Survival analysis is concerned with finding models to predict the survival of patients or to assess the efficacy of a clinical treatment. A key part of the model-building process is the selection of the predictor variables. It is standard to use a stepwise procedure guided by a series of significance tests to select a single model, and then to make inference conditionally on the selected model. However, this ignores model uncertainty, which can be substantial. We review the standard Bayesian model averaging solution to this problem and extend it to survival analysis, introducing partial Bayes factors to do so for the Cox proportional hazards model. In two examples, taking account of model uncertainty enhances predictive performance, to an extent that could be clinically useful. 1 Introduction From 1974 to 1984 the Mayo Clinic conducted a double-blinded randomized clinical trial involving 312 patients to compare the drug DPCA with a placebo in the treatment of primary biliary cirrhosis...

Statistical inference in long-horizon event studies has been hampered by the fact that abnormal returns are neither normally distributed nor independent. This study presents a new approach to inference that overcomes these difficulties and dominates other popular testing methods. I illustrate the use of the methodology by examining the long-horizon returns of initial public offerings ~IPOs!. I find that the Fama and French ~1993! three-factor model is inconsistent with the observed long-horizon price performance of these IPOs, whereas a characteristic-based model cannot be rejected. RECENT EMPIRICAL STUDIES IN FINANCE document systematic long-run abnormal price reactions subsequent to numerous corporate activities. 1 Since these results imply that stock prices react with a long delay to publicly available information, they appear to be at odds with the Efficient Markets Hypothesis ~EMH!. Long-run event studies, however, are subject to serious statistical difficulties

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

Elicitation is a key task for subjectivist Bayesians. While skeptics hold that it cannot (or perhaps should not) be done, in practice it brings statisticians closer to their clients and subjectmatter-expert colleagues. This paper reviews the state-of-the-art, reflecting the experience of statisticians informed by the fruits of a long line of psychological research into how people represent uncertain information cognitively, and how they respond to questions about that information. In a discussion of the elicitation process, the first issue to address is what it means for an elicitation to be successful, i.e. what criteria should be employed? Our answer is that a successful elicitation faithfully represents the opinion of the person being elicited. It is not necessarily “true ” in some objectivistic sense, and cannot be judged that way. We see elicitation as simply part of the process of statistical modeling. Indeed in a hierarchical model it is ambiguous at which point the likelihood ends and the prior begins. Thus the same kinds of judgment that inform statistical modeling in general also inform elicitation of prior distributions.

The authors consider the problem of Bayesian variable selection for proportional hazards regression models with right censored data. They propose a semi-parametric approach in which a nonparametric prior is specified for the baseline hazard rate and a fully parametric prior is specified for the regression coe#cients. For the baseline hazard, they use a discrete gamma process prior, and for the regression coe#cients and the model space, they propose a semi-automatic parametric informative prior specification that focuses on the observables rather than the parameters. To implement the methodology, they propose a Markov chain Monte Carlo method to compute the posterior model probabilities. Examples using simulated and real data are given to demonstrate the methodology. R ESUM E Les auteurs abordent d'un point de vue bayesien le problemedelaselection de variables dans les modeles de regression des risques proportionnels en presence de censure a droite. Ils proposent une approche semi-p...

Both knowledge-based systems and statistical models are typically concerned with making predictions about future observables. Here we focus on assessment of predictive performance and provide two techniques for improving the predictive performance of Bayesian graphical models. First, we present Bayesian model averaging, a technique for accounting for model uncertainty. Second, we describe a technique for eliciting a prior distribution for competing models from domain experts. We explore the predictive performance of both techniques in the context of a urological diagnostic problem. KEYWORDS: Prediction; Bayesian graphical model; Bayesian network; Decomposable model; Model uncertainty; Elicitation. 1 Introduction Both statistical methods and knowledge-based systems are typically concerned with combining information from various sources to make inferences about prospective measurements. Inevitably, to combine information, we must make modeling assumptions. It follows that we should car...

To account for the input-model and input-parameter uncertainties inherent in many simulations as well as the usual stochastic uncertainty, we present a Bayesian input-modeling technique that yields improved point and confidence-interval estimators for a selected posterior mean response. Exploiting prior information to specify the prior probabilities of the postulated input models and the associated prior input-parameter distributions, we use sample data to compute the posterior input-model and input-parameter distributions. Our Bayesian simulation replication algorithm involves: (i) estimating parameter uncertainty by randomly sampling the posterior input-parameter distributions; (ii) estimating stochastic uncertainty by running independent replications of the simulation using each set of input-model parameters sampled in (i); and (iii) estimating input-model uncertainty by weighting the responses generated in (ii) using the corresponding posterior input-model probabilities. Sampling effort is allocated among input models to minimize final point-estimator variance subject to a computing-budget constraint. A queueing simulation demonstrates the advantages of this approach.

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.

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In this paper, we use multivariate logistic regression models to incorporate correlation among binary response data. Our objective is to develop a variable subset selection procedure to identify important covariates in predicting correlated binary responses using a Bayesian approach. In order to incorporate available prior information, we propose a class of informative prior distributions on the model parameters and on the model space. The propriety of the proposed informative prior is investigated in detail. Novel computational algorithms are also developed for sampling from the posterior distribution as well as for computing posterior model probabilities. Finally, a simulated data example and a real data example from a prostate cancer study are used to illustrate the proposed methodology.