Double-loop quasi-Monte Carlo estimator for nested integration

Evaluating the expected information gain (EIG) is a critical task in many areas of computational science and statistics, necessitating the approximation of nested integrals. Available techniques for this problem based on quasi-Monte Carlo (QMC) methods have focused on enhancing the efficiency of either the inner or outer integral approximation. In this work, we introduce a novel approach that extends the scope of these efforts to address inner and outer expectations simultaneously. Leveraging the principles of Owen's scrambling of digital nets, we develop a randomized QMC (rQMC) method that improves the convergence behavior of the approximation of nested integrals. We also indicate how to combine this methodology with importance sampling to address a measure concentration arising in the inner integral. Our method capitalizes on the unique structure of nested expectations to offer a more efficient approximation mechanism. By incorporating Owen's scrambling techniques, we handle integrands exhibiting infinite variation in the Hardy--Krause sense, paving the way for theoretically sound error estimates. As the main contribution of this work, we derive asymptotic error bounds for the bias and variance of our estimator, along with regularity conditions under which these error bounds can be attained. In addition, we provide nearly optimal sample sizes for the rQMC approximations, which are helpful for the actual numerical implementations. Moreover, we verify the quality of our estimator through numerical experiments in the context of EIG estimation. Specifically, we compare the computational efficiency of our rQMC method against standard nested MC integration across two case studies: one in thermo-mechanics and the other in pharmacokinetics. These examples highlight our approach's computational savings and enhanced applicability.