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

We introduce a novel Bayesian approach for quantitative learning for
graphical log-linear marginal models. These models belong to curved exponential
families that are difficult to handle from a Bayesian perspective. 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 Markov Chain Monte Carlo (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 such models that handles these problems. While the
prior is expressed in terms of the marginal log-linear interactions, we build an
MCMC algorithm that employs a proposal on the probability parameter space.
The corresponding proposal on the marginal log-linear interactions is obtained
via parameter transformation. We 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.