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 Jeffreys prior mass for the precise hypothesis on the separability of probability and utility.

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Abstract: We consider various properties of Bayesian inferences related to repeated sampling interpretations, when we have a proper prior. While these can be seen as particularly relevant when the prior is diffuse, we argue that it is generally reasonable to consider such properties as part of our assessment of Bayesian inferences. We discuss the logical implications for how repeated sampling properties should be assessed when we have a proper prior. We develop optimal Bayesian repeated sampling inferences using a generalized idea of what it means for a credible region to contain a false value and discuss the practical use of this idea for error assessment and experimental design. We present results that connect Bayes factors with optimal inferences and develop a generalized concept of unbiasedness for credible regions. Further, we consider the effect of reparameterizations on hpd-like credible regions and argue that one reparameterization is most relevant, when repeated sampling properties and the prior are taken into account. Key words and phrases: repeated sampling properties, relative surprise inferences, prior-data conflict, Bayes factors, relative belief ratios. 1

We show how the Full Bayesian Significance Test (FBST) can be used as a model selection criterion. The FBST was presented in Pereira and Stern as a coherent Bayesian significance test.

Measures of surprise refer to quantifications of the degree of incompatibility of data with some hypothesized model H 0 without any reference to alternative models. Traditional measures of surprise have been the p-values, which are however known to grossly overestimate the evidence against H 0 . Strict Bayesian analysis calls for an explicit specification of all possible alternatives to H 0 so Bayesians have not made routine use of measures of surprise. In this report we CRITICALLY REVIEw the proposals that have been made in this regard. We propose new modifications, stress the connections with robust Bayesian analysis and discuss the choice of suitable predictive distributions which allow surprise measures to play their intended role in the presence of nuisance parameters. We recommend either the use of appropriate likelihoodratio type measures or else the careful calibration of p-values so that they are closer to Bayesian answers. Key words and phrases. Bayes factors; Bayesian p-values; Bayesian robustness; Conditioning; Model checking; Predictive distributions. 1.

New application of the Full Bayesian Significance Test (FBST) for precise hypotheses is presented. The FBST is an alternative to significance tests or, equivalently, to p-values. In the FBST we compute the evidence of the precise hypothesis. This evidence is the complement of the probability of a credible set "tangent" to the sub-manifold (of the parameter space) that defines the null hypothesis. We use the FBST to compare coefficients of variation, in applications arising in finance and industrial engineering.

Inferences that arise from loss functions determined by the prior are considered and it is shown that these lead to limiting Bayes rules that are closely connected with likelihood. The procedures obtained via these loss functions are invariant under reparameterizations and are Bayesian unbiased or limits of Bayesian unbiased inferences. These inferences serve as well-supported alternatives to MAP-based inferences. Key words and phrases: loss functions, relative surprise, lowest posterior risk region, Bayesian unbiasedness. 1

Abstract: We consider various properties of Bayesian inferences related to repeated sampling interpretations, when we have a proper prior. While these can be seen as particularly relevant when the prior is diffuse, we argue that it is generally reasonable to consider such properties as part of our assessment of Bayesian inferences. We discuss the logical implications for how repeated sampling properties should be assessed when we have a proper prior. We develop optimal Bayesian repeated sampling inferences using a generalized idea of what it means for a credible region to contain a false value and discuss the practical use of this idea for error assessment and experimental design. We present results that connect Bayes factors with optimal inferences and develop a generalized concept of unbiasedness for credible regions. Further, we consider the effect of reparameterizations on hpd-like credible regions and argue that one reparameterization is most relevant, when repeated sampling properties and the prior are taken into account. Key words and phrases: repeated sampling properties, relative surprise inferences, prior-data conflict, Bayes factors, relative belief ratios. 1

Relative surprise inferences are based on how beliefs change from a priori to a posteriori. These inferences can be seen to be based on the posterior distribution of the integrated likelihood and, as such, are invariant under relabellings of the parameter of interest. In this paper we demonstrate that relative surprise inferences possess an optimality property. Further, computational techniques are developed for implementing these inferences that are applicable whenever we have algorithms to sample from the prior and posterior distributions. 1

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