The sample size required in importance sampling

The goal of importance sampling is to estimate the expected value of a given function with respect to a probability measure ν using a random sample of size n drawn from a different probability measure μ. If the two measures μ and ν are nearly singular with respect to each other, which is often the case in practice, the sample size required for accurate estimation is large. In this article it is shown that in a fairly general setting, a sample of size approximately exp(D(ν||μ)) is necessary and sufficient for accurate estimation by importance sampling, where D(ν||μ) is the Kullback–Leibler divergence of μ from ν. In particular, the required sample size exhibits a kind of cut-off on the logarithmic scale. The theory is applied to obtain a fairly general formula for the sample size required in importance sampling for exponential families (Gibbs measures). We also show that the standard variance-based diagnostic for convergence of importance sampling is fundamentally problematic. An alternative diagnostic that provably works in certain situations is suggested. 1. Theory Let μ and ν be two probability measures on a set X equipped with some sigma-algebra. Suppose that ν is absolutely continuous with respect to μ. Let ρ be the probability density of ν with respect to μ. Let X1, X2, . . . be a sequence of X -valued random variables with law μ. Let f : X → R be a measurable function. Suppose that our goal is to evaluate the intergal