Multilevel Monte Carlo estimation of the expected value of sample information

TL;DR

This paper introduces a multilevel Monte Carlo method to efficiently estimate the expected value of sample information, significantly reducing computational costs compared to traditional nested Monte Carlo approaches.

Contribution

The paper proposes a novel MLMC estimator for EVSI that achieves optimal computational complexity by using a hierarchy of inner sample sizes, improving efficiency over existing methods.

Findings

Achieves optimal $O( ext{error}^{-2})$ computational complexity.

Demonstrates significant computational savings in numerical experiments.

Provides theoretical proof of improved convergence properties.

Abstract

We study Monte Carlo estimation of the expected value of sample information (EVSI) which measures the expected benefit of gaining additional information for decision making under uncertainty. EVSI is defined as a nested expectation in which an outer expectation is taken with respect to one random variable $Y$ and an inner conditional expectation with respect to the other random variable $\theta$. Although the nested (Markov chain) Monte Carlo estimator has been often used in this context, a root-mean-square accuracy of $\varepsilon$ is achieved notoriously at a cost of $O(\varepsilon^{-2-1/\alpha})$, where $\alpha$ denotes the order of convergence of the bias and is typically between $1/2$ and $1$. In this article we propose a novel efficient Monte Carlo estimator of EVSI by applying a multilevel Monte Carlo (MLMC) method. Instead of fixing the number of inner samples for $\theta$ as done in the nested Monte Carlo estimator, we consider a geometric progression on the number of inner samples, which yields a hierarchy of estimators on the inner conditional expectation with increasing approximation levels. Based on an elementary telescoping sum, our MLMC estimator is given by a sum of the Monte Carlo estimates of the differences between successive approximation levels on the inner conditional expectation. We show, under a set of assumptions on decision and information models, that successive approximation levels are tightly coupled, which directly proves that our MLMC estimator improves the necessary computational cost to optimal $O(\varepsilon^{-2})$. Numerical experiments confirm the considerable computational savings as compared to the nested Monte Carlo estimator.