On the use of Markovian stick-breaking priors

Abstract

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 ${\sum }_{i\ge 1}{P}_i{\delta }_{T_i}$ 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.

18 pages, 4 figures. To appear in AMS Contemporary Math Volume in honor of M.M. Rao