Reference functions to decrease errors in Monte Carlo integrals

Abstract

Given a set of points generated by Monte Carlo, the accuracy of the Monte Carlo integral of a function f;( x) performed using these points can be improved if there are “reference functions”, i.e. other functions that can be integrated both analytically and by Monte Carlo using the same points. This statement is shown to be true. Formulae are developed to make practical use of it. Guidelines are given as how to choose reference functions, recommending that there be a linear combination of these functions that approximates the function f( x) as well as possible. Not only the statistical error but also some systematic errors due to biases in the Monte Carlo generator are reduced. Monte Carlo integrations using reference functions may be considered as bridging the gap between numerical and usual Monte Carlo integrations.