Conditioning for variance reduction in estimating the sensitivity of simulations

Abstract

We consider first a discrete event static system that is to be simulated at values of a parameter or vector of parametersθ. The system is assumed driven by an inputX, where typicallyX is a vector of variables whose densityf _θ (x) depends on the parameterθ. For the purpose of optimizing, finding roots, or graphing the expected performanceE _θ L(X) for performance measureL, it is useful to estimate not only the expected value but also its gradient. An unbiased estimator for the latter is the score function estimator

$$L(X)S(\theta ) = L(X)\frac{\partial }{{\partial \theta }}\ln f_\theta (x).$$

This estimator and likelihood ratio analogues typically require variance reduction, and we consider conditioning on the value of the score function for this purpose. The efficiency gains due to performing the Monte Carlo conditionally can be very large. Extension to discrete event dynamic systems such as theM/G/1 queue and other more complicated systems is considered.