No abstract

In this paper, the relationship between the e-value of a complex hypothesis, H , and those of its constituent elementary hypotheses, H j , j = 1...k, is analyzed, in the independent setup. The e-value of a hypothesis H , ev(H), is a Bayesian epistemic, credibility or truth value defined under the Full Bayesian Significance Testing (FBST) mathematical apparatus. The questions addressed concern the important issue of how the truth value of H , and the truth function of the corresponding FBST structure M , relate to the truth values of its elementary constituents, H j , and to the truth functions of their corresponding FBST structures M j , respectively.

The full Bayesian signiÿcance test (FBST) for precise hypotheses is presented, with some illustrative applications. In the FBST we compute the evidence against the precise hypothesis. We discuss some of the theoretical properties of the FBST, and provide an invariant formulation for coordinate transformations, provided a reference density has been established. This evidence is the probability of the highest relative surprise set, "tangential" to the sub-manifold (of the parameter space) that deÿnes the null hypothesis.

A Bayesian measure of evidence for precise hypotheses is presented. The intention is to give a Bayesian alternative to significance tests or, equivalently, to p-values. In fact, a set is defined in the parameter space and the posterior probability, its credibility, is evaluated. This set is the "Highest Posterior Density Region" that is "tangent" to the set that defines the null hypothesis. Our measure of evidence is the complement of the credibility of the "tangent" region.

This paper has three main objectives: (a) Discuss the formal analogy between some important symmetry-invariance arguments used in physics, probability and statistics. Specifically, we will focus on Noether's theorem in physics, the maximum entropy principle in probability theory, and de Finetti-type theorems in Bayesian statistics; (b) Discuss the epistemological and ontological implications of these theorems, as they are interpreted in physics and statistics. Specifically, we will focus on the positivist (in physics) or subjective (in statistics) interpretations vs. objective interpretations that are suggested by symmetry and invariance arguments; (c) Introduce the cognitive constructivism epistemological framework as a solution that overcomes the realism-subjectivism dilemma and its pitfalls. The work of the physicist and philosopher Max Born will be particularly important in our discussion.