Nested Sampling for Uncertainty Quantification and Rare Event Estimation

TL;DR

Nested Sampling is a numerical method that efficiently estimates Bayesian evidence and rare event probabilities by transforming high-dimensional integrals into one-dimensional integrals, with proven validity for complex integrands.

Contribution

The paper introduces a rigorous justification for Nested Sampling's effectiveness in handling integrands with plateaus and demonstrates its practical utility in rare event probability estimation.

Findings

Validated Nested Sampling for integrands with plateaus

Effective in estimating rare event probabilities

Transforms high-dimensional integrals into one-dimensional form

Abstract

Nested Sampling is a method for computing the Bayesian evidence, also called the marginal likelihood, which is the integral of the likelihood with respect to the prior. More generally, it is a numerical probabilistic quadrature rule. The main idea of Nested Sampling is to replace a high-dimensional likelihood integral over parameter space with an integral over the unit line by employing a push-forward with respect to a suitable transformation. Practically, a set of active samples ascends the level sets of the integrand function, with the measure contraction of the super-level sets being statistically estimated. We justify the validity of this approach for integrands with non-negligible plateaus, and demonstrate Nested Sampling's practical effectiveness in estimating the (log-)probability of rare events.