In [10], a `Markovian stick-breaking process which generalizes the Dirichlet process $(mu, theta)$ with respect to a discrete base space ${mathfrak X}$ was introduced. In particular, a sample from from the `Markovian stick-breaking processs may be represented in stick-breaking form $sum_{igeq 1} P_i delta_{T_i}$ where ${T_i}$ is a stationary, irreducible Markov chain on ${mathfrak X}$ 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 motivation in [10] 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.
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