Nested integral estimation using median randomized quasi-Monte Carlo

Overview

Abstract:

Nested integrals of the form $\int f\left(\int g(\bs{y},\bs{x})\di{}\bs{x}\right)\di{}\bs{y}$, where $f$ is a nonlinear function, arise in various fields such as Bayesian experimental design, medical decision making, and computational finance. We develop a novel double-loop median randomized quasi-Monte Carlo (RQMC) estimator to address such integrals. It was recently demonstrated that the median RQMC method achieves super-polynomial convergence rates under certain smoothness assumptions. We show that super-polynomial convergence rates are also possible in the nested setting for a simple polynomial example. The expected information gain of an experiment is used in Bayesian optimal experimental design to quantify the utility of an experiment. For this application, super-polynomial convergence rates are achievable for the inner integral. For the outer integral, we demonstrate that the median RQMC method achieves higher algebraic rates than standard mean-based RQMC methods through an importance sampling scheme.