Bayesian Inference with Probability Matrix Decomposition Models

Probability Mcatrix Decompositioni models mtay bve uised to model observed binary associations between two sets of elements. More specifically, to explain observed associations betweeni two elements, it is assumed that B laitent Bernoulli variables are realized for each element and that these variables are subsequently mapped into an observed data point accordingg to a prespecijied dererministic rule. In this papet; we present a fully Bayesian analysis for the PMD model makintg use of the Gibbs sampler. 7his approach is shown to yield three dislinct advantages: (a) in addition to posterior mean1 estim'lates it yields (/ 1 o<)% posterior intervals for the parameters. (b) it allows for an investigation of kypothesi7-ed indeterminacies in the model's parameters and for thle visualization of the best possible reduction oJ' the posterior distribution in a low-dimnensional space, and (c) it allows Jfr a broad range of goodness-of fit tests, making use of the technique of posterior predictive checks. To illustrate the approach, we applied the PMI) model to opinions of respondents of diferent countries concerning the possibility of contracting AID)S in a specific sitizationi.