Multilevel Monte Carlo estimation of the expected value of sample information
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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 θ. Although the nested (Markov chain) Monte Carlo estimator has been often used in this context, a root-mean-square accuracy of ε is achieved notoriously at a cost of O(ε^-2-1/α), where α 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 θ 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(ε^-2). Numerical experiments confirm the considerable computational savings as compared to the nested Monte Carlo estimator.
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