Markov Stick-breaking Processes

TL;DR

This paper introduces Markov stick-breaking processes, a novel class of Bayesian nonparametric models with Markov-dependent lengths, expanding the theoretical understanding and practical applicability of stick-breaking constructions.

Contribution

It explores Markov-dependent length variables in stick-breaking processes, establishing conditions for properness, support, and invariance properties, and identifies subclasses including Dirichlet and Pitman-Yor processes.

Findings

Established conditions for proper species sampling processes.

Proved full topological support for Markov stick-breaking processes.

Identified subclasses with properties including Dirichlet and Pitman-Yor processes.

Abstract

Stick-breaking has a long history and is one of the most popular procedures for constructing random discrete distributions in Statistics and Machine Learning. In particular, due to their intuitive construction and computational tractability they are ubiquitous in modern Bayesian nonparametric inference. Most widely used models, such as the Dirichlet and the Pitman-Yor processes, rely on iid or independent length variables. Here we pursue a completely unexplored research direction by considering Markov length variables and investigate the corresponding general class of stick-breaking processes, which we term Markov stick-breaking processes. We establish conditions under which the associated species sampling process is proper and the distribution of a Markov stick-breaking process has full topological support, two fundamental desiderata for Bayesian nonparametric models. We also analyze the stochastic ordering of the weights and provide a new characterization of the Pitman-Yor process as the only stick-breaking process invariant under size-biased permutations, under mild conditions. Moreover, we identify two notable subclasses of Markov stick-breaking processes that enjoy appealing properties and include Dirichlet, Pitman-Yor and Geometric priors as special cases. Our findings include distributional results enabling posterior inference algorithms and methodological insights.