Towards Reconciliation between Bayesian and Frequentist Reasoning

Abstract

A theory of quantitative inference about the parameters of sampling distributions is constructed deductively by following very general rules, referred to as the Cox-Polya-Jaynes Desiderata. The inferences are made in terms of probability distributions that are assigned to the parameters. The Desiderata, focusing primarily on consistency of the plausible reasoning, lead to unique assignments of these probabilities in the case of sampling distributions that are invariant under Lie groups. In the scalar cases, e.g. in the case of inferring a single location or scale parameter, the requirement for logical consistency is equivalent to the requirement for calibration: the consistent probability distributions are automatically also the ones with the exact calibration and vice versa. This equivalence speaks in favour of reconciliation between the Bayesian and Frequentist schools of reasoning.