjugate prior for Gaussian models). These results link the existence of a locally specified joint prior to the solvability of a related marginal problem over the cliques of the graph. Copyright 2007 Board of the Foundation of the Scandinavian Journal of Statistics..

Computational Aspects Related to Inference in Gaussian Graphical Models With the G-Wishart Prior

“…This sampling method is called the block Gibbs sampler and was originally discussed by Piccioni (2000). Asci and Piccioni (2007) developed it explicitly by exploiting the theory of exponential families with cuts. Here a cut is a clique of G.…”

“…The Laplace approximation works well because the mode of a G-Wishart distribution can be efficiently and accurately determined using the iterative proportional scaling algorithm (Speed and Kiiveri 1986). In our Bayesian framework estimation of K and is performed by sampling from the G-Wishart posterior distribution using the block Gibbs sampler algorithm (Asci and Piccioni 2007). We show that the combination of our new stochastic search and our algorithm for estimating the marginal likelihood associated with a G-Wishart conjugate prior represents a computationally efficient method for rapidly exploring regions of high posterior probability of Gaussian graphical models.…”

The mode oriented stochastic search (MOSS) algorithm for log-linear models with conjugate priors

“…Piccioni (2000) exploits the theory of regular exponential families with cuts to formally construct a Gibbs sampler algorithm for sampling from their natural conjugate densities. Asci and Piccioni (2007) give an extension to improper target distributions. The Bayesian IPF is a particular case of the Gibbs sampler for blocks of natural parameters of multivariate Table 7.…”

“…In this section we generalize to arbitrary contingency tables the version of Bayesian IPF for binary data described in Asci and Piccioni (2007). The algorithm starts with a random set of…”

“…This in turn allows us to compute the Laplace approximation to the marginal likelihood of hierarchial log-linear models. Another advantage of the conjugate prior of Massam et al (2008) is that it is the conjugate prior to an exponential family, hence sampling from the posterior distribution of the log-linear parameters can be done using the Bayesian iterate proportional fitting algorithm originally proposed by Asci and Piccioni (2007).…”