Optimal Nested Simulation Experiment Design via Likelihood Ratio Method

Nested simulation arises frequently in {risk management} or uncertainty quantification problems, where the performance measure is a function of the simulation output mean conditional on the outer scenario. The standard nested simulation samples $M$ outer scenarios and runs $N$ inner replications at each. We propose a new experiment design framework for a problem whose inner replication's inputs are generated from distributions parameterized by the outer scenario. This structure lets us pool replications from an outer scenario to estimate another scenario's conditional mean via the likelihood ratio method. We formulate a bi-level optimization problem to decide not only which of $M$ outer scenarios to simulate and how many times to replicate at each, but also how to pool these replications such that the total simulation effort is minimized while achieving a target level of {precision}. The resulting optimal design requires far less simulation effort than $MN$. We provide asymptotic analyses on the convergence rates of the performance measure estimators computed from the experiment design. Empirical results show that our experiment design reduces the simulation effort by orders of magnitude compared to the standard nested simulation and outperforms a state-of-the-art regression-based design that pools replications via regression.