  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.
