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

TL;DR

This paper introduces a novel Bayesian MCMC method for graphical log-linear marginal models, effectively handling the complex curved exponential family structure and parameter constraints to improve quantitative learning.

Contribution

It develops an automatic, efficient MCMC algorithm that uses probability space proposals and parameter transformations, addressing key challenges in Bayesian inference for these models.

Findings

The method successfully samples from the posterior distribution in simulated data.

Application to real data demonstrates improved inference accuracy.

The approach maintains parameter constraints and model compatibility during sampling.

Abstract

Bayesian methods for graphical log-linear marginal models have not been developed in the same extent as traditional frequentist approaches. In this work, we introduce a novel Bayesian approach for quantitative learning for such models. These models 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 that employs a proposal on the probability parameter space. The corresponding proposal on the marginal log-linear interactions is obtained via parameter transformation. 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.