4. Bayesian Inference

This chapter delves into Bayesian inference for discrete regression chain graph models, particularly those that are Markov equivalent to bi-directed graphs. It explores the application of conjugate analysis, which is limited to specific graphical configurations, and the necessity of Markov Chain Monte Carlo (MCMC) techniques for broader applications. The discussion encompasses model specification and estimation, focusing on probability and marginal log-linear parameterizations of the model. Real data applications are used to illustrate the previous issues, and the complexities involved in prior specification, a critical aspect of the Bayesian framework, are thoroughly examined. The chapter also addresses model comparison, highlighting the use of Bayes factors and the importance of compatible priors to ensure coherence in the inference process. Additionally, it discusses the Markov equivalence between regression graphs and bi-directed graphs, providing insights into the interpretation and application of these models. The chapter concludes with a detailed analysis of Bayesian inference for Coppen's data, demonstrating the practical application of the discussed methodologies. The chapter also explores the use of marginal log-linear parameterization, which offers a more straightforward approach to specifying sub-models based on a priori knowledge. This approach accommodates linear constraints on the marginal log-linear parameter space, providing a more intuitive and mathematically tractable framework for model specification and the integration of prior information. The chapter concludes with a discussion on the flexibility of the presented methodologies, which can be extended to regression graph models Markov equivalent to either DAGs or bi-directed graphs, enhancing their utility across a diverse array of graphical configurations.