Imprecise Monte Carlo simulation and iterative importance sampling for the estimation of lower previsions

TL;DR

This paper introduces a theoretical framework for estimating lower previsions using Monte Carlo and importance sampling, including iterative methods to improve accuracy, with applications demonstrated on the imprecise Dirichlet model.

Contribution

It develops a new theoretical approach linking estimator consistency to Glivenko-Cantelli classes and proposes an iterative importance sampling method for better estimation of lower previsions.

Findings

Bounded bias and proved consistency for sub-Gaussian processes

Proposed a new upper estimator for confidence intervals

Demonstrated improved performance on the imprecise Dirichlet model

Abstract

We develop a theoretical framework for studying numerical estimation of lower previsions, generally applicable to two-level Monte Carlo methods, importance sampling methods, and a wide range of other sampling methods one might devise. We link consistency of these estimators to Glivenko-Cantelli classes, and for the sub-Gaussian case we show how the correlation structure of this process can be used to bound the bias and prove consistency. We also propose a new upper estimator, which can be used along with the standard lower estimator, in order to provide a simple confidence interval. As a case study of this framework, we then discuss how importance sampling can be exploited to provide accurate numerical estimates of lower previsions. We propose an iterative importance sampling method to drastically improve the performance of imprecise importance sampling. We demonstrate our results on the imprecise Dirichlet model.