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This paper examines the implications of the market selection hypothesis on the accuracy of the probabilities implied by equilibrium prices and on the “learning” mechanism of markets. I use the standard machinery of dynamic general equilibrium models to generate a rich class of probabilities, price probabilities, and discuss their properties. This class includes the Bayes’ rule and known non-Bayesian rules. If the prior support is well-specified, I prove that all members of this class perform as well as Bayes’ rule in terms of likelihood. If the prior support is misspecified in that the Bayesian prior does not converge, I demonstrate that some members of price probabilities significantly outperform Bayes’. Because these members are never worse and sometimes better than Bayes, my result challenges the prevailing opinion that Bayes’ rule is the only rational way to learn.

One-sided Bayes factors are obtained for the association in 2 by 2 tables, under three different sampling designs. These Bayes factors are relevant when one has a prior believe about the direction of an association. It is further shown how prior knowledge can be incorporated in determining the support of a positive association against the hypothesis of row-column independence.

Empirical Bayes estimates of the local false discovery rate can reflect uncertainty about the estimated prior by supplementing their Bayesian posterior probabilities with confidence levels as posterior probabilities. This use of coherent fiducial inference with hierarchical models generates set estimators that propagate uncertainty to varying degrees. Some of the set estimates approach estimates from plug-in empirical Bayes methods for high numbers of comparisons and can come close to the usual confidence sets given a sufficiently low number of comparisons.

In Bayesian model selection or testing problems, default priors are typically improper; that is, the resulting Bayes factor is not well defined. To circumvent this problem, two methodologies, namely, intrinsic and fractional Bayes factors are proposed and developed. Further, these two Bayes factors are asymptotically equivalent to the ordinary Bayes factors computed with proper priors called intrinsic priors. However, it seems that there are some necessary conditions to satisfy asymptotic equivalence. Such conditions are derived and justified in this article and illustrative examples are provided. Simulations are performed to demonstrate the results.

Much of science is (rightly or wrongly) driven by hypothesis testing. Even in situations where the hypothesis testing paradigm is correct, the common practice of basing inferences solely on p-values has been under intense criticism for over 50 years. We propose, as an alternative, the use of the odds of a correct rejection of the null hypothesis to incorrect rejection. Both pre-experimental versions (involving the power and Type I error) and post-experimental versions (depending on the actual data) are considered. Implementations are provided that range from depending only on the p-value to consideration of full Bayesian analysis. A surprise is that all implementations -- even the full Bayesian analysis -- have complete frequentist justification. Versions of our proposal can be implemented that require only minor modifications to existing practices yet overcome some of their most severe shortcomings., Comment: 62-00, 62-01, 62A01

Informative hypotheses are increasingly being used in psychological sciences because they adequately capture researchers' theories and expectations. In the Bayesian framework, the evaluation of informative hypotheses often makes use of default Bayes factors such as the fractional Bayes factor. This paper approximates and adjusts the fractional Bayes factor such that it can be used to evaluate informative hypotheses in general statistical models. In the fractional Bayes factor a fraction parameter must be specified which controls the amount of information in the data used for specifying an implicit prior. The remaining fraction is used for testing the informative hypotheses. We discuss different choices of this parameter and present a scheme for setting it. Furthermore, a software package is described which computes the approximated adjusted fractional Bayes factor. Using this software package, psychological researchers can evaluate informative hypotheses by means of Bayes factors in an easy manner. Two empirical examples are used to illustrate the procedure.

This paper investigates the classical type I and type II error probabilities of default Bayes factors for a Bayesian t test. Default Bayes factors quantify the relative evidence between the null hypothesis and the unrestricted alternative hypothesis without needing to specify prior distributions for the unknown parameters based on one's prior beliefs. It is shown that in most typical situations in psychological research (when either observing no, small, medium or large effects) default Bayes factors are asymmetric in information, i.e., they result in unequal error probabilities. The tendency to either prefer the null hypothesis or the alternative hypothesis varies for different default Bayes factors. Although this asymmetry in information is a natural property of a Bayes factor, severe cases of asymmetry may be undesirable in a default setting because the underlying default priors are not a translation of one's prior beliefs. A calibration scheme is presented to make a default Bayes factor symmetric in information under certain conditions. (C) 2015 Elsevier Inc. All rights reserved.

