A Fully Bayesian Analysis of Multivariate Latent Class Models with an Application to Metric Conjoint Analysis

In this paper we head for a fully Bayesian analysis of the latent class model with a priori unknown number of classes. Estimation is carried out by means of Markov Chain Monte Carlo (MCMC) methods. We deal explicitely with the consequences the unidentifiability of this type of model has on MCMC estimation. Joint Bayesian estimation of all latent variables, model parameters, and parameters determining the probability law of the latent process is carried out by a new MCMC method called permutation sampling. In a first run we use the random permutation sampler to sample from the unconstrained posterior. We will demonstrate that a lot of important information, such as e.g. estimates of the subject-specific regression coefficients, is available from such an unidentified model. The MCMC output of the random permutation sampler is explored in order to find suitable identifiability constraints. In a second run we use the permutation sampler to sample from the constrained posterior by imposing identifiablity constraints. The unknown number of classes is determined by formal Bayesian model comparison through exact model likelihoods. We apply a new method of computing model likelihoods for latent class models which is based on the method of bridge sampling. The approach is applied to simulated data and to data from a metric conjoint analysis in the Austrian mineral water market.