Sergio Wechsler

Can a significance test be genuinely Bayesian

The Full Bayesian Signicance 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.

On the Bayesianity of Pereira-Stern tests

C. Pereira and J. Stern have recently introduced a measure of evidence of a precise hypothesis consisting of the posterior probability of the set of points having smaller density than the supremum over the hypothesis. The related procedure is seen to be a Bayes test for specific loss functions. The nature of such loss functions and their relation to stylised inference problems are investigated. The dependence of the loss function on the sample is also discussed as well as the consequence of the introduction of Jeffrey’s prior mass for the precise hypothesis on the separability of probability and utility.

Birnbaum’s Theorem Redux

We revisit Birnbaum’s results on the Likelihood Principle, reorganizing and making a few formal changes which address some of the criticisms at the original development. The meaning of the results for different theories of statistical inference is exemplified, and the role of the Likelihood Principle as means to assess the consistency of those theories is emphasized.

A Bayesian Decision-Theoretic Approach to Logically-Consistent Hypothesis Testing

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.

A note on extendibility and predictivistic inference in finite populations

The usual finite population model—where information provided by a subset of units is used to reduce uncertainty about functions of the complete population list of values—is explored from a predictivistic point of view. Under this approach, only operationally meaningful quantities (operational parameters) are considered and therefore no superpopulation parameters are involved. This paper addresses the estimation of both population total and maximum based on uniformity and/or exchangeability judgments on finite sequences of random variables. A central point of this paper is that there are contexts in which the superpopulation approach cannot be employed in inferential problems in finite populations. There are circumstances in which the prior distributions for the operational parameters cannot be obtained from any superpopulation model. Conditions for the extendibility to infinite populations are also established for some models, as this approach may ease the inferential problem.

The Gambler’s Fallacy: A Bayesian Approach

We study the problem of prediction in sequences of binary random variables. The models are then considered vis‐a‐vis the Gambler’s Fallacy. Another model in which the Gambler’s Fallacy need not be a fallacy is presented. The results may contribute for the judgment of how reasonable the assumption of infinite exchangeability is relative to typical human perception.

Exchangeability and the law of maturity

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.

Rain dance: the role of randomization in clinical trials

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A Bayesian Look at Nonidentifiability: A Simple Example

This article discusses the concept of identifiability in simple probability calculus. Emphasis is given to Bayesian solutions. In particular, we compare Bayes and maximum likelihood estimators. We advocate adoption of informative prior probabilities for the Bayesian operation in place of diffuse or reference priors. We also discuss the concept of identifying functions.

On the Estimation of Process Parameters in the Taguchi’s Approach to the On‐line Control Procedure for Attributes

Under the model proposed by Nayebpour and Woodall [5] for Taguchi’s on‐line control procedure for attributes, estimators for the process parameter vector are derived both from the Classical (maximum likelihood) and Bayesian standpoints. The likelihood function is generated by the detection time of the first defective item under the control procedure. Under the Classical standpoint, a case of nonidentifiability is disclosed. Under the Bayesian standpoint, posterior probability distributions for the process parameters are determined by taking into account independent beta prior distributions.