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.

The law of maturity is the belief that less-observed events are becoming mature and, therefore, more likely to occur in the future. Previous studies have shown that the assumption of infinite exchangeability contradicts the law of maturity. In particular, it has been shown that infinite exchangeability contradicts probabilistic descriptions of the law of maturity such as the gambler’s belief and the belief in maturity. We show that the weaker assumption of finite exchangeability is compatible with both the gambler’s belief and (...) belief in maturity. We provide sufficient conditions under which these beliefs hold under finite exchangeability. These conditions are illustrated with commonly used parametric models. (shrink)

It is always possible to construct a real function f, given random quantities X and Y with continuous distribution functions F and G, respectively, in such a way that f(X) and f(Y), also random quantities, have both the same distribution function, say H. This result of De Finetti introduces an alternative way to somehow describe the `opinion' of a group of experts about a continuous random quantity by the construction of Fields of coincidence of opinions (FCO). A Field of coincidence (...) of opinions is a finite union of intervals where the opinions of the experts coincide with respect to that quantity of interest. We speculate on (dis)advantages of Fields of Opinion compared to usual `probability' measures of a group and on their relation with a continuous version of the well-known Allais' paradox. (shrink)

It is always possible to construct a real function φ, given random quantities X and Y with continuous distribution functions F and G, respectively, in such a way that φ(X) and φ(Y), also random quantities, have both the same distribution function, say H. This result of De Finetti introduces an alternative way to somehow describe the `opinion' of a group of experts about a continuous random quantity by the construction of Fields of coincidence of opinions (FCO). A Field of coincidence (...) of opinions is a finite union of intervals where the opinions of the experts coincide with respect to that quantity of interest. We speculate on (dis)advantages of Fields of Opinion compared to usual `probability' measures of a group and on their relation with a continuous version of the well-known Allais' paradox. (shrink)

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 θ. 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 θ 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 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 θ is uniformly distributed in, for in this case there is a paradox. (shrink)

. The uniform prior distribution is often seen as a mathematical description of noninformativeness. This paper uses the well-known Three Prisoners Paradox to examine the impossibility of maintaining noninformativeness throughout hierarchization. The Paradox has been solved by Bayesian conditioning over the choice made by the Warder when asked to name a prisoner who will be shot. We generalize the paradox to situations of N prisoners, k executions and m announcements made by the Warder. We then extend the consequences of hierarchically (...) placing uniform and symmetrical priors for the probability p of the Warder naming Prisoner B, say. We prove that breaks of indifference and neutrality caused by assignment of uniform and symmetrical priors in lieu of degenerate indifference probabilities hold in general. Speaking of unknown probabilities or of probability of a probability must be forbidden as meaningless. Bruno de Finetti, 1977 I regard the use of hierarchical chains as a technique helping you to sharpen your subjective probabilities. I. J. Good, 1981. (shrink)

The main purpose of this thesis is to explore de Finetti's ideas and contributions to decision theory. Such ideas are not as well-known as his work on probability. ;The first part of the work is placed in a unisubjective decision-making context. It starts by including a discussion on predictivism, an approach to statistics which de Finetti insisted on and which has only recently been rediscovered and advocated. ;The second part is placed in the context of group, or multisubjective, decision-making. This (...) area is well-known for its negative results, starting with Arrow's impossibility theorem. While being aware of such difficulties, de Finetti made important contributions to this field, which he saw as the appropriate context to approach basic concepts such as "objectivity" and Wald's admissibility theorem. Such concepts, together with exchangeability, illuminate the Bayesian position by also making connection with the non-Bayesian approach. ;The last part of the work starts with a critical discussion of classical methods which are unacceptable for Bayesians. Finally, an important application problem--environmental stress screening--is solved from a Bayesian predictivist point of view. (shrink)