Truncated Random Measures

TL;DR

This paper systematically organizes, analyzes, and compares sequential representations of completely random measures (CRMs), providing new truncation error bounds and computational insights to improve Bayesian nonparametric inference.

Contribution

It introduces a unified framework for sequential CRM representations, offers novel truncation error analyses, and compares their computational efficiencies, advancing Bayesian nonparametric methods.

Findings

Provided new truncation error bounds for sequential CRM representations

Compared computational complexities of different representations

Enabled straightforward analysis of CRMs in Bayesian inference

Abstract

Completely random measures (CRMs) and their normalizations are a rich source of Bayesian nonparametric priors. Examples include the beta, gamma, and Dirichlet processes. In this paper we detail two major classes of sequential CRM representations---series representations and superposition representations---within which we organize both novel and existing sequential representations that can be used for simulation and posterior inference. These two classes and their constituent representations subsume existing ones that have previously been developed in an ad hoc manner for specific processes. Since a complete infinite-dimensional CRM cannot be used explicitly for computation, sequential representations are often truncated for tractability. We provide truncation error analyses for each type of sequential representation, as well as their normalized versions, thereby generalizing and improving upon existing truncation error bounds in the literature. We analyze the computational complexity of the sequential representations, which in conjunction with our error bounds allows us to directly compare representations and discuss their relative efficiency. We include numerous applications of our theoretical results to commonly-used (normalized) CRMs, demonstrating that our results enable a straightforward representation and analysis of CRMs that has not previously been available in a Bayesian nonparametric context.