On the use of Markovian stick-breaking priors

Recently, a 'Markovian stick-breaking' process which generalizes the Dirichlet process (mu, theta) with respect to a discrete base space X was introduced. In particular, a sample from from the 'Markovian stick-breaking' processs may be represented in stick-breaking form Sigma(i >= 1) P-i delta(Ti) where {T-i} is a stationary, irreducible Markov chain on X with stationary distribution mu, instead of i.i.d. {T-i} each distributed as mu as in the Dirichlet case, and {P-i} is a GEM(theta) residual allocation sequence. Although the previous motivation was to relate these Markovian stick-breaking processes to empirical distributional limits of types of simulated annealing chains, these processes may also be thought of as a class of priors in statistical problems. The aim of this work in this context is to identify the posterior distribution and to explore the role of the Markovian structure of {T-i} in some inference test cases.