In a 1935 paper, and in his book Theory of Probability, Jeffreys developed a methodology for quantifying the evidence in favor of a scientific theory. The centerpiece was a number, now called the Bayes factor, which is the posterior odds of the null hypothesis when the prior probability on the null is one-half. Although there has been much discussion of Bayesian hypothesis testing in the context of criticism of P -values, less attention has been given to the Bayes factor as a practical tool of applied statistics. In this paper we review and discuss the uses of Bayes factors in the context of five scientific applications in genetics, sports, ecology, sociology and psychology.

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.

In the specification of linear regression models it is common to indicate a list of candidate variables from which a subset enters the model with nonzero coefficients. In some cases any combination of variables may enter, but in others certain necessary conditions must be satisfied: e.g., in time series applications it is common to allow a lagged variable only if all shorter lags for the same variable also enter. This paper interprets this specification as a mixed continuous-discrete prior distribution for coefficient values. It then utilizes a Gibbs sampler to construct posterior moments. It is shown how this method can incorporate sign constraints and provide posterior probabilities for all possible subsets of regressors. The methods are illustrated using some standard data sets.

Variable selection in the linear regression model takes many apparent faces from both frequentist and Bayesian standpoints. In this paper we introduce a variable selection method referred to as a rescaled spike and slab model. We study the importance of prior hierarchical specifications and draw connections to frequentist generalized ridge regression estimation. Specifically, we study the usefulness of continuous bimodal priors to model hypervariance parameters, and the effect scaling has on the posterior mean through its relationship to penalization. Several model selection strategies, some frequentist and some Bayesian in nature, are developed and studied theoretically. We demonstrate the importance of selective shrinkage for effective variable selection in terms of risk misclassification, and show this is achieved using the posterior from a rescaled spike and slab model. We also show how to verify a procedure’s ability to reduce model uncertainty in finite samples using a specialized forward selection strategy. Using this tool, we illustrate the effectiveness of rescaled spike and slab models in reducing model uncertainty. 1. Introduction. We

This paper considers the class of sequential probit models in relation to other models for ordinal data. Hierarchical and other extensions of the model are proposed for applications involving discrete time (grouped) survival data. Computationally practical Markov chain Monte Carlo algorithms are developed for the fitting of these models. The ideas and methods are illustrated in detail with a real data example on the length of hospital stay for patients undergoing heart surgery. A notable aspect of this analysis is the comparison, based on marginal likelihoods and training sample priors, of several non-nested models, such as the sequential model, the cumulative ordinal model and Weibull and log-logistic models. Keywords: Bayes factor; Discrete hazard function; Gibbs sampling; Marginal likelihood; Metropolis-Hastings algorithm; Non-nested models; Sequential probit; Training sample prior; Model comparison. 1 Introduction Ordinal response data is generally analyzed using the cumulative o...

this article we propose a new method to solve #i#. We do this by focusing on observables, requiring only a few easily interpretable prior parameters to be speci#ed. These same parameter speci#cations can also be used to solve #ii# as proposed in L&I.

This paper considers the fitting, criticism and comparison of three ordinal regression models -- the cumulative, sequential and two-step models. Efficient algorithms based on Markov chain Monte Carlo methods are developed for each model. In the case of the cumulative model, a new Metropolis-Hastings procedure to sample the cut points is proposed. This procedure relies on a simple transformation of the cut-points that leaves the transformed cut-points unordered. For comparing these models, we develop a coherent approach based on marginal likelihoods and Bayes factors. To help in the assignment of prior distributions to regression parameters and the cut-points, different methods for forming and representing prior beliefs are provided. One set of methods is based on the idea of a training sample and a prior imaginary sample. Another method is based on the direct assessment of distributions on the multinomial response, followed by change of variable to a distribution on the parameters of t...

ABSTRACT: The “business cycle ” is a fundamental, yet elusive concept in macroeconomics. In this paper, we consider the problem of measuring the business cycle. First, we argue for the ‘output-gap ’ view that the business cycle corresponds to transitory deviations in economic activity away from a permanent or “trend ” level. Then, we investigate the extent to which a general model-based approach to estimating trend and cycle for the U.S. economy leads to measures of the business cycle that reflect models versus the data. We find empirical support for a nonlinear time series model that produces a business cycle measure with an asymmetric shape across NBER expansion and recession phases. Specifically, this business cycle measure suggests that recessions are periods of relatively large and negative transitory fluctuations in output. However, several close competitors to the nonlinear model produce business cycle measures of widely differing shapes and magnitudes. Given this model-based uncertainty, we construct a model-averaged measure of the business cycle. The model-averaged measure also displays an asymmetric shape and is closely related to other measures of economic

this paper, we propose two variable and transformation selection procedures on the predictor variables in the linear model. The first procedure is a simultaneous variable and transformation selection procedure. For data sets with many predictors, a stepwise variable selection procedure is also presented. The procedures are based on Bayesian model selection criteria introduced by Ibrahim and Laud (1994) and Laud and Ibrahim (1995). Several examples are given to illustrate the methodology.

In the specification of linear regression models it is common to indicate a list of candidate variables from which a subset enters the model with nonzero coefficients. In some cases any combination of variables may enter, but in others certain necessary conditions must be satisfied: e.g., in time series applications it is common to allow a lagged variable only if all shorter lags for the same variable also enter. This paper interprets this specification as a mixed continuous-discrete prior distribution for coefficient values. It then utilizes a Gibbs sampler to construct posterior moments. It is shown how this method can incorporate sign constraints and provide posterior probabilities for all possible subsets of regressors. The methods are illustrated using some standard data sets. Partial financial support from NSF grant SES-9210070 is gratefully acknowledged. Thanks to Rob McCulloch for providing some of the data used in this paper. This research was conducted while the author was a...