Bayesian inference with probability matrix decomposition models

Probability Matrix Decomposition models may be used to model observed binary associations between two sets of elements. More specifically, to explain observed associations between two elements, it is assumed that B latent Bernoulli variables are realized for each element and that these variables are subsequently mapped into an observed data point according to a prespecified deterministic rule. In the present paper, a fully Bayesian analysis for the PMD model is presented making use of the Gibbs sampler. This approach is shown to yield three distinct advantages: (a) in addition to posterior mean estimates it yields (1-alpha)% posterior intervals for the parameters, (b) it allows for an investigation of hypothesized indeterminacies in the model's parameters and for the visualization of the best possible reduction of the posterior distribution in a low-dimensional space, and (c) it allows for a broad range of goodness-of-fit tests, making use of the technique of posterior predictive checks. To illustrate the approach, the PMD model is applied to opinions of respondents of different countries concerning the possibility of contracting AIDS in a specific situation