Properties of Nested Sampling

TL;DR

Nested sampling is a Monte Carlo method for estimating marginal likelihoods, with error properties that improve with more samples, but its efficiency depends on problem dimension, and new extensions can improve its practicality.

Contribution

The paper establishes the statistical properties of nested sampling, including error rates and asymptotic behavior, and proposes an extension to improve its applicability.

Findings

Error vanishes at the Monte Carlo rate

Asymptotic variance grows linearly with dimension

Extension avoids MCMC for simulated points

Abstract

Nested sampling is a simulation method for approximating marginal likelihoods proposed by Skilling (2006). We establish that nested sampling has an approximation error that vanishes at the standard Monte Carlo rate and that this error is asymptotically Gaussian. We show that the asymptotic variance of the nested sampling approximation typically grows linearly with the dimension of the parameter. We discuss the applicability and efficiency of nested sampling in realistic problems, and we compare it with two current methods for computing marginal likelihood. We propose an extension that avoids resorting to Markov chain Monte Carlo to obtain the simulated points.