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Pamminger and Fruwirth-Schnatter (2010) considered a Bayesian approach tomodel-based clustering of categorical time series assuming a fixed number ofclusters. But the popular methods for selecting the number of clusters, forexample, the Bayes Information Criterion (BIC), turned out to have severeproblems in the categorical time series context.  In this paper, we circumvent the difficulties of choosing the number ofclusters by adopting the Bayesian semiparametric mixture model approachintroduced by Bhattacharya (2008), who assume that the number of clusters is arandom 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 ofconvergence of the underlying Markov chain Monte Carlo (MCMC) approach.Importantly, within our main perfect simulation algorithm, there arose thenecessity to simulate perfectly from the joint distribution of a set ofcontinuous random variables with log-concave full conditional densities. Wepropose and develop a novel and efficient perfect simulation methodology forjoint distributions with log-concave full conditionals. This perfect samplingmethodology is of independent interest as well since in a very large andimportant class of Bayesian applications the full conditionals turn out to belog-concave.  We will consider application of our model and methodology to the Austrianwage mobility data, also analysed by Pamminger and Fruwirth-Schnatter (2010),and adopting the methods developed in Mukhopadhyay et al. (2011), Mukhopadhyayet al. (2012), will obtain the posterior modes of clusterings and also thedesired highest posterior distribution credible regions of the posteriordistribution of clusterings.

Clustering Categorical Time Series into Unknown Number of Clusters: A Perfect Simulation based Approach