The calibration of P-values,posterior Bayes factors and the AIC from the posterior distribution of the likelihood

The posterior distribution of the likelihood is used to interpret the evidential meaning of P-values, posterior Bayes factors and Akaike's information criterion when comparing point null hypotheses with composite alternatives. Asymptotic arguments lead to simple re-calibrations of these criteria in terms of posterior tail probabilities of the likelihood ratio. (Prior) Bayes factors cannot be calibrated in this way as they are model-specific.     

A bayesian alternative to the sign test in the presence of ties

Bayesian alternatives to the sign test are proposed which incorporate the number of ties observed. These alternatives arise from different strategies in dealing with the number of ties. One strategy is incorporating the true proportion of ties into the hypotheses of interest. The Bayesian methods are compared to each other and to the typical sign test in a simulation study. Also, the new methods are compared to another version of the sign test proposed by Coakley and Heise (1996). This new version of the sign test was shown to perform especially well in situations where the probability of observing a tie is very high. Although one of the Bayesian methods appears to perform best overall in the simulation study, its performance is not dominating and the easy to use typical sign test generally performs very well.     

Bayes factor in testing precise hypotheses

Substitution of a mixed prior distribution by a continuous one for the point null hypothesis testing problem is discussed. Conditions are established in order to approximate the Bayes factors for the two problems. Besides, trough this approximation an assignation of priorprobabilities is suggested.     

A Comparison of Bayes Factors for Separated Models: Some Simulation Results

Some alternative Bayes Factors: Intrinsic, Posterior, and Fractional have been proposed to overcome the difficulties presented when prior information is weak and improper prior are used. Additional difficulties also appear when the models are separated or non nested. This article presents both simulation results and some illustrative examples analysis comparing these alternative Bayes factors to discriminate among the Lognormal, the Weibull, the Gamma, and the Exponential distributions. Simulation results are obtained for different sample sizes generated from the distributions. Results from simulations indicates that these alternative Bayes factors are useful for comparing non nested models. The simulations also show some similar behavior and that when both models are true they choose the simplest model. Some illustrative example are also presented.     

Simpson's paradox and the Bayes factor

A counter-example presented by Lindley to Aitkin's posterior Bayes factor is examined. The paradoxical feature of the counter-example is found to be a simple case of Simpson's paradox.     

Bayes factors for goodness of fit testing

We propose the use of the generalized fractional Bayes factor for testing fit in multinomial models. This is a non-asymptotic method that can be used to quantify the evidence for or against a sub-model. We give expressions for the generalized fractional Bayes factor and we study its properties. In particular, we show that the generalized fractional Bayes factor has better properties than the fractional Bayes factor.     

Bayesian change-point problem using Bayes factor with hierarchical prior distribution

We consider the hierarchical Bayesian models of change-point problem in a sequence of random variables having either normal population or skew-normal population. Further, we consider the problem of detecting an influential point concerning change point using Bayes factors. Our proposed models are illustrated with the real data example, the annual flow volume data of Nile River at Aswan from 1871 to 1970. The result using our proposed models indicated the largest influential observation in the year 1888 among outliers. We have shown that it is useful to measure the influence of observations on Bayes factors. Here, we consider omitting single observation as well.     

Bayesian testing of an exponential point null hypothesis

The problem of testing a point null hypothesis involving an exponential mean is The problem of testing a point null hypothesis involving an exponential mean is usual interpretation of P-values as evidence against precise hypotheses is faulty. As in Berger and Delampady (1986) and Berger and Sellke (1987), lower bounds on Bayesian measures of evidence over wide classes of priors are found emphasizing the conflict between posterior probabilities and P-values. A hierarchical Bayes approach is also considered as an alternative to computing lower bounds and “automatic” Bayesian significance tests which further illustrates the point that P-values are highly misleading measures of evidence for tests of point null hypotheses.     

Robustness of the shrinkage estimator for the relative potency in the combination of multivariate bioassays

ABSTRACT

This article investigates the robustness of the shrinkage Bayesian estimator for the relative potency parameter in the combinations of multivariate bioassays proposed in Chen et al. (1999 Chen, D.G., Carter, E.M., Hubert, J.J., Kim, P.T. (1999). Empirical Bayesian estimation for combinations of multivariate bioassays. Biometrics 55(4):1035–1043. [Google Scholar]), which incorporated prior information on the model parameters based on Jeffreys’ rules. This investigation is carried out for the families of t-distribution and Cauchy-distribution based on the characteristics of bioassay theory since the t-distribution approaches the normal distribution which is the most commonly used distribution in the applications of bioassay as the degrees of freedom increases and the t-distribution approaches the Cauchy-distribution as the degrees of freedom approaches 1 which is also an important distribution in bioassay. A real data is used to illustrate the application of this investigation. This analysis further supports the application of the shrinkage Bayesian estimator to the theory of bioassay along with the empirical Bayesian estimator.     

Adaptive wavelet empirical Bayes estimation of a location or a scale parameter

Assume that in independent two-dimensional random vectors (X₁,θ₁),…,(X_n,θ_n), each θ_i is distributed according to some unknown prior density function g. Also, given θ_i=θ, X_i has the conditional density function q(x−θ), x,θ(−∞,∞) (a location parameter case), or θ⁻¹q(x/θ), x,θ(0,∞) (a scale parameter case). In each pair the first component is observable, but the second is not. After the (n+1)th pair (X_n+1,θ_n+1) is obtained, the objective is to construct an empirical Bayes (EB) estimator of θ. In this paper we derive the EB estimators of θ based on a wavelet approximation with Meyer-type wavelets. We show that these estimators provide adaptation not only in the case when g belongs to the Sobolev space H with an unknown , but also when g is supersmooth.     

