Symmetry properties and asymmetry evaluation of Bayesian Confirmation Measures

Bayesian Confirmation Measures (BCMs) are used to assess the degree to which an evidence (or premise) E supports or contradicts an hypothesis (or conclusion) H, making use of prior probability Pr(H), posterior probability Pr(H|E) and of probability of evidence Pr(E). In the literature many BCMs have been defined with the consequent need for their comparison. For this purpose, various criteria have been proposed and some of these refer to symmetry properties. We relate the set of possible symmetries of BCMs, via an isomorphism, to the dihedral group of symmetries of the square. In this way it is possible to identify 10 subsets of symmetries that can coexist, for each subset we suggest a representative BCM, defining at this aim two new BCMs. The structure of the subgroups of the dihedral group allows also to provide an algorithm that simplifies the verification of the symmetry properties. Addressing the debate on which symmetry properties should be considered as desirable and which should not, we define asymmetry measures for BCMs. In fact, different BCMs that do not satisfy a specific symmetry property may exhibit different levels of asymmetry, this way resulting more (less) desirable. The evidence for the practical use of the approach is given through the numerical evaluation of the asymmetry degrees of some BCMs, showing this way how it is possible to discover some of their characteristics, similarities and differences.