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.

To compute a Bayes factor for testing H 0 : / = / 0 in the presence of a nuisance parameter fi, priors under the null and alternative hypotheses must be chosen. As in Bayesian estimation, an important problem has been to define automatic or "reference" methods for determining priors based only on the structure of the model. In this paper we apply the heuristic device of taking the amount of information in the prior on / equal to the amount of information in a single observation. Then, after transforming fi to be "null orthogonal" to /, we take the marginal priors on fi to be equal under the null and alternative hypotheses. Doing so, and taking the prior on / to be Normal, we find that the log of the Bayes factor may be approximated by the Schwarz criterion with an error of order O(n \Gamma1=2 ), rather than the usual error of order O(1). This result suggests the Schwarz criterion should provide sensible approximate solutions to Bayesian testing problems, at least when the hypothese...

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

Structured additive regression models are perhaps the most commonly used class of models in statistical applications. It includes, among others, (generalised) linear models, (generalised) additive models, smoothing-spline models, state-space models, semiparametric regression, spatial and spatio-temporal models, log-Gaussian Cox-processes, geostatistical and geoadditive models. In this paper we consider approximate Bayesian inference in a popular subset of structured additive regression models, latent Gaussian models, where the latent field is Gaussian, controlled by a few hyperparameters and with non-Gaussian response variables. The posterior marginals are not available in closed form due to the non-Gaussian response variables. For such models, Markov chain Monte Carlo methods can be implemented, but they are not without problems, both in terms of convergence and computational time. In some practical applications, the extent of these problems is such that Markov chain Monte Carlo is simply not an appropriate tool for routine analysis. We show that, by using an integrated nested Laplace approximation and its simplified version, we can directly compute very accurate approximations to the posterior marginals. The main benefit of these approximations

Introduction To motivate the methods described in this chapter, consider the following inference problem in astronomy (Soubiran, 1993). Until fairly recently, it has been believed that the Galaxy consists of two stellar populations, the disk and the halo. More recently, it has been hypothesized that there are in fact three stellar populations, the old (or thin) disk, the thick disk, and the halo, distinguished by their spatial distributions, their velocities, and their metallicities. These hypotheses have different implications for theories of the formation of the Galaxy. Some of the evidence for deciding whether there are two or three populations is shown in Figure 1, which shows radial and rotational velocities for n = 2; 370 stars. A natural model for this situation is a mixture model with J components, namely y i = J X j=1 ae j

Region-based image segmentation methods require some criterion for determining when to merge regions. This paper presents a novel approach by introducing a Bayesian probability of homogeneity in a general statistical context. Our approach does not require parameter estimation, and is therefore particularly beneficial for cases in which estimation-based methods are most prone to error: when little information is contained in some of the regions and, therefore, parameter estimates are unreliable. We apply this formulation to three distinct parametric model families that have been used in past segmentation schemes: implicit polynomial surfaces, parametric polynomial surfaces, and Gaussian Markov random fields. We present results on a variety of real range and intensity images. 1 Introduction The problem of image segmentation, partitioning an image into a set of homogeneous regions, is a fundamental problem in computer vision. Approaches to the segmentation problem can be grouped...

Epidemiological studies for assessing risk factors often use logistic regression, log-linear models, or other generalized linear models. They involve many decisions, including the choice and coding of risk factors and control variables. It is common practice to select independent variables using a series of significance tests and to choose the way variables are coded somewhat arbitrarily. The overall properties of such a procedure are not well understood, and conditioning on a single model ignores model uncertainty, leading to underestimation of uncertainty about quantities of interest (QUOIs). We describe a Bayesian modeling strategy that formalizes the model selection process and propagates model uncertainty through to inference about QUOIs. Each possible combination of modeling decisions defines a different model, and the models are compared using Bayes factors. Inference about a QUOI is based on an average of its posterior distributions under the individual models, weighted by thei...

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...

The goal of traditional probabilistic approaches to image segmentation has been to derive a single, optimal segmentation, given statistical models for the image formation process. In this paper, we describe a new probabilistic approach to segmentation, in which the goal is to derive a set of plausible segmentation hypotheses and their corresponding probabilities. Because the space of possible image segmentations is too large to represent explicitly, we present a representation scheme that allows the implicit representation of large sets of segmentation hypotheses that have low probability. We then derive a probabilistic mechanism for applying Bayesian, model-based evidence to guide the construction of this representation. One key to our approach is a general Bayesian method for determining the posterior probability that the union of regions is homogeneous, given that the individual regions are homogeneous. This method does not rely on estimation, and properly treats the issu...