Point Process-based Monte Carlo estimation

TL;DR

This paper introduces a unified, unbiased Monte Carlo framework for estimating expectations of complex random variables, extending nested sampling to unbounded and heavy-tailed cases with improved variance properties and parallelization strategies.

Contribution

It extends nested sampling to unbounded and heavy-tailed variables, providing an unbiased estimator with finite variance and a fixed computational budget, supported by theoretical analysis and parallel implementation.

Findings

Unbiased estimator with finite variance for unbounded variables.

Random truncation preserves unbiasedness with controlled variance increase.

Parallel nested sampling significantly reduces computational cost.

Abstract

This paper addresses the issue of estimating the expectation of a real-valued random variable of the form $X = g(\mathbf{U})$ where $g$ is a deterministic function and $\mathbf{U}$ can be a random finite- or infinite-dimensional vector. Using recent results on rare event simulation, we propose a unified framework for dealing with both probability and mean estimation for such random variables, \emph{i.e.} linking algorithms such as Tootsie Pop Algorithm (TPA) or Last Particle Algorithm with nested sampling. Especially, it extends nested sampling as follows: first the random variable $X$ does not need to be bounded any more: it gives the principle of an ideal estimator with an infinite number of terms that is unbiased and always better than a classical Monte Carlo estimator -- in particular it has a finite variance as soon as there exists $k \in \mathbb{R} > 1$ such that $\operatorname{E}[X^k] < \infty$. Moreover we address the issue of nested sampling termination and show that a random truncation of the sum can preserve unbiasedness while increasing the variance only by a factor up to 2 compared to the ideal case. We also build an unbiased estimator with fixed computational budget which supports a Central Limit Theorem and discuss parallel implementation of nested sampling, which can dramatically reduce its computational cost. Finally we extensively study the case where $X$ is heavy-tailed.