Monte Carlo Estimation for Imprecise Probabilities: Basic Properties

TL;DR

This paper explores Monte Carlo methods for estimating lower bounds of expectations under imprecise probabilities, analyzing bias, consistency, and practical estimation techniques.

Contribution

It introduces theoretical insights into bias and consistency of Monte Carlo estimators for imprecise probabilities, with practical proof techniques and counterexamples.

Findings

Bias is negative and diminishes with larger samples

Proposed techniques can establish strong consistency

An example demonstrates potential inconsistency

Abstract

We describe Monte Carlo methods for estimating lower envelopes of expectations of real random variables. We prove that the estimation bias is negative and that its absolute value shrinks with increasing sample size. We discuss fairly practical techniques for proving strong consistency of the estimators and use these to prove the consistency of an example in the literature. We also provide an example where there is no consistency.