In objective Bayesian model selection, no single criterion has emerged as dominant in defining objective prior distributions. Indeed, many criteria have been separately proposed and utilized to propose differing prior choices. We first formalize the most general and compelling of the various criteria that have been suggested, together with a new criterion. We then illustrate the potential of these criteria in determining objective model selection priors by considering their application to the problem of variable selection in normal linear models. This results in a new model selection objective prior with a number of compelling properties.

A key question in evaluation of computer models is Does the computer model adequately represent reality? A six-step process for computer model validation is set out in Bayarri et al. [Technometrics 49 (2007) 138--154] (and briefly summarized below), based on comparison of computer model runs with field data of the process being modeled. The methodology is particularly suited to treating the major issues associated with the validation process: quantifying multiple sources of error and uncertainty in computer models; combining multiple sources of information; and being able to adapt to different, but related scenarios. Two complications that frequently arise in practice are the need to deal with highly irregular functional data and the need to acknowledge and incorporate uncertainty in the inputs. We develop methodology to deal with both complications. A key part of the approach utilizes a wavelet representation of the functional data, applies a hierarchical version of the scalar validation methodology to the wavelet coefficients, and transforms back, to ultimately compare computer model output with field output. The generality of the methodology is only limited by the capability of a combination of computational tools and the appropriateness of decompositions of the sort (wavelets) employed here. The methods and analyses we present are illustrated with a test bed dynamic stress analysis for a particular engineering system.Comment: Published in at http://dx.doi.org/10.1214/009053607000000163 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

In this paper, we consider that observations Y come from a general normal linear model and that it is desired to test a simplifying (null) hypothesis about the parameters. We approach this problem from an objective Bayesian, model selection perspective. Crucial ingredients for this approach are 'proper objective priors' to be used for deriving the Bayes factors. Jeffreys-Zellner-Siow priors have shown to have good properties for testing null hypotheses defined by specific values of the parameters in full rank linear models. We extend these priors to deal with general hypotheses in general linear models, not necessarily full rank. The resulting priors, which we call 'conventional priors', are expressed as a generalization of recently introduced 'partially informative distributions'. The corresponding Bayes factors are fully automatic, easy to compute and very reasonable. The methodology is illustrated for two popular problems: the change point problem and the equality of treatments effects problem. We compare the conventional priors derived for these problems with other objective Bayesian proposals like the intrinsic priors. It is concluded that both priors behave similarly although interesting subtle differences arise. Finally, we accommodate the conventional priors to deal with non nested model selection as well as multiple model comparison.

One important aspect of Bayesian model selection is how to deal with huge model spaces, since the exhaustive enumeration of all the models entertained is not feasible and inferences have to be based on the very small proportion of models visited. This is the case for the variable selection problem with a moderately large number of possible explanatory variables considered in this article. We review some of the strategies proposed in the literature, from a theoretical point of view using arguments of sampling theory and in practical terms using several examples with a known answer. All our results seem to indicate that sampling methods with frequency-based estimators outperform searching methods with renormalized estimators. Supplementary materials for this article are available online.

There is abundant empirical literature that focuses on whether energy consumption is a critical driver of economic growth. The evolution of this literature has largely consisted of attempts to solve the problems and answer the criticisms arising from earlier studies. One of the most common criticisms is that previous work concentrates on the bivariate relationship, energy consumption-economic growth. Many authors try to overcome this critique using control variables.However, the choice of these variables has been ad hoc, made according to the subjective economic rationale of the authors. Our contribution to this literature is to apply a robust probabilistic model to select the explanatory variables from a large set of potential candidates in the case of the US from 1949 to 2010, not only for an aggregate analysis but also for a sector analysis. The results highlight the critical role of public spending and energy intensity in the explanation of growth. Furthermore, since the study reveals different explanatory variables for each sector, it indicates the importance of policy decisions specifically aimed