Equidistribution testing with Bayes factors and the ECT

John Skilling’ s Nested Sampling algorithm is a numerical method for fitting models to data in the Bayesian setting, producing estimates of the Bayesian Evidence Z and Information ℋ as well as posterior samples. A central step in the process is the generation of a new random sample from the (typically uniform) prior distribution subject to the constraint that the new prior sample’s likelihood is greater than a current likelihood threshold. One way to test a generation method - the “outside in” approach - is to incorporate it in a Nested Sampling algorithm and compare the resulting model estimates with known cases. Another way - the “inside out” approach - is to validate the uniformity of prior samples produced by the new method before its incorporation in a Nested Sampling system. Using the “inside out” approach, we show that E T Jaynes’ Entropy Concentration Theorem (ECT) and a Bayes Factor test of a particular type provide sensitive tests of uniformity in irregular 2D regions.