Online Quantification of Input Model Uncertainty by Two-Layer Importance Sampling

Stochastic simulation has been widely used to analyze the performance of complex stochastic systems and facilitate decision making in those systems. Stochastic simulation is driven by the input model, which is a collection of probability distributions that model the stochasticity in the system. The input model is usually estimated using a finite amount of data, which introduces the so-called input model uncertainty to the simulation output. How to quantify input uncertainty has been studied extensively, and many methods have been proposed for the batch data setting, i.e., when all the data are available at once. However, methods for "streaming data" arriving sequentially in time are still in demand, despite that streaming data have become increasingly prevalent in modern applications. To fill this gap, we propose a two-layer importance sampling framework that incorporates streaming data for online input uncertainty quantification. Under this framework, we develop two algorithms that suit different application scenarios: the first scenario is when data come at a fast speed and there is no time for any new simulation in between updates; the second is when data come at a moderate speed and a few but limited simulations are allowed at each time stage. We prove the consistency and asymptotic convergence rate results, which theoretically show the efficiency of our proposed approach. We further demonstrate the proposed algorithms on a numerical example of the news vendor problem.