In [10], a `Markovian stick-breaking' process which generalizes the Dirichlet process $(μ, θ)$ 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 $\sum_{i\geq 1} P_i δ_{T_i}$ where $\{T_i\}$ is a stationary, irreducible Markov chain on ${\mathfrak X}$ with stationary distribution $μ$, instead of i.i.d. $\{T_i\}$ each distributed as $μ$ as in the Dirichlet case, and $\{P_i\}$ is a GEM$(θ)$ 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.