Bayes theorem is the normative method for revising probability forecasts using new information. However, for unaided forecasters its application can be difficult, effortful, opaque and even counter-intuitive. The study proposes two simple heuristics for approximating Bayes formula while yielding accurate decisions. Their performance was assessed where a decision is made on which of two events is most probable and where a choice is made between an option yielding an intermediate utility for something that is certain or for a gamble which will result in either a worse or better utility (“certainty or risk” decisions). For “most probable event” decisions the first heuristic always results in the correct decision when the reliability of the new information does not depend on which event will occur. In other cases, the second heuristic typically led to the correct decision for about 95% of “most probable event” decisions and 86% of “certainty or risk” decisions.

A large literature suggests that many individuals do not apply Bayes’ Rule when making decisions that depend on them correctly pooling prior information and sample data. We replicate and extend a classic experimental study of Bayesian updating from psychology, employing the methods of experimental economics, with careful controls for the confounding effects of risk aversion. Our results show that risk aversion significantly alters inferences on deviations from Bayes’ Rule.

How much more likely is a positive test to be found in a patient with a pathological condition than in a patient without it? How much more likely is a negative test to be found in a patient without...

The main objective of this paper is to find a close link between the adaptive level of significance, presented here, and the sample size. We, statisticians, know of the inconsistency, or paradox, in the current classical tests of significance that are based on p-value statistics that is compared to the canonical significance levels (10%, 5% and 1%): "Raise the sample to reject the null hypothesis" is the recommendation of some ill-advised scientists! This paper will show that it is possible to eliminate this problem of significance tests. The Bayesian Lindley's paradox – "increase the sample to accept the hypothesis" – also disappears. Obviously, we present here only the beginning of a possible prominent research. The intention is to extend its use to more complex applications such as survival analysis, reliability tests and other areas. The main tools used here are the Bayes Factor and the extended Neyman-Pearson Lemma.

To apply Bayes Rule when forming expectations agents must correctly assess the inherent “process variance” of each relevant cue, and place a larger weight on those cues with a smaller process variance. We test this notion by analysing subjective probabilities inferred from odds on the outcomes of tennis matches, exploiting natural variation in process variance related to the format with which tennis matches are played. Our results are consistent with “process variance neglect”, i.e., agents are not adjusting their ex-ante probabilities sufficiently according to process variance, and thus violate Bayes Rule. The resulting biases in expectations are costly.

One of the main challenges facing potential users of Bayes factors as an inferential technique is the difficulty of computing them. We highlight a useful relationship that allows certain order-restricted and sign-restricted Bayes factors, such as one-sided Bayes factor tests, to be computed with ease.

A new method is proposed for testing multiple hypotheses with equality and inequality constraints on the parameters of interest. The method is based on the fractional Bayes factor with a modification that the updated prior is centered on the boundary of the constrained parameter space under investigation. The resulting prior adjusted default Bayes factors work as an “Ockham’s razor” when testing inequality constrained hypotheses, which is not the case for the fractional Bayes factor. Two different types of prior adjusted default Bayes factors are considered. In the first type, the updated prior is based on imaginary training data. Analytical and numerical examples show that this criterion converges fastest to a true inequality constrained hypothesis. In the second type, the updated prior is based on empirical training data. This second criterion only outperforms the fractional Bayes factor in the case of small samples.Keywords: Fractional Bayes factor, Ockham’s razor, (In)equality constraints, Updated prior

Several issues are discussed when testing inequality constrained hypotheses using a Bayesian approach. First, the complexity (or size) of the inequality constrained parameter spaces can be ignored. This is the case when using the posterior probability that the inequality constraints of a hypothesis hold, Bayes factors based on non-informative improper priors, and partial Bayes factors based on posterior priors. Second, the Bayes factor may not be invariant for linear one-to-one transformations of the data. This can be observed when using balanced priors which are centred on the boundary of the constrained parameter space with a diagonal covariance structure. Third, the information paradox can be observed. When testing inequality constrained hypotheses, the information paradox occurs when the Bayes factor of an inequality constrained hypothesis against its complement converges to a constant as the evidence for the first hypothesis accumulates while keeping the sample size fixed. This paradox occurs when using Zellner's g prior as a result of too much prior shrinkage. Therefore, two new methods are proposed that avoid these issues. First, partial Bayes factors are proposed based on transformed minimal training samples. These training samples result in posterior priors that are centred on the boundary of the constrained parameter space with the same covariance structure as in the sample. Second, a g prior approach is proposed by letting g go to infinity. This is possible because the Jeffreys–Lindley paradox is not an issue when testing inequality constrained hypotheses. A simulation study indicated that the Bayes factor based on this g prior approach converges fastest to the true inequality constrained hypothesis.

