A Poisson process model for Monte Carlo

TL;DR

This paper introduces a Poisson process framework that unifies perturbation and accept-reject Monte Carlo methods, enabling analysis and generalization of sampling techniques over infinite spaces.

Contribution

It presents a Poisson process model that unifies different Monte Carlo sampling approaches and extends perturbation methods to infinite spaces using Gumbel processes.

Findings

Unified Poisson process framework for Monte Carlo methods

Generalization of perturbation trick to infinite spaces

Analysis of A* sampling and OS* methods within the model

Abstract

Simulating samples from arbitrary probability distributions is a major research program of statistical computing. Recent work has shown promise in an old idea, that sampling from a discrete distribution can be accomplished by perturbing and maximizing its mass function. Yet, it has not been clearly explained how this research project relates to more traditional ideas in the Monte Carlo literature. This chapter addresses that need by identifying a Poisson process model that unifies the perturbation and accept-reject views of Monte Carlo simulation. Many existing methods can be analyzed in this framework. The chapter reviews Poisson processes and defines a Poisson process model for Monte Carlo methods. This model is used to generalize the perturbation trick to infinite spaces by constructing Gumbel processes, random functions whose maxima are located at samples over infinite spaces. The model is also used to analyze A* sampling and OS*, two methods from distinct Monte Carlo families.