Decision-making under uncertainty: using MLMC for efficient estimation of EVPPI

In this paper we develop a very efficient approach to the Monte Carlo estimation of the expected value of partial perfect information (EVPPI) that measures the average benefit of knowing the value of a subset of uncertain parameters involved in a decision model. The calculation of EVPPI is inherently a nested expectation problem, with an outer expectation with respect to one random variable $X$ and an inner conditional expectation with respect to the other random variable $Y$. We tackle this problem by using a Multilevel Monte Carlo (MLMC) method (Giles 2008) in which the number of inner samples for $Y$ increases geometrically with level, so that the accuracy of estimating the inner conditional expectation improves and the cost also increases with level. We construct an antithetic MLMC estimator and provide sufficient assumptions on a decision model under which the antithetic property of the estimator is well exploited, and consequently a root-mean-square accuracy of $\varepsilon$ can be achieved at a cost of $O(\varepsilon^{-2})$. Numerical results confirm the considerable computational savings compared to the standard, nested Monte Carlo method for some simple testcases and a more realistic medical application.