Probability Based Independence Sampler for Bayesian Quantitative Learning in Graphical Log-Linear Marginal Models

Bayesian methods for graphical log-linear marginal models has not been developed in the same extend as traditional frequentist approaches. In this work; we introduce a novel Bayesian approach for quantitative learning for such models. They belong to curved exponential families that are difficult to handle from a Bayesian perspective. Furthermore; the likelihood cannot be analytically expressed as a function of the marginal log-linear interactions; but only in terms of cell counts or probabilities. Posterior distributions cannot be directly obtained; and MCMC methods are needed. Finally; a well- defined model requires parameter values that lead to compatible marginal probabilities. Hence; any MCMC should account for this important restriction. We construct a fully automatic and efficient MCMC strategy for quantitative learning for graphical log-linear marginal models that handles these problems. While the prior is expressed in terms of the marginal log-linear interactions; we build an MCMC algorithm which employs a proposal on the probability parameter space. The corresponding proposal on the marginal log-linear interactions is obtained via parameter transformations. By this strategy; we achieve to move within the desired target space. At each step we directly work with well- defined probability distributions. Moreover; we can exploit a conditional conjugate setup to build an efficient proposal on probability parameters. The proposed methodology is illustrated by a simulation study and a real dataset