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.

The Full Bayesian Significance Test, FBST, is extensively reviewed. Its test statistic, a genuine Bayesian measure of evidence, is discussed in detail. Its behavior in some problems of statistical inference like testing for independence in contingency tables is discussed.

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 e-value or epistemic value, ev(H), measures the statistical significance of H, a hypothesis about the parameter θ of a Bayesian model. The e-value is obtained by a probability-possibility transformation of the model's posterior measure, p(θ), and can, in turn, be used to define the FBST or Full Bayesian Significance Test. This article investigates the relation of this novel approach to more standard probability-possibility transformations. In particular, we show how and why the e-value focus on or conforms with s(θ) = p(θ)/r(θ), the model's surprise function relative to the reference density r(θ), while it keeps itself consistent with the model's posterior probability measure. In addition, we investigate traditional objections raised in decision theoretic Bayesian statistics against measures of significance engendered by probability-possibility transformations.

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.