  Latent class analysis is used to perform model based clustering for
multivariate categorical responses. Selection of the variables most relevant
for clustering is an important task which can affect the quality of clustering
considerably. This work considers a Bayesian approach for selecting the number
of clusters and the best clustering variables. The main idea is to reformulate
the problem of group and variable selection as a probabilistically driven
search over a large discrete space using Markov chain Monte Carlo (MCMC)
methods. This approach results in estimates of degree of relevance of each
variable for clustering along with posterior probability for the number of
clusters. Bayes factors can then be easily calculated, and a suitable model
chosen in a principled manner. Both selection tasks are carried out
simultaneously using an MCMC approach based on a collapsed Gibbs sampling
method, whereby several model parameters are integrated from the model,
substantially improving computational performance. Approaches for estimating
posterior marginal probabilities of class membership, variable inclusion and
number of groups are proposed, and post-hoc procedures for parameter and
uncertainty estimation are outlined. The approach is tested on simulated and
real data.