We use Bayes’ estimator of a consumer-surplus probit model to study the relevance of the prior in a discrete choice model. We take random subsamples of varying sizes of stated preference data regarding ultra-low emission vehicle purchases in California and focus on the willingness-to-pay for improvements in driving range. Prior information is obtained from a meta-analysis of consumer valuation of driving range. We find the posterior distribution of the willingness-to-pay using a tight and a weakly informative prior, and also analyze the nonparametric estimates of the posterior compare these with the likelihood function of the problem. It is found that the weight of the prior is relevant for very small samples, but for standard sample sizes the prior vanishes. Thus, the Bayes estimator of a static discrete choice model is in general equivalent to the maximum likelihood estimator, although for some intermediate sample sizes the prior provides more realistic values.

In this engaging article, Zhang and Zhang (2013) (henceforth ZZ) assembled examples from clinical trials to make a compelling case for medical science's need to measure the evidence for composite h...

We consider the use of default priors in the Bayes methodology for seeking information concerning the true value of a parameter. By default prior, we mean the mathematical prior as initiated by Bayes [Philos. Trans. R. Soc. Lond. 53 (1763) 370–418] and pursued by Laplace [Theorie Analytique des Probabilites (1812) Courcier], Jeffreys [Theory of Probability (1961) Clarendon Press], Bernardo [J. Roy. Statist. Soc. Ser. B 41 (1979) 113–147] and many more, and then recently viewed as “potentially dangerous” [Science 340 (2013) 1177–1178] and “potentially useful” [Science 341 (2013) 1452]. We do not mean, however, the genuine prior [Science 340 (2013) 1177–1178] that has an empirical reference and would invoke standard frequency modelling. And we do not mean the subjective or opinion prior that an individual might have and would be viewed as specific to that individual. A mathematical prior has no referenced frequency information, but on occasion is known otherwise to lead to repetition properties called confidence. We investigate the presence of such supportive property, and ask can Bayes give reliability for other than the particular parameter weightings chosen for the conditional calculation. Thus, does the methodology have reproducibility? Or is it a leap of faith. For sample-space analysis, recent higher-order likelihood methods with regular models show that third-order accuracy is widely available using profile contours [In Past, Present and Future of Statistical Science (2014) 237– 252 CRC Press]. But for parameter-space analysis, accuracy is widely limited to first order. An exception arises with a scalar full parameter and the use of the scalar Jeffreys [J. Roy. Statist. Soc. Ser. B 25 (1963) 318–329]. But for vector full parameter even with a scalar interest parameter, difficulties have long been known [J. Roy. Statist. Soc. Ser. B 35 (1973) 189–233] and with parameter curvature, accuracy beyond first order can be unavailable [Statist. Sci. 26 (2011) 299–316].We show, however, that calculations on the parameter space can give full second-order information for a chosen scalar interest parameter; these calculations, however, require a Jeffreys prior that is used fully restricted to the one-dimensional profile for that interest parameter. Such a prior is effectively data-dependent and parameter-dependent and is focally restricted to the one-dimensional contour; these priors fall outside the usual Bayes approach and yet with substantial calculations can still give less than frequency analysis. We provide simple examples using discrete extensions of Jeffreys prior. These serve as counter-examples to general claims that Bayes can offer accuracy for statistical inference. To obtain this accuracy with Bayes, more effort is required compared to recent likelihood methods, which still remain more accurate. And with vector full parameters, accuracy beyond first order is routinely not available, as a change in parameter curvature causes Bayes and frequentist values to change in opposite direction, yet frequentist has full reproducibility. An alternative is to view default Bayes as an exploratory technique and then ask does it do as it overtly claims? Is it reproducible as understood in contemporary science? The posterior gives a distribution for an interest parameter and, thereby, a quantile for the interest parameter; an oracle could record whether it was left or right of the true value. If the average split in evaluative repetitions is in accord with the nominal level, then the approach is providing accuracy. And if not, then what is up, other than performance specific to the parameter frequencies in the prior. No one has answers although speculative claims abound.

