  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.
