Conjugate and conditional conjugate Bayesian analysis of discrete graphical models of marginal independence

A conjugate and conditional conjugate Bayesian analysis is presented for bi-directed discrete graphical models, which are used to describe and estimate marginal associations between categorical variables. To achieve this, each bi-directed graph is re-expressed by a Markov equivalent, over the observed margin, directed acyclic graph (DAG). This DAG equivalent model is obtained using the same vertex set or with the addition of some latent variables when required. It is characterised by a minimal set of marginal and conditional probability parameters. Hence compatible priors based on products of Dirichlet distributions can be applied. 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, the marginal likelihood is estimated using Chib’s estimator. Additional implementation details including identifiability of such models are discussed. Moreover, analytic details concerning the computation of the posterior distributions of the marginal log-linear parameters are provided. The computation is achieved via a simple transformation of the simulated values of the probability parameters of the bi-directed model under study. The marginal log-linear parameterisation provides a straightforward interpretation in terms of log-odds ratios on specific marginals quantifying the associations between variables involved in the corresponding marginal. The proposed methodology is illustrated using a popular 4-way dataset.