Nested Sampling for Uncertainty Quantification and Rare Event Estimation

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 superlevel sets being statistically estimated. We justify the validity of this approach for integrands with nonnegligible plateaus and demonstrate nested sampling’s practical effectiveness in estimating the (log-)probability of rare events.