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.

We consider the problem of model selection and accounting for model uncertainty in high-dimensional contingency tables, motivated by expert system applications. The approach most used currently is a stepwise strategy guided by tests based on approximate asymptotic P-values leading to the selection of a single model; inference is then conditional on the selected model. The sampling properties of such a strategy are complex, and the failure to take account of model uncertainty leads to underestimation of uncertainty about quantities of interest. In principle, a panacea is provided by the standard Bayesian formalism which averages the posterior distributions of the quantity of interest under each of the models, weighted by their posterior model probabilities. Furthermore, this approach is optimal in the sense of maximising predictive ability. However, this has not been used in practice because computing the posterior model probabilities is hard and the number of models is very large (often greater than 1011). We argue that the standard Bayesian formalism is unsatisfactory and we propose an alternative Bayesian approach that, we contend, takes full account of the true model uncertainty byaveraging overamuch smaller set of models. An efficient search algorithm is developed for nding these models. We consider two classes of graphical models that arise in expert systems: the recursive causal models and the decomposable

Ways of obtaining approximate Bayes factors for generalized linear models are described, based on the Laplace method for integrals. I propose a new approximation which uses only the output of standard computer programs such as GUM; this appears to be quite accurate. A reference set of proper priors is suggested, both to represent the situation where there is not much prior information, and to assess the sensitivity of the results to the prior distribution. The methods can be used when the dispersion parameter is unknown, when there is overdispersion, to compare link functions, and to compare error distributions and variance functions. The methods can be used to implement the Bayesian approach to accounting for model uncertainty. I describe an application to inference about relative risks in the presence of control factors where model uncertainty is large and important. Software to implement the

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. The points we emphasize are:- from Jeffreys's Bayesian point of view, the purpose of hypothesis testing is to evaluate the evidence in favor of a scientific theory;- Bayes factors offer a way of evaluating evidence in favor ofa null hypothesis;- Bayes factors provide a way of incorporating external information into the evaluation of evidence about a hypothesis;- Bayes factors are very general, and do not require alternative models to be nested;- several techniques are available for computing Bayes factors, including asymptotic approximations which are easy to compute using the output from standard packages that maximize likelihoods;- in "non-standard " statistical models that do not satisfy common regularity conditions, it can be technically simpler to calculate Bayes factors than to derive non-Bayesian significance

When the Gibbs sampler is used to estimate posterior distributions (Gelfand and Smith, 1990), the question of how many iterations are required is central to its implementation. When interest focuses on quantiles of functionals of the posterior distribution, we describe an easily-implemented method for determining the total number of iterations required, and also the number of initial iterations that should be discarded to allow for "burn-in". The method uses only the Gibbs iterates themselves, and does not, for example, require external specification of characteristics of the posterior density. Here the method is described for the situation where one long run is generated, but it can also be easily applied if there are several runs from different starting points. It also applies more generally to Markov chain Monte Carlo schemes other than the Gibbs sampler. It can also be used when several quantiles are to be estimated, when the quantities of interest are probabilities rath...

Our research empirically analyzes consumer behavior at Internet shopbots — sites that allow consumers to make “one-click ” price comparisons for product offerings from multiple retailers. By allowing researchers to observe exactly what information the consumer is shown and their search behavior in response to this information, shopbot data has unique strengths for analyzing consumer behavior. Furthermore, the method in which the data is displayed to consumers lends itself to a utility-based evaluation process, consistent with econometric analysis techniques. While price is an important determinant of customer choice, we find that, even among shopbot consumers, branded retailers and retailers a consumer visited previously hold significant price advantages in head-to-head price comparisons. Further, customers are very sensitive to how the total price is allocated among the item price, the shipping cost, and tax, and are also quite sensitive to the ordinal ranking of retailer offerings with respect to price. We also find that consumers use brand as a proxy for a retailer’s credibility with regard to non-contractible aspects of the product bundle such as shipping time. In each case our models accurately predict consumer behavior out of sample, suggesting

A Bayesian approach to model selection for structural equation models is outlined. This enables us to compare individual models, nested or non-nested, and also to search through the (perhaps vast) set of possible models for the best ones. The approach selects several models rather than just one, when appropriate, and so enables us to take account, both informally and formally, of uncertainty about model structure when making inferences about quantities of interest. The approach tends to select simpler models than strategies based on multiple P-value-based tests. It may thus help to overcome the criticism of structural

Introduction In order to use Markov chain Monte Carlo, MCMC, it is necessary to determine how long the simulation needs to be run. It is also a good idea to discard a number of initial "burnin " simulations, since from an arbitrary starting point it would be unlikely that the initial simulations came from the stationary distribution intended for the Markov chain. Also, consecutive simulations from Markov chains are dependent, sometimes highly so. Since saving all simulations can require a large amount of storage, researchers using MCMC sometimes prefer saving only every third, fifth, tenth, etc. simulation, especially if the chain is highly dependent. This is sometimes referred to as thinning the chain. While neither burn-in nor thinning are mandatory practices, they both reduce the amount of data saved from a MCMC run. In this chapter, we outline a way of determining in advance the number of iterations needed for a given level of precision in a MCMC algorithm.

In simple one-dimensional stochastic processes it is feasible to model change points explicitly and to make inference about them. I have found that the Bayesian approach produces results more easily than non-Bayesian approaches. It has the advantages of relative technical simplicity, theoretical optimality, and of allowing a formal comparison between abrupt and gradual descriptions of change. When it can be assumed that there is at most one changepoint, this is especially simple. This is illustrated in the context of Poisson point processes. A simple approximation is introduced that is applicable to a wide range of problems in which the change point model can be written as a regression or generalized linear model. When the number of change points is unknown, the Bayesian approach proceeds most naturally by state-space modeling or "hidden Markov chains". The general ideas of this are briefly reviewed, particularly the multi-process Kalman filter. I then describe the application of these...

A general methodology is presented for finding suitable Poisson log-linear models with applications to multiway contingency tables. Mixtures of multivariate normal distributions are used to model prior opinion when a subset of the regression vector is believed to be nonzero. This prior distribution is studied for two and three-way contingency tables, in which the regression coefficients are interpretable in terms of odds-ratios in the table. Efficient and accurate schemes are proposed for calculating the posterior model probabilities. The methods are illustrated for a large number of two-way simulated tables and for two three-way tables. These methods appear to be useful in selecting the best log-linear model and in estimating parameters of interest that reflect uncertainty in the true model. Key words and phrases: Bayes factors, Laplace method, Gibbs sampling, Model selection, Odds ratios. AMS subject classifications: Primary 62H17, 62F15, 62J12. 1 Introduction 1.1 Bayesian testing...