Recently, a ‘Markovian stick-breaking’ process which generalizes the Dirichlet process
(
μ
,
θ
)
(\mu , \theta )
with respect to a discrete base space
X
\mathfrak {X}
was introduced. In particular, a sample from from the ‘Markovian stick-breaking’ processs may be represented in stick-breaking form
∑
i
≥
1
P
i
δ
T
i
\sum _{i\geq 1} P_i \delta _{T_i}
where
{
T
i
}
\{T_i\}
is a stationary, irreducible Markov chain on
X
\mathfrak {X}
with stationary distribution
μ
\mu
, instead of i.i.d.
{
T
i
}
\{T_i\}
each distributed as
μ
\mu
as in the Dirichlet case, and
{
P
i
}
\{P_i\}
is a GEM
(
θ
)
(\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
}
\{T_i\}
in some inference test cases.