A tree perspective on stick-breaking models in covariate-dependent mixtures

TL;DR

This paper introduces a tree-based perspective on stick-breaking models in covariate-dependent mixtures, proposing balanced tree structures to improve prior assumptions, posterior uncertainty, and computational efficiency.

Contribution

It generalizes traditional SB models by exploring alternative tree topologies, especially balanced trees, to address limitations of the lopsided tree structure.

Findings

Balanced trees mitigate undesirable properties of SB models.

Tree topology influences prior assumptions and posterior uncertainty.

Balanced structures improve computational effectiveness.

Abstract

Stick-breaking (SB) processes are often adopted in Bayesian mixture models for generating mixing weights. When covariates influence the sizes of clusters, SB mixtures are particularly convenient as they can leverage their connection to binary regression to ease both the specification of covariate effects and posterior computation. Existing SB models are typically constructed based on continually breaking a single remaining piece of the unit stick. We view this from a dyadic tree perspective in terms of a lopsided bifurcating tree that extends only in one side. We show that two unsavory characteristics of SB models are in fact largely due to this lopsided tree structure. We consider a generalized class of SB models with alternative bifurcating tree structures and examine the influence of the underlying tree topology on the resulting Bayesian analysis in terms of prior assumptions, posterior uncertainty, and computational effectiveness. In particular, we provide evidence that a balanced tree topology, which corresponds to continually breaking all remaining pieces of the unit stick, can resolve or mitigate these undesirable properties of SB models that rely on a lopsided tree.