Computation of normalizing constants in geoacoustic Bayesian inference.

Abstract

This paper considers approaches to computing normalizing constants (Z) in Bayesian inference problems. Bayes’ theorem combines the likelihood function, model prior, and Z to form the posterior probability density (PPD). Z (also known as evidence) is difficult to compute for general problems and a common approach is to avoid its computation entirely by calculating an unnormalized estimate of the PPD which is sufficient for moment estimates. However, estimating the normalized PPD, including Z, allows for moment estimates as well as quantifying the likelihood of the model parametrization. This is commonly referred to as model selection and poses a natural way to quantifying the most appropriate model parametrization for a given data set (Bayesian razor). Several approaches for computing Z have been developed in the statistics community, some of which are applied here to the geoacoustic inference problem. Annealed importance sampling follows an annealing approach and computes weighted averages along cooling trajectories. Nested sampling uses a likelihood constraint to move from the prior mass to the posterior. Both methods also give parameter estimates which are compared to Metropolis–Hastings results. [Work supported by the Office of Naval Research.]