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

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

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

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...

I would like to thank David L. Weakliem (1999 [this issue]) for a thought-provoking discussion of the basis of the Bayesian information criterion (BIC). We may be in closer agreement than one might think from reading his article. When writing about Bayesian model selection for social researchers, I focused on the BIC approximation on the grounds that it is easily implemented and often reasonable, and simplifies the exposition of an already technical topic. As Weakliem says, BIC corresponds to one of many possible priors, although I will argue that this prior is such as to make BIC appropriate for baseline reference use and reporting, albeit not necessarily always appropriate for drawing final conclusions. When writing about the same subject for statistical journals, however, I have paid considerable attention to the choice of priors for Bayes factors. I thank Weakliem for bringing this subtle but important topic to the attention of sociologists. In 1986, I proposed replacing P values by Bayes factors as the basis for hypothesis testing and model selection in social research, and I suggested BIC as a simple and convenient, albeit crude, approximation. Since then, a great deal has been learned about Bayes factors in general, and about BIC in particular. Weakliem seems to agree that the Bayes factor framework is a useful one for hypothesis testing and model selection; his concern is with how the Bayes factors are to be evaluated. Weakliem makes two main points about the BIC approximation. The first is that BIC yields an approximation to Bayes factors that corresponds closely to a particular prior (the unit information prior) on

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The aim of my PhD is to develop novel algorithms to allow an Autonomous Underwater Vehicle (AUV) to locate hydrothermal vents on the ocean floor. Hydrothermal vents are tectonically-driven outgassings of mineral-rich superheated water, and they produce a chemical-advecting plume that can be detected from kilometres away. Finding vents is challenging firstly because detecting a chemical tracer from a plume gives very little information on the bearing or range to the source, and secondly because tracers from different vents combine in an additive way, and there is no a priori way of telling how many vents have contributed to a measured signal. I have decomposed the task of finding vents into a mapping problem, where a probabilistic map of nearby vents is constructed, and a planning problem, which uses the uncertain map to determine actions the AUV should take to allow it to find as many vents as possible on a mission, subject to the limited power resources it has. Both problems will require the devel-opment of new methods to solve them. The mapping problem is novel because sensors do not provide even an approximate range to their target, there are potentially multiple targets, and

Null-hypothesis significance testing remains the standard inferential tool in cognitive science despite its serious disadvantages. Primary among these is the fact that the resulting probability value does not tell the researcher what he or she usually wants to know: how probable is a hypothesis, given the obtained data? Inspired by developments presented by Wagenmakers (2007), I provide a tutorial on a Bayesian model-selection approach that requires only a simple transformation of sum of squares values generated by the standard analysis of variance. This approach generates a graded level of evidence regarding which model (e.g., effect absent [null hypothesis] vs. effect present [alternative hypothesis]) is more strongly supported by the data. This method also obviates admonitions never to speak of accepting the null hypothesis. An Excel worksheet for computing the Bayesian analysis is provided as supplemental material. The widespread use of null-hypothesis significance testing (NHST) in psychological research has withstood numerous rounds of debate (e.g., Chow, 1998; Cohen,

This paper examines the factors driving community tax incentives for industry recruitment. The empirical results show that spatial competition among Georgia counties is an important factor determining their propensity to use fiscal incentives. The proximity effect is robust and diminishes with distance. In addition, we find that county governments use fiscal incentives to compensate for higher taxes and that higher income jurisdictions do not tend to forestall nonresidential development (and its attendant externalities). Interestingly, neither economic diversification nor government form affects development policies. It also appears that fiscally troubled local governments cannot sustain the short term costs of aggressively recruiting industry in order to garner the long term benefits.