Bayes factors provide a symmetrical measure of evidence for one model versus another (e.g. H1 versus H0) in order to relate theory to data. These properties help solve some (but not all) of the problems underlying the credibility crisis in psychology. The symmetry of the measure of evidence means that there can be evidence for H0 just as much as for H1; or the Bayes factor may indicate insufficient evidence either way. PP-values cannot make this three-way distinction. Thus, Bayes factors indicate when the data count against a theory (and when they count for nothing); and thus they indicate when replications actually support H0 or H1 (in ways that power cannot). There is every reason to publish evidence supporting the null as going against it, because the evidence can be measured to be just as strong either way (thus the published record can be more balanced). Bayes factors can be BB-hacked but they mitigate the problem because a) they allow evidence in either direction so people will be less tempted to hack in just one direction; b) as a measure of evidence they are insensitive to the stopping rule; c) families of tests cannot be arbitrarily defined; and d) falsely implying a contrast is planned rather than post hoc becomes irrelevant (though the value of pre-registration is not mitigated).

Despite its normative appeal and widespread use, Bayes’ rule has two well-known limitations: first, it does not predict how agents should react to an information to which they assigned probability zero; second, a sizable empirical evidence documents how agents systematically deviate from its prescriptions by overreacting to information to which they assigned a positive but small probability. By replacing Dynamic Consistency with a novel axiom, Dynamic Coherence, we characterize an alternative updating rule that is not subject to these limitations, but at the same time coincides with Bayes’ rule for “normal” events. In particular, we model an agent with a utility function over consequences, a prior over priors ρ, and a threshold. In the first period she chooses the prior that maximizes the prior over priors ρ--a’ la maximum likelihood. As new information is revealed: if the chosen prior assigns to this information a probability above the threshold, she follows Bayes’ rule and updates it. Otherwise, she goes back to her prior over priors ρ, updates it using Bayes’ rule, and then chooses the new prior that maximizes the updated ρ. We also extend our analysis to the case of ambiguity aversion.

We consider an Empirical Bayes method to correct for the Winner's Curse phenomenon in genome-wide association studies. Our method utilizes the collective distribution of all odds ratios (ORs) to determine the appropriate correction for a particular single-nucleotide polymorphism (SNP). We can show that this approach is squared error optimal provided that this collective distribution is accurately estimated in its tails. To improve the performance when correcting the OR estimates for the most highly associated SNPs, we develop a second estimator that adaptively combines the Empirical Bayes estimator with a previously considered Conditional Likelihood estimator. The applications of these methods to both simulated and real data suggest improved performance in reducing selection bias.

In this article, multiple forecasts given as probabilities of events are aggregated using two assumptions: calibration and conditional independence. The forecasts are treated as data and the aggregation is based on Bayes's theorem. A measure of discrimination is given and the behavior of the aggregated posterior probability is examined as the number of forecasters grows without bound. The work is motivated by recent research efforts employing large numbers of individual forecasts.

An objective Bayesian model selection procedure is proposed for the one way analysis of variance under homoscedasticity. Bayes factors for the usual default prior distributions are not well defined and thus Bayes factors for intrinsic priors are used instead. The intrinsic priors depend on a training sample which is typically a unique random vector. However, for the homoscedastic ANOVA it is not the case. Nevertheless, we are able to illustrate that the Bayes factors for the intrinsic priors are not sensitive to the minimal training sample chosen; furthermore, we propose an alternative pooled prior that yields similar Bayes factors. To compute these Bayes factors Bayesian computing methods are required when the sample sizes of the involved populations are large. Finally, a one to one relationship—which we call the calibration curve—between the posterior probability of the null hypothesis and the classical $$p$$ value is found, thus allowing comparisons between these two measures of evidence. The behavior of the calibration curve as a function of the sample size is studied and conclusions relating both procedures are stated.

