Stick-breaking processes, clumping, and Markov chain occupation laws

Abstract

We consider the connections among 'clumped' residual allocation models (RAMs), a general class of stick-breaking processes including Dirichlet processes, and the occupation laws of certain discrete space timeinhomogeneous Markov chains related to simulated annealing and other applications. An intermediate structure is introduced in a given RAM, where proportions between successive indices in a list are added or clumped together to form another RAM. In particular, when the initial RAM is a Griffiths-Engen-McCloskey (GEM) sequence and the indices are given by the random times that an auxiliary Markov chain jumps away from its current state, the joint law of the intermediate RAM and the locations visited in the sojourns is given in terms of a 'disordered' GEM sequence, and an induced Markov chain. Through this joint law, we identify a large class of 'stick breaking' processes as the limits of empirical occupation measures for associated time-inhomogeneous Markov chains.1.1. GEM and Dirichlet measures. Consider the infinite-dimensional simplex ∆ ∞ of all all discrete (probability) distributions on N = {1, 2, . . .}. A residual

On the use of Markovian stick-breaking priors

“…We note, in this definition, the distribution of ν does not depend on the choice of θ ≥ θ G , say as its moments by Corollary 5 below depend only on G; see also [10] for more discussion. We remark also, when Q is a 'constant' stochastic matrix with common rows µ, then {T j } j≥1 is an i.i.d.…”

“…We remark also, when Q is a 'constant' stochastic matrix with common rows µ, then {T j } j≥1 is an i.i.d. sequence with common distribution µ and so the Markovian stick-breaking measure ν reduces to the Dirichlet distribution with parameters (θ, µ); see [10] for further remarks.…”

“…14], [20], among others. In [10], another generalization where {T j } j≥1 is a Markov chain was introduced: Let G = {G i,j : i, j ∈ X} be a generator matrix, that is G i,j ≥ 0 for i = j and G i,i = − j =i G i,j , which is irreducible with suitably bounded entries, and has µ as its stationary distribution. Let also Q = I + G/θ be a Markov transition kernel on X with stationary distribution µ.…”

“…Although it was shown in [10], [11] that such Markovian stick-breaking processes connect to the limiting empirical distribution of certain simulated annealing chains, it is natural to consider their use as priors in statistical problems, the aim of this article. We first give a formula for the moments of the Markovian stick-breaking process in Theorem 4.…”

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In [10], a 'Markovian stick-breaking' process which generalizes the Dirichlet process (µ, θ) with respect to a discrete base space 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 where {T i } is a stationary, irreducible Markov chain on X with stationary distribution µ, instead of i.i.d.

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