The nested sampling identity for marginal likelihood estimation can be understood in two steps: the first of which is to observe that the expectation value of a non-negative random variable is identical to the integral of its survival function from 0 to its maximum value (proof is via ‘integration by parts’); the second step being to flip the axes of the integration to run from 1 to 0 (the range of the survival function). Having first learnt of this identity from the nested sampling literature it’s always interesting to me to see it pop up in quite unexpected places. Today’s example comes from a proof in Amanda Turner’s old masters thesis in which it is shown that if a sequence of probability measures on a metric space, converges to a target measure, , under the Prohorov metric then it converges in distribution as well. The first step of the nested sampling identity is used here to transform the expectation of a bounded continuous function, , into a limit over closed sets, , to which the definition of convergence under the Prohorov metric can be applied. I.e.:
Of course, such applications of this identity well & truly pre-date the nested sampling algorithm (and they appear in all the classic stats textbooks: Billingsley, Feller, etc), but for me they will always be called the nested sampling identity.