 
Summary: Using Supervised Learning to Improve Monte Carlo Integral
Estimation
Brendan Tracey
Stanford University, Stanford, CA 94305
David Wolpert
NASA Ames Research Center, Moffett Field, CA 94035
Juan J. Alonso
Stanford University, Stanford, CA 94305
Monte Carlo (MC) techniques are often used to estimate integrals of a multivariate function using ran
domly generated samples of the function. In light of the increasing interest in uncertainty quantification and
robust design applications in aerospace engineering, the calculation of expected values of such functions (e.g.
performance measures) becomes important. However, MC techniques often suffer from high variance and slow
convergence as the number of samples increases. In this paper we present Stacked Monte Carlo (StackMC),
a new method for postprocessing an existing set of MC samples to improve the associated integral estimate.
StackMC is based on the supervised learning techniques of fitting functions and cross validation. It should
reduce the variance of any type of Monte Carlo integral estimate (simple sampling, importance sampling,
quasiMonte Carlo, MCMC, etc.) without adding bias. We report on an extensive set of experiments confirm
ing that the StackMC estimate of an integral is more accurate than both the associated unprocessed Monte
Carlo estimate and an estimate based on a functional fit to the MC samples. These experiments run over a
wide variety of integration spaces, numbers of sample points, dimensions, and fitting functions. In particular,