We study a Bayes factor based on a robust test statistic, MAX3, for case-control genetic association studies. MAX3 is the maximum of three trend tests derived under the recessive, additive and dominant genetic models, respectively. The proposed Bayes factor, denoted as BFM, models the asymptotic distributions of MAX3 under the null and alternative hypotheses. It is compared to an existing Bayes factor based on Bayesian model averaging (BMA). Through simulation studies, we show that both BFM and BMA are robust under genetic model uncertainty. They both depend on specifying the prior distribution for the genetic model, and have similar performance when common objective priors are used. When the prior places a large probability on the true (wrong) genetic model, BMA (BFM) is more powerful. Applications to real data from a genome-wide association study are presented to illustrate their use and show their sensitivity under genetic model uncertainty.

In retrospective studies, odds ratio is often used as the measure of association. Under independent beta prior assumption, the exact posterior distribution of odds ratio given a single 2 × 2 table has been derived in the literature. However, independence between risks within the same study may be an oversimplified assumption because cases and controls in the same study are likely to share some common factors and thus to be correlated. Furthermore, in a meta-analysis of case–control studies, investigators usually have multiple 2 × 2 tables. In this article, we first extend the published results on a single 2 × 2 table to allow within study prior correlation while retaining the advantage of closed-form posterior formula, and then extend the results to multiple 2 × 2 tables and regression setting. The hyperparameters, including within study correlation, are estimated via an empirical Bayes approach. The overall odds ratio and the exact posterior distribution of the study-specific odds ratio are inferred based on the estimated hyperparameters. We conduct simulation studies to verify our exact posterior distribution formulas and investigate the finite sample properties of the inference for the overall odds ratio. The results are illustrated through a twin study for genetic heritability and a meta-analysis for the association between the N-acetyltransferase 2 (NAT2) acetylation status and colorectal cancer.

This article deals with robust Bayes estimation, namely Stable action, Conditional Gamma-Minimax action and Posterior Regret Gamma-Minimax action. We extend the results of Meczarski (1993) and obtain the exact actions in each case. The results are applied for a family of one-parameter exponential families introduced by Morris (1983) as an extension of the results of Boratynska (1997).

Current status data are a type of interval-censored event time data in which all the individuals are either left or right censored. For example, our motivation is drawn from a cross-sectional study, which measured whether or not fibroid onset had occurred by the age of an ultrasound exam for each woman. We propose a semiparametric Bayesian proportional odds model in which the baseline event time distribution is estimated nonparametrically by using adaptive monotone splines in a logistic regression model and the potential risk factors are included in the parametric part of the mean structure. The proposed approach has the advantage of being straightforward to implement using a simple and efficient Gibbs sampler, whereas alternative semiparametric Bayes’ event time models encounter problems for current status data. The model is generalized to allow systematic underreporting in a subset of the data, and the methods are applied to an epidemiologic study of uterine fibroids.

This paper presents a complete, choice-based, axiomatic Bayesian decision theory. It introduces a new choice set consisting of information-contingent plans for choosing actions and bets and subjective expected utility model with effect-dependent utility functions and action-dependent subjective probabilities which, in conjunction with the updating of the probabilities using Bayes’ rule, gives rise to a unique prior and a set of action-dependent posterior probabilities representing the decision maker’s prior and posterior beliefs.

The paper develops multivariate limited translation empirical Bayes estimators of the normal mean vector which serve as a compromise between the empirical Bayes and the maximum likelihood estimators. These compromise estimators perform better than the regular empirical Bayes estimators, in a frequentist sense, when there is wide departure of an individual observation from the grand average.

In this article, we consider Bayes prediction in a finite population under the simple location error-in-variables superpopulation model. Bayes predictor of the finite population mean under Zellner's balanced loss function and the corresponding relative losses and relative savings loss are derived. The prior distribution of the unknown location parameter of the model is assumed to have a non-normal distribution belonging to the class of Edgeworth series distributions. Effects of non normality of the “true” prior distribution and that of a possible misspecification of the loss function on the Bayes predictor are illustrated for a hypothetical population.

