On the use of Markovian stick-breaking priors

TL;DR

This paper explores Markovian stick-breaking priors, generalizing Dirichlet processes by incorporating Markov chain structures, and investigates their posterior distributions and inference properties.

Contribution

It identifies the posterior distribution for Markovian stick-breaking priors and examines how the Markov structure influences statistical inference.

Findings

Derived the posterior distribution for the process

Showed the impact of Markovian structure on inference

Connected to empirical distributional limits of simulated annealing

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 $\sum_{i\geq 1} P_i \delta_{T_i}$ 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.