Validation of Bayesian posterior distributions using a multidimensional Kolmogorov–Smirnov test

Abstract

We extend the Kolmogorov–Smirnov (K–S) test to multiple dimensions by suggesting a |$\mathbb {R}^n \rightarrow [0,1]$| mapping based on the probability content of the highest probability density region of the reference distribution under consideration; this mapping reduces the problem back to the one-dimensional case to which the standard K–S test may be applied. The universal character of this mapping also allows us to introduce a simple, yet general, method for the validation of Bayesian posterior distributions of any dimensionality. This new approach goes beyond validating software implementations; it provides a sensitive test for all assumptions, explicit or implicit, that underlie the inference. In particular, the method assesses whether the inferred posterior distribution is a truthful representation of the actual constraints on the model parameters. We illustrate our multidimensional K–S test by applying it to a simple two-dimensional Gaussian toy problem, and demonstrate our method for posterior validation in the real-world astrophysical application of estimating the physical parameters of galaxy clusters parameters from their Sunyaev–Zel'dovich effect in microwave background data. In the latter example, we show that the method can validate the entire Bayesian inference process across a varied population of objects for which the derived posteriors are different in each case.