A hierarchical Bayes analysis for one-shot device testing experiment under the assumption of exponentiality

The article considers a two-stage hierarchical Bayes technique to analyze a dataset coming from a “one-shot” device testing experiment. The development is based on the assumption of exponential model for the lifetimes with failure rate regressed according to the Cox proportional hazards model. The Bayes implementation is done through a Gibbs–Metropolis hybridization scheme that easily entertains the missing data cases as well. Lastly, numerical illustration is provided based on a real data example on electro-explosive devices. The results show that the Bayesian method performs considerably well for such type of experiments.     

The application of robust Bayesian analysis to hypothesis testing and Occam's Razor

Summary Robust Bayesian analysis deals simultaneously with a class of possible prior distributions, instead of a single distribution. This paper concentrates on the surprising results that can be obtained when applying the theory to problems of testing precise hypotheses when the “objective” class of prior distributions is assumed. First, an example is given demonstrating the serious inadequacy of P-values for this problem. Next, it is shown how the approach can provide statistical quantification of Occam's Razor, the famous principle of science that advocates choice of the simpler of two hypothetical explanations of data. Finally, the theory is applied to multinomial testing. Research supported by the National Science Foundation, Grant DMS-8923071, and by NASA Contract NAS5-29285 for the hubble Space Telescope.     

Bayes and Empirical Bayes Estimation of the Parameter n in a Binomial Distribution

The problem of estimating the total number of trials n in a binomial distribution is reconsidered in this article for both cases of known and unknown probability of success p from the Bayesian viewpoint. Bayes and empirical Bayes point estimates for n are proposed under the assumption of a left-truncated prior distribution for n and a beta prior distribution for p. Simulation studies are provided in this article in order to compare the proposed estimate with the most familiar n estimates.     

A Short Note on Almost Sure Convergence of Bayes Factors in the General Set-Up

ABSTRACT

Although there is a significant literature on the asymptotic theory of Bayes factor, the set-ups considered are usually specialized and often involves independent and identically distributed data. Even in such specialized cases, mostly weak consistency results are available. In this article, for the first time ever, we derive the almost sure convergence theory of Bayes factor in the general set-up that includes even dependent data and misspecified models. Somewhat surprisingly, the key to the proof of such a general theory is a simple application of a result of Shalizi to a well-known identity satisfied by the Bayes factor. Supplementary materials for this article are available online.     

Bayes estimators in the presence of a guess value

The paper deals with the problem of parameter estimation in the presence of a guess value and attempts to justify the use of Bayes estimators as an alternative to ordinary shrinkage estimators. Finally, certain Bayes estimators of exponential parameters are obtained under type II censoring, and these are compared with the corresponding MLEs and ordinary shrinkage estimators using a Monte Carlo study.     

Bayes and Empirical Bayes Inference in Changepoint Problems

ABSTRACT

In the case of the random design nonparametric regression, the double smoothing technique is applied to estimate the multivariate regression function. The proposed estimator has desirable properties in both the finite sample and the asymptotic cases. In the finite sample case, it has bounded conditional (and unconditional) bias and variance. On the other hand, in the asymptotic case, it has the same mean square error as the local linear estimator in Fan (Design-Adaptive Nonparametric Regression. Journal of the American Statistical Association 1992, 87, 998–1004; Local Linear Regression Smoothers and Their Minimax Efficiencies. Annals of Statistics 1993, 21, 196–216). Simulation studies demonstrate that the proposed estimator is better than the local linear estimator, because it has a smaller sample mean integrated square error and gives smoother estimates.     

A bayesian approach for testing periodicity of dichotomous observations

A Bayesian approach is utilized to test for periodicity in a dichotomous time series. Dichotomous data arise in a variety of circumstances when a variable takes on only two possible values. Conjugate and noninformative priors are considered as well as a hierarchical Bayes approach; the latter is considered the superior Bayes methodology. The situation of stochastic period lengths is also discussed. The generalization to the multinomial model is investigated to allow for the case that a variable takes on more than two possible values. In all cases decisions are made based on a Bayes factor. The proposed procedures are demonstrated on earthquake data in the central Virginia seismic zone     

Bayes and empirical bayes estimation of the probability that z > x + y

Let X, Y and Z be independent random variables with common unknown distribution F. Using the Dirichlet process prior for F and squared erro loss function, the Bayes and empirical Bayes estimators of the parameters λ(F). the probability that Z > X + Y, are derived. The limiting Bayes estimator of λ(F) under some conditions on the parameter of the process is shown to be asymptotically normal. The aysmptotic optimality of the empirical Bayes estimator of λ(F) is established. When X, Y and Z have support on the positive real line, these results are derived for randomly right censored data. This problem relates to testing whether than used discussed by Hollander and Proshcan (1972) and Chen, Hollander and Langberg (1983).     

Choosing the link function and accounting for link uncertainty in generalized linear models using Bayes factors

One important component of model selection using generalized linear models (GLM) is the choice of a link function. We propose using approximate Bayes factors to assess the improvement in fit over a GLM with canonical link when a parametric link family is used. The approximate Bayes factors are calculated using the Laplace approximations given in [32], together with a reference set of prior distributions. This methodology can be used to differentiate between different parametric link families, as well as allowing one to jointly select the link family and the independent variables. This involves comparing nonnested models and so standard significance tests cannot be used. The approach also accounts explicitly for uncertainty about the link function. The methods are illustrated using parametric link families studied in [12] for two data sets involving binomial responses. The first author was supported by Sonderforschungsbereich 386 Statistische Analyse Diskreter Strukturen, and the second author by NIH Grant 1R01CA094212-01 and ONR Grant N00014-01-10745.