We discuss three learning rules for generalized Bayesian updating of an imprecise probability: (a modified version of) the generalized Bayes’ rule, the maximum likelihood update rule (after Gilboa and Schmeidler) and a newly developed hybrid rule. We investigate the general methodology for a special class of multivariate probability measures with prescribed marginals but arbitrary correlation structure. Both the choice and analysis of this class are motivated by expert interviews that we conducted with modelers in the field of climatic change.We argue that both updating rules from the literature have strong limitations, the generalized Bayes’ rule is too conservative, i.e., too inclusive, while the maximum likelihood update rule being too exclusive, adding spurious information. As a powerful extension we introduce a new rule for Bayesian updating of an imprecise measure: a “weighted likelihood update method,” which bases Bayesian updating on the whole set of priors but weights the influence of its members. We study the different rules in the case of bivariate Gaussian priors. Our investigation shows that the new rule combines certain attractive features of the generalized Bayes’ rule and the maximum likelihood update rule. In this article we aim at highlighting the sequence of not yet fully resolved statistical issues a practitioner on complex mechanistic models would face when updating imprecise prior knowledge.

This article examines the consistency, interpretation and application of Bayes factors constructed from standard test statistics. Primary conclusions are that Bayes factors based on multinomial and normal test statistics are consistent for suitable choices of the hyperparameters used to specify alternative hypotheses, and that such constructions can be extended to obtain consistent Bayes factors based on likelihood ratio statistics. A connection between Bayes factors based on likelihood ratio statistics and the Bayesian information criterion is exposed, as is a connection between Bayes factors based on F statistics and parametric Bayes factors based on normal-inverse gamma models. Similarly, Bayes factors based on chi-squared statistics for multinomial data are shown to provide accurate approximations to Bayes factors based on multinomial/Dirichlet models. An illustration of how the simple form of these Bayes factors can be exploited to generate easily interpretable summaries of the experimental ‘weight of evidence’ is provided.

The discussion following Bem's ( 2011 ) psi research highlights that applications of the Bayes factor in psychological research are not without problems. The first problem is the omission to translate subjective prior knowledge into subjective prior distributions. In the words of Savage ( 1961 ): "they make the Bayesian omelet without breaking the Bayesian egg." The second problem occurs if the Bayesian egg is not broken: the omission to choose default prior distributions such that the ensuing inferences are well calibrated. The third problem is the adherence to inadequate rules for the interpretation of the size of the Bayes factor. The current paper will elaborate these problems and show how to avoid them using the basic hypotheses and statistical model used in the first experiment described in Bem ( 2011 ). It will be argued that a thorough investigation of these problems in the context of more encompassing hypotheses and statistical models is called for if Bayesian psychologists want to add a well-founded Bayes factor to the tool kit of psychological researchers.

This paper is concerned with a Bayes prediction problem in the exponential distribution under random censorship. Using censored samples, we work out a prediction interval for a sum of interest which consists of some future samples. Differing from the general Bayes approach, we do not specify the prior distribution of the parameter, and only a first moment condition on the prior is assumed. Simulation studies are conducted to exhibit the coverage probabilities of the prediction interval.

This article addresses the problem of heterogeneity among various studies to be combined in a meta-analysis. We adopt quasi-empirical Bayes methodology to predict the odds ratios for each study. As a result, the predicted odds ratios are pulled toward the estimated common odds ratio of the various studies under consideration. With strong heterogeneity among the studies, we jointly consider the display of the 95% CIs of the ORs and a Dixon's test (1950) for “outliers” to exclude the “extreme” estimated ORs. We demonstrate the effectiveness of our methodology based on the data analyzed by Thompson and Pocock (1987) demonstrating the power of the new approach to meta-analysis to find statistical agreement in what looks like great disagreement via a chi-squared test. We believe our technique (i.e., minimum mean-square sense) will go a long way toward increasing the trustworthiness of meta-analysis.

A variety of pseudo-Bayes factors have been proposed, based on using part of the data to update an improper prior, and using the remainder of the data to compute the Bayes factor. A number of these approaches are of a bootstrap or cross-validation nature, with some type of average being taken over the data used for updating. Asymptotic characteristics of a number of these pseudo-Bayes factors are discussed, and it is shown how many behave quite differently from ordinary Bayes factors. It is also shown that arguments of predictive optimality, based on simply inserting the empirical distribution in place of the 'true predictive distribution', can be misleading; the particular example of this that is studied is the argument given in Bernardo and Smith [1994. Bayesian Theory. Wiley, Chichester] to the effect that the geometric intrinsic Bayes factor has an optimal predictive property.

