In contrast to a posterior analysis given a particular sampling model, posterior model probabilities in the context of model uncertainty are typically rather sensitive to the specification of the prior. In particular, “diffuse” priors on model-specific parameters can lead to quite unexpected consequences. Here we focus on the practically relevant situation where we need to entertain a (large) number of sampling models and we have (or wish to use) little or no subjective prior information. We aim at providing an “automatic” or “benchmark” prior structure that can be used in such cases. We focus on the Normal linear regression model with uncertainty in the choice of regressors. We propose a partly noninformative prior structure related to a Natural Conjugate g-prior specification, where the amount of subjective information requested from the user is limited to the choice of a single scalar hyperparameter g0j. The consequences of different choices for g0j are examined. We investigate theoretical properties, such as consistency of the implied Bayesian procedure. Links with classical information criteria are provided. More importantly, we examine the finite sample implications of several choices of g0j in a simulation study. The use of the MC3 algorithm of Madigan and York (1995), combined with efficient coding in Fortran, makes it feasible to conduct large simulations. In addition to posterior criteria, we shall also compare the predictive performance of different priors. A classic example concerning the economics of crime will also be provided and contrasted with results in the literature. The main findings of the paper will lead us to propose a “benchmark” prior specification in a linear regression context with model uncertainty.

Standard statistical practice ignores model uncertainty. Data analysts typically select a model from some class of models and then proceed as if the selected model had generated the data. This approach ignores the uncertainty in model selection, leading to over-con dent inferences and decisions that are more risky than one thinks they are. Bayesian model averaging (BMA) provides a coherent mechanism for accounting for this model uncertainty. Several methods for implementing BMA haverecently emerged. We discuss these methods and present anumber of examples. In these examples, BMA provides improved out-of-sample predictive performance. We also provide a catalogue of

Covariate and confounder selection in case-control studies is most commonly carried out using either a two-step method or a stepwise variable selection method in logistic regression. Inference is then carried out conditionally on the selected model, but this ignores the model uncertainty implicit in the variable selection process, and so underestimates uncertainty about relative risks. We report on a simulation study designed to be similar to actual case-control studies. This shows that p-values computed after variable selection can greatly overstate the strength of conclusions. For example, for our simulated case-control studies with 1,000 subjects, of variables declared to be "significant" with p-values between.01 and.05, only 49 % actually were risk factors when stepwise variable selection was used. We propose Bayesian model averaging as a formal way of taking account of model uncertainty in case-control studies. This yields an easily interpreted summary, the posterior probability that a variable is a risk factor, and our simulation study indicates this to be reasonably well calibrated in the situations simulated. The methods are applied and compared

The paper summarizes some important results at the intersection of the fields of Bayesian statistics and stochastic simulation. Two statistical analysis issues for stochastic simulation are discussed in further detail from a Bayesian perspective. First, a review of recent work in input distribution selection is presented. Then, a new Bayesian formulation for the problem of output analysis for a single system is presented. A key feature is analyzing simulation output as a random variable whose parameters are an unknown function of the simulation's inputs. The distribution of those parameters is inferred from simulation output via Bayesian response-surface methods. A brief summary of Bayesian inference and decision making is included for reference.

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We investigate the Bayesian Information Criterion (BIC) for variable selection in models for censored survival data. Kass and Wasserman (1995) showed that BIC provides a close approximation to the Bayes factor when a unit-information prior on the parameter space is used. We propose a revision of the penalty term in BIC so that it is de ned in terms of the number of uncensored events instead of the number of observations. For the simplest censored data model, that of exponential distributions of survival times (i.e. a constant hazard rate), this revision results in a better approximation to the exact Bayes factor based on a conjugate unit-information prior. In the Cox proportional hazards regression model, we propose de ning BIC in terms of the maximized partial likelihood. Using the number of deaths rather than the number of individuals in the BIC penalty term corresponds to a more realistic prior on the parameter space, and is shown to improve predictive performance for assessing stroke risk in the Cardiovascular Health Study.

The need to explore model uncertainty in linear regression models with many predictors has motivated improvements in Markov chain Monte Carlo sampling algorithms for Bayesian variable selection. Currently used sampling algorithms for Bayesian variable selection may perform poorly when there are severe multicollinearities among the predictors. This article describes a new sampling method based on an analogy with the Swendsen-Wang algorithm for the Ising model, and which can give substantial improvements over alternative sampling schemes in the presence of multicollinearity. In linear regression with a given set of potential predictors we can index possible models by a binary parameter vector that indicates which of the predictors are included or excluded. By thinking of the posterior distribution of this parameter as a binary spatial field, we can use auxiliary variable methods inspired by the Swendsen-Wang algorithm for the Ising model to sample from the posterior where dependence among parameters is reduced by conditioning on auxiliary variables. Performance of the method is described for both simulated and real data.

Standard survival analysis can be given a neural interpretation in terms of a multi-layered perceptron (MLP) with exponential transfer functions. More hidden units accommodate more complex relationships. The neural interpretation suggests a Bayesian analysis, which allows one to introduce sensible priors and to sample from the posterior. We also propose a method for computing p-values from the obtained ensemble of networks, because, in the end, this is the kind of information medical experts are familiar with. We apply our methods on a database regarding patients with ovarian cancer. 1 Introduction The goal of survival analysis (in medical terms) is to estimate the chances of a patient 's survival as a function of time, given the medical information available on this patient. A well-known way to conduct such an analysis, is the proportional hazards method designed by Cox [1]. In this method the hazard function h(t; x), which estimates the probability density of death occurring at tim...

We investigate country heterogeneity in cross-country growth regressions. In contrast to the previous literature that focuses on low-income countries, this study also highlights growth determinants in high-income (OECD) countries. We introduce Iterative Bayesian Model Averaging (IBMA) to address not only potential parameter heterogeneity, but also the model uncertainty inherent in growth regressions. IBMA is essential to our estimation because the simultaneous consideration of model uncertainty and parameter heterogeneity in standard growth regressions increases the number of candidate regressors beyond the processing capacity of ordinary BMA algorithms. Our analysis generates three results that strongly support different dimensions of parameter heterogeneity. First, while a large number of regressors can be identified as growth determinants in Non-OECD countries, the same regressors are irrelevant for OECD countries. Second, Non-OECD countries and the global sample feature only a handful of common growth determinants. Third, and most devastatingly, the long list of variables included in popular cross-country datasets does not contain regressors that begin to satisfactorily characterize the basic

We study a flexible class of non-proportional hazard function regression models in which the influence of the covariates splits into the sum of a parametric part and a time-dependent nonparametric part. We develop a method of covariate selection for the parametric part by adjusting for the implicit fitting of the nonparametric part. Our approach is based on the general model selection methodology of Barron, Birge and Massart, suitably adapted to the censored semiparametric regression framework. Asymptotic consistency of the proposed covariate selection method is established, leading to asymptotically normal estimators of both parametric and nonparametric parts of the model in the presence of covariate selection. The approach is applied to a real data set and a simulation study is presented.