Multilevel Monte Carlo estimation of the expected value of sample information

Tomohoko Hironaka, Mike Giles, Takashi Goda, Howard H Z Thom

Research output: Contribution to journalArticle (Academic Journal)peer-review


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
Original languageEnglish
Pages (from-to)1236–1259
Number of pages24
JournalSIAM/ASA Journal on Uncertainty Quantification
Issue number3
Early online date30 Sep 2020
Publication statusE-pub ahead of print - 30 Sep 2020


  • expected value of sample information
  • multilevel Monte Carlo
  • nested expectations
  • decision making under uncertainty

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