A new evidence measure for Bayesian Inference

In the last sixty years there have been several attempts to build a measure of evidence that covers, in a Bayesian context, the role that the p-value has played in the frequentist setting. A prominent example is the decision test based on the Bayes Factor. Worth to mention it is also the e-value, another Bayesian evidence measure on which the Full Bayesian Significance Test procedure is based. The aim of this thesis is to make a contribution to the Bayesian testing procedure of precise hypotheses for parametric models. To this end we propose a Bayesian measure of evidence, called Bayesian Discrepancy Measure, which gives an absolute evaluation of the suitability of a hypothesis H in light of the prior knowledge about the parameter and the observed data. The starting point is the idea that a hypothesis may be more or less supported by the available evidence contained in the posterior distribution. Since reference is made to a precise hypothesis H and no alternative against this hypothesis is considered, we do not adopt the approach whereby there is no test that can lead to the rejection of a hypothesis except by comparing it with an alternative one (the Bayes factor in the Bayesian perspective and Neyman-Pearson-Wald in the frequentist one). The proposed measure of evidence has the desired properties of invariance under reparametrisations and consistency for large samples. In this thesis we also show that it is possible to construct a testing procedure, based on the Bayesian Discrepancy Measure, that allows to compare parameter functions from two independent populations. This approach is flexible, as it can be adapted to take into account different distributions and different parameter transformations. Moreover, we address the problem of comparing k parameters, or their transformations, from inde- pendent populations. This is not a simple extension of the procedure for two popula- tions, due to the geometry of the hypothesis and the parameter space, and it therefore demands a separate discussion. In conclusion, this methodologies enables us to tackle some problems that are not yet covered in the literature.