this paper, we will only consider undirected graphical models. For details of Bayesian model selection for directed graphical models see Madigan et al (1995). An (undirected) graphical model is determined by a set of conditional independence constraints of the form `fl 1 is independent of fl 2 conditional on all other fl i 2 C'. Graphical models are so called because they can each be represented as a graph with vertex set C and an edge between each pair fl 1 and fl 2 unless fl 1 and fl 2 are conditionally independent as described above. Darroch, Lauritzen and Speed (1980) show that each graphical log-linear model is hierarchical, with generators given by the cliques (complete subgraphs) of the graph. The total number of possible graphical models is clearly given by 2 (

Abstract not found

In Bayesian analysis of multi-way contingency tables, the selection of a prior distribution for either the log-linear parameters or the cell probabilities parameters is a major challenge. In this paper, we define a flexible family of conjugate priors for the wide class of discrete hierarchical log-linear models, which includes the class of graphical models. These priors are defined as the Diaconis–Ylvisaker conjugate priors on the log-linear parameters subject to “baseline constraints ” under multinomial sampling. We also derive the induced prior on the cell probabilities and show that the induced prior is a generalization of the hyper Dirichlet prior. We show that this prior has several desirable properties and illustrate its usefulness by identifying the most probable decomposable, graphical and hierarchical log-linear models for a six-way contingency table. 1. Introduction. We

this article we approximate the rate of convergence of the Gibbs sampler by a normal approximation of the target distribution. Based on this approximation, we consider many implementational issues for the Gibbs sampler, e.g., updating strategy, parameterization and blocking. We give theoretical results to justify our approximation and illustrate our methods in a number of realistic examples. Key words: Correlation Structure; Gaussian distribution; Generalized linear models; Gibbs sampler; Markov chain Monte Carlo; Parameterization; Random scan; Rates of convergence.

We develop computational strategies for extended maximum likelihood estimation, as defined in Rinaldo (2006), for general classes of log-linear models of widespred use, under Poisson and product-multinomial sampling schemes. We derive numerically efficient procedures for generating and manipulating design matrices and we propose various algorithms for computing the extended maximum likelihood estimates of the expectations of the cell counts. These algorithms allow to identify the set of estimable cell means for any given observable table and can be used for modifying traditional goodness-of-fit tests to accommodate for a nonexistent MLE. We describe and take advantage of the connections between extended maximum likelihood

We consider the specification of prior distributions for Bayesian model comparison, focusing on regression-type models. We propose a particular joint specification of the prior distribution across models so that sensitivity of posterior model probabilities to the dispersion of prior distributions for the parameters of individual models (Lindley’s paradox) is diminished. We illustrate the behavior of inferential and predictive posterior quantities in linear and log-linear regressions under our proposed prior densities with a series of simulated and real data examples.

We develop computational strategies for extended maximum likelihood estimation, as defined in Rinaldo (2006), for general classes of log-linear models of widespred use, under Poisson and product-multinomial sampling schemes. We derive numerically efficient procedures for generating and manipulating design matrices and we propose various algorithms for computing the extended maximum likelihood estimates of the expectations of the cell counts. These algorithms allow to identify the set of estimable cell means for any given observable table and can be used for modifying traditional goodness-of-fit tests to accommodate for a nonexistent MLE. We describe and take advantage of the connections between extended maximum likelihood

www.elsevier.com/locate/jspi Data augmentation in multi-way contingency tables with fixed marginal totals

journal homepage: www.elsevier.com/locate/stamet The mode oriented stochastic search (MOSS) algorithm for log-linear models with conjugate priors