A Differential Geometric Approach to Bayesian Marginalization

Abstract

The statistical tool box offers a wide range of inference techniques to choose from; regrettably, the reliability of these methods in the sense of ‘reproducibility of frequency properties’ can often be unclear or even ignored. We examine this issue for default Bayes methods and develop a prior that leads to full second-order inference for any regular scalar parameter of interest in presence of nuisance parameters; the new prior is Jeffreys based. Also, in parallel, we show that such second-order accuracy is widely unavailable for vector parameters of interest through the Bayesian framework, unless the interest parameter has a special linearity. Detailed examples, including simulations, are presented and discussed.