This paper reviews the literature on Bayesian experimental design, both for linear and nonlinear models. A unified view of the topic is presented by putting experimental design in a decision theoretic framework. This framework justifies many optimality criteria, and opens new possibilities. Various design criteria become part of a single, coherent approach.

this article we propose three criteria that can be used to address model selection. These emphasize observables rather than parameters and are based on a certain Bayesian predictive density. They have a unifying basis that is simple and interpretable,are free of asymptotic de#nitions,and allow the incorporation of prior information. Moreover,two of these criteria are readily calibrated.

The problem of variable selection is one of the most pervasive model selection problems in statistical applications. Often referred to as the problem of subset selection, it arises when one wants to model the relationship between a variable of interest and a subset of potential explanatory variables or predictors, but there is uncertainty about which subset to use. This vignette reviews some of the key developments which have led to the wide variety of approaches for this problem. 1

this paper, we assume that there are a finite number of possible true models. For each possible model m, we define the random (vector) variable \Theta m whose values correspond to the possible values of the parameters for m. We encode our uncertainty about \Theta m using the probability distribution p(\Theta m jm). In this paper, we assume that p(\Theta m jm) is a probability density function. Given random sample D, we compute the posterior distributions for M and each \Theta m

... In this article we present the general outlook and discuss general families of elaborations for use in practice; the exponential connection elaboration plays a key role. We then describe model elaborations for use in diagnosing: departures from normality, goodness of fit in generalized linear models, and variable selection in regression and outlier detection. We illustrate our approach with two applications.

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.

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