Nested sampling statistical errors

TL;DR

This paper analyzes statistical errors in nested sampling algorithms, providing analytical insights and numerical validation, including the relationship between different error estimates and their behavior in various scenarios.

Contribution

It offers new analytical results linking information-theoretic and moment-based error estimates in nested sampling, and clarifies the relationship between single-run and repeated-run uncertainties.

Findings

Leading terms in error estimates match analytically.

Uncertainty in single runs approximates repeated-run standard deviation.

In some cases, NS uncertainties can grow without bound.

Abstract

Nested sampling (NS) is a popular algorithm for Bayesian computation. We investigate statistical errors in NS both analytically and numerically. We show two analytic results. First, we show that the leading terms in Skilling's expression using information theory match the leading terms in Keeton's expression from an analysis of moments. This approximate agreement was previously only known numerically and was somewhat mysterious. Second, we show that the uncertainty in single NS runs approximately equals the standard deviation in repeated NS runs. Whilst intuitive, this was previously taken for granted. We close by investigating our results and their assumptions in several numerical examples, including cases in which NS uncertainties increase without bound.