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.

The classical chi-square test of goodness-of-fit compares the hypothesis that data arise from some parametric family of distributions, against the nonparametric alternative that they arise from some other distribution. However, the chi-square test requires continuous data to be grouped into arbitrary categories. Furthermore, as the test is based upon an approximation, it can only be used if there is su#cient data. In practice, these requirements are often wasteful of information and overly restrictive. The authors explore the use of the fractional Bayes factor to obtain a Bayesian alternative to the chi-square test when no specific prior information is available. They consider the extent to which their methodology can handle small data sets and continuous data without arbitrary grouping. R ESUM E Le test classique d'ajustement du khi-deux confronte l'hypothese que les observations proviennent d'une famille parametrique de lois a l'hypothese non parametrique qu'elles sont issues d'un...

We propose the use of the generalized fractional Bayes factor for testing fit in multinomial models. This is a non-asymptotic method that can be used to quantify the evidence for or against a sub-model. We give expressions for the generalized fractional Bayes factor and we study its properties. In particular, we show that the generalized fractional Bayes factor has better properties than the fractional Bayes factor. Keywords: generalized fractional Bayes factor, Dirichlet process, Beta-Stacy process. 1. Introduction. In this paper we propose a Bayesian method for testing fit in multinomial models. Specifically, we will use the Bayes factor for evaluating the evidence for or against a null sub-model of the multinomial. The advantages of using a Bayesian approach for this problem are that it does not rely