This work addresses an important issue regarding the performance of simultaneous test procedures: the construction of multiple tests that at the same time are optimal from a statistical perspective and that also yield logically-consistent results that are easy to communicate to practitioners of statistical methods. For instance, if hypothesis A implies hypothesis B, is it possible to create optimal testing procedures that reject A whenever they reject B? Unfortunately, several standard testing procedures fail in having such logical consistency. Although this has been deeply investigated under a frequentist perspective, the literature lacks analyses under a Bayesian paradigm. In this work, we contribute to the discussion by investigating three rational relationships under a Bayesian decision-theoretic standpoint: coherence, invertibility and union consonance. We characterize and illustrate through simple examples optimal Bayes tests that fulfill each of these requisites separately. We also explore how far one can go by putting these requirements together. We show that although fairly intuitive tests satisfy both coherence and invertibility, no Bayesian testing scheme meets the desiderata as a whole, strengthening the understanding that logical consistency cannot be combined with statistical optimality in general. Finally, we associate Bayesian hypothesis testing with Bayes point estimation procedures. We prove the performance of logically-consistent hypothesis testing by means of a Bayes point estimator to be optimal only under very restrictive conditions.

In An Introduction to Probability Theory and its Applications, W. Feller established a way of ending the St. Petersburg paradox by the introduction of an entrance fee, and provided it for the case in which the game is played with a fair coin. A natural generalization of his method is to establish the entrance fee for the case in which the probability of heads is $\theta$ $(0<\theta <1/2)$ . The deduction of those fees is the main result of Section 2. We then propose a Bayesian approach to the problem. When the probability of heads is $\theta$ $(1/2<\theta <1)$ the expected gain of the St. Petersburg game is finite, therefore there is no paradox. However, if one takes $\theta$ as a random variable assuming values in $(1/2,1)$ the paradox may hold, which is counter-intuitive. In Section 3 we determine necessary conditions for the absence of paradox in the Bayesian approach and in Section 4 we establish the entrance fee for the case in which $\theta$ is uniformly distributed in $(1/2,1)$ , for in this case there is a paradox.

In his best-selling book An Introduction to Probability Theory and its Applications, W. Feller established a way of ending the St. Petersburg Paradox by the introduction of an entrance fee, and provided it for the case in which the game is played with a fair coin. A natural generalization of his method is to establish the entrance fee for the case in which the probability of head is θ (0 < θ < 1/2). The deduction of those fees is the main result of Section 2. We then propose a Bayesian approach to the problem. When the probability of head is θ (1/2 < θ < 1) the expected gain of the St. Petersburg Game is finite, therefore there is no paradox. However, if one takes θ as a random variable assuming values in (1/2,1) the paradox may hold, what is counter-intuitive. On Section 3 we determine a necessary and sufficient condition for the absence of paradox on the Bayesian approach and on Section 4 we establish the entrance fee for the case in which θ is uniformly distributed in (1/2,1), for in this case there is paradox.