Conjugate and Conditional Conjugate Bayesian Analysis of Discrete Graphical Models of Marginal Independence

We propose a conjugate and conditional conjugate Bayesian analysis of models of marginal independence with a bi-directed graph representation. We work with Markov equivalent directed acyclic graphs (DAGs) obtained using the same vertex set with the addition of some latent vertices when required. The DAG equivalent model is characterised by a minimal set of marginal and conditional probability parameters. This allows us to use compatible prior distributions based on products of Dirichlet distributions. For models with DAG representation on the same vertex set, the posterior distribution and the marginal likelihood is analytically available, while for the remaining ones a data augmentation scheme introducing additional latent variables is required. For the latter, we estimate the marginal likelihood using Chib's (1995) estimator. Additional implementation details including identifiability of such models is discussed. For all models, we also provide methodology for the computation of the posterior distributions of the marginal log-linear parameters based on a simple transformation of the simulated values of the probability parameters. We illustrate our method using a popular 4-way dataset.