A moment-matching Ferguson and Klass algorithm

TL;DR

This paper introduces a moment-matching algorithm for the Ferguson and Klass series representation of completely random measures, enabling precise truncation and improved sampling in Bayesian nonparametric models.

Contribution

It proposes a novel moment-matching criterion to quantify and control the approximation error in the Ferguson and Klass sampling algorithm.

Findings

The algorithm effectively determines truncation levels for desired accuracy.

Implementation on various models demonstrates improved sampling precision.

The method provides a systematic way to balance computational cost and accuracy.

Abstract

Completely random measures (CRM) represent the key building block of a wide variety of popular stochastic models and play a pivotal role in modern Bayesian Nonparametrics. A popular representation of CRMs as a random series with decreasing jumps is due to Ferguson and Klass (1972). This can immediately be turned into an algorithm for sampling realizations of CRMs or more elaborate models involving transformed CRMs. However, concrete implementation requires to truncate the random series at some threshold resulting in an approximation error. The goal of this paper is to quantify the quality of the approximation by a moment-matching criterion, which consists in evaluating a measure of discrepancy between actual moments and moments based on the simulation output. Seen as a function of the truncation level, the methodology can be used to determine the truncation level needed to reach a certain level of precision. The resulting moment-matching \FK algorithm is then implemented and illustrated on several popular Bayesian nonparametric models.