Multilevel Monte Carlo estimation of the expected value of sample information

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 $\epsilon$ is achieved notoriously at a cost of $O({\epsilon }^{-2-1/\alpha })$, 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 $O({\epsilon }^{-2})$. Numerical experiments confirm the considerable computational savings as compared to the nested Monte Carlo estimator.

References in corpus (5)

• Curvature, concentration and error estimates for Markov chain Monte Carlo
• Estimating the Expected Value of Partial Perfect Information in Health Economic Evaluations using Integrated Nested Laplace Approximation
• Nonasymptotic bounds on the estimation error of MCMC algorithms
• Decision-making under uncertainty: using MLMC for efficient estimation of EVPPI
• Multilevel Monte Carlo estimation of expected information gains

Cited by in corpus (4)

• Unbiased MLMC stochastic gradient-based optimization of Bayesian experimental designs
• Constructing unbiased gradient estimators with finite variance for conditional stochastic optimization
• Unbiased Optimal Stopping via the MUSE
• An efficient estimation of nested expectations without conditional sampling