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Importance Sampling
Abstract
Importance sampling is a technique for estimating integrals of the form E[φ(X)], where X has a density g that may be difficult to sample from or otherwise handle. The simplest Monte Carlo method would consist of generating i.i.d. samples X_1,...,X_n from g and averaging the values φ(X_i). By contrast, Importance Sampling (IS) consists of using samples Y_1,...,Y_n from another density, say f, and averaging the reweighted values φ(Y_i) * g(Y_i)/f(Y_i).
This works provided that f is positive when g is positive, but there is no free lunch: the price to pay can be high variance and poor accuracy.
In this expository talk, I will (1) explain when and how to use IS, with practical examples from chemistry and combinatorics; (2) explain how to measure the accuracy of IS and determine the sample size n needed for an accurate estimation of E[φ(X)], locally for a specific φ; and (3) explain how to measure this accuracy globally using Wasserstein distances.
