On the use of Markovian stick-breaking priors

Abstract

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.