Bayesian solutions to non-Bayesian detection problems: Unification through fusion

Abstract

In 1950 Abraham Wald proved that every admissible statistical decision rule is either a Bayesian procedure or the limit of a sequence of such procedures. He thus provided a decision-theoretic justification for the use of Bayesian inference, even for non-Bayesian problems. It is often assumed that his result also justified the use of Bayesian priors to solve such problems. However, the principles one should use for defining the values of prior probabilities have been controversial for decades, especially when applied to epistemic unknowns. Now a new approach indirectly assigns values to the quantities usually interpreted as priors by imposing design constraints on a detection algorithm. No assumptions about prior "states of belief are necessary. The result shows how Wald's theorem can accommodate both Bayesian and non-Bayesian problems. The unification is mediated by the fusion of clairvoyant detectors.