In this paper, we highlight properties of Bayesian models in which the prior puts positive mass on all Kullback–Leibler neighborhoods of all densities. These properties are concerned with model choice via the Bayes factor, density estimation and the maximization of expected utility for decision problems. In four illustrations we focus on the Bayes factor and show that whatever models are being compared, the [log(Bayes factor)]/[sample size] converges to a non-random number which has a nice interpretation. A parametric versus semiparametric model comparison provides a fifth illustration.

When testing a point null hypothesis versus an alternative that is vaguely specified, a Bayesian test usually proceeds by putting a non-parametric prior on the alternative and then computing a Bayes factor based on the observations. This paper addresses the question of consistency, that is, whether the Bayes factor is correctly indicative of the null or the alternative as the sample size increases. We establish several consistency results in the affirmative under fairly general conditions. Consistency of Bayes factors for testing a point null versus a parametric alternative has long been known. The results here can also be viewed as the non-parametric extension of the parametric counterpart.

The proportional odds model provides a powerful tool for analysing ordered categorical data and setting sample size, although for many clinical trials its validity is questionable. The purpose of this paper is to present a new class of constrained odds models which includes the proportional odds model. The efficient score and Fisher's information are derived from the profile likelihood for the constrained odds model. These results are new even for the special case of proportional odds where the resulting statistics define the Mann-Whitney test. A strategy is described involving selecting one of these models in advance, requiring assumptions as strong as those underlying proportional odds, but allowing a choice of such models. The accuracy of the new procedure and its power are evaluated.

In the last few years, there has been an increasing interest for default Bayes methods for hypothesis testing and model selection. The availability of such methods is potentially very useful in mixture models, where the elicitation process on the (unknown number of) parameters is usually rather difficult. Two recent yet already popular approaches, namely intrinsic Bayes factor (J. Amer. Statist. Assoc. 91 (1996) 109) and fractional Bayes factor (J. Roy. Statist. Soc. Ser. B 57 (1995) 99), have been proven quite successful in generating sensible prior distributions, to compute actual Bayes factors. From a theoretical viewpoint, the application of these methods to a mixture model selection problem involves two difficulties. The first is the choice of a “good” default prior for the mixture. The second problem is related to the fact that, for improper default priors, the prior predictive distribution of the data need not exist; In this paper, we argue that the problem of choosing among mixture models can be reduced to the problem of comparing models with simpler structures. It is shown that these simpler models can be compared via standard default Bayesian methods.

From the orthodox statistical analysis of probabilities of data under null hypothesis we move into the more rarefied spheres of probabilities of meaningful scientific hypotheses. Bayes theorem is explained from the ground up and tied to basic axioms of probability theory. Scientifically useful versions of the Bayes’ theorem (both for hypothesis testing and for parameter estimation) are given and explained with a view to their uses and limits. Bayesian statistics is compared with frequentist statistics, likelihoodism, and intuitive approaches to data analysis. Bayesianism as a philosophy of science is contrasted with Bayesianism as a probability logic. Bayes’ theorem is used to demonstrate the limits that are put on scientific interpretability by the relative probabilities of experimental bias and truth of the scientific hypothesis of interest, and it is used to illustrate “why most published research findings are false.”

It has become increasingly important to understand the asymptotic behavior of the Bayes factor for model selection in general statistical models. In this paper, we discuss recent results on Bayes factor consistency in semiparametric regression problems where observations are independent but not identically distributed. Specifically, we deal with the model selection problem in the context of partial linear models in which the regression function is assumed to be the additive form of the parametric component and the nonparametric component using Gaussian process priors, and Bayes factor consistency is investigated for choosing between the parametric model and the semiparametric alternative.

There have been considerable amounts of work regarding the development of various default Bayes factors in model selection and hypothesis testing. Two commonly used criteria, the intrinsic Bayes factor and the fractional Bayes factor are compared to test two independent normal means and variances. We also derive several intrinsic priors whose Bayes factors are asymptotically equivalent to the respective Bayes factors. We demonstrate our results in simulated datasets.

Γ-minimax theory deals with the problem of selecting decision rules under partial prior information on the distribution of the unknown parameter. The selected rule is the one minimizing the supremum of the payoff (Bayes and regret risk in a frequentist approach, and posterior expected loss and regret in a Bayesian one) over the priors in Γ. Keywords: minimax principle; Bayes principle; classes of priors