  Pamminger and Fruwirth-Schnatter (2010) considered a Bayesian approach to
model-based clustering of categorical time series assuming a fixed number of
clusters. But the popular methods for selecting the number of clusters, for
example, the Bayes Information Criterion (BIC), turned out to have severe
problems in the categorical time series context.
  In this paper, we circumvent the difficulties of choosing the number of
clusters by adopting the Bayesian semiparametric mixture model approach
introduced by Bhattacharya (2008), who assume that the number of clusters is a
random quantity, but is bounded above by a (possibly large) number of clusters.
We adopt the perfect simulation approach of Mukhopadhyay and Bhattacharya
(2012) for posterior simulation for completely solving the problems of
convergence of the underlying Markov chain Monte Carlo (MCMC) approach.
Importantly, within our main perfect simulation algorithm, there arose the
necessity to simulate perfectly from the joint distribution of a set of
continuous random variables with log-concave full conditional densities. We
propose and develop a novel and efficient perfect simulation methodology for
joint distributions with log-concave full conditionals. This perfect sampling
methodology is of independent interest as well since in a very large and
important class of Bayesian applications the full conditionals turn out to be
log-concave.
  We will consider application of our model and methodology to the Austrian
wage mobility data, also analysed by Pamminger and Fruwirth-Schnatter (2010),
and adopting the methods developed in Mukhopadhyay et al. (2011), Mukhopadhyay
et al. (2012), will obtain the posterior modes of clusterings and also the
desired highest posterior distribution credible regions of the posterior
distribution of clusterings.
