On the choice of the prior distribution in hypergeometric sampling

Information in a statistical procedure arising from sources other than sampling is called prior information, and its incorporation into the procedure forms the basis of the Bayesian approach to statistics. Under hypergeometric sampling, methodology is developed which quantifies the amount of information provided by the sample data relative to that provided by the prior distribution and allows for a ranking of prior distributions with respect to conservativeness, where conservatism refers to restraint of extraneous information embedded in any prior distribution. The most conservative prior distribution from a specified class (each member of which carries the available prior information) is that prior distribution within the class over which the likelihood function has the greatest average domination. Four different families of prior distributions are developed by considering a Bayesian approach to the formation of lots. The most conservative prior distribution from each of the four families of prior distributions is determined and compared for the situation when no prior information is available. The results of the comparison advocate the use of the Polya (beta-binomial) prior distribution in hypergeometric sampling.     

Bayesian Transformation Models for Multivariate Survival Data

In this paper, we propose a general class of Gamma frailty transformation models for multivariate survival data. The transformation class includes the commonly used proportional hazards and proportional odds models. The proposed class also includes a family of cure rate models. Under an improper prior for the parameters, we establish propriety of the posterior distribution. A novel Gibbs sampling algorithm is developed for sampling from the observed data posterior distribution. A simulation study is conducted to examine the properties of the proposed methodology. An application to a data set from a cord blood transplantation study is also reported.     

Non parametric Bayesian analysis of the two-sample problem with censoring

Testing for differences between two groups is a fundamental problem in statistics, and due to developments in Bayesian non parametrics and semiparametrics there has been renewed interest in approaches to this problem. Here we describe a new approach to developing such tests and introduce a class of such tests that take advantage of developments in Bayesian non parametric computing. This class of tests uses the connection between the Dirichlet process (DP) prior and the Wilcoxon rank sum test but extends this idea to the DP mixture prior. Here tests are developed that have appropriate frequentist sampling procedures for large samples but have the potential to outperform the usual frequentist tests. Extensions to interval and right censoring are considered and an application to a high-dimensional data set obtained from an RNA-Seq investigation demonstrates the practical utility of the method.     

Estimation of a poisson mean using discrete mixtures of conjugate priors

The problem of estimating a Poisson mean is considered using incomplete prior information. The user is only able to assess two fractiles of the prior distribution. A class of mixture distributions is constructed to model this prior information; variation within this class primarily occurs in the tail region where little prior information exists. The posterior analysis using the mixture class is attractive computationally and compares favorably with the conjugate posterior analysis.     

Bayesian estimation of incomplete data using conditionally specified priors

In this article, a class of conjugate prior for estimating incomplete count data based on a broad class of conjugate prior distributions is presented. The new class of prior distributions arises from a conditional perspective, making use of the conditional specification methodology and can be considered as the generalization of the form of prior distributions that have been used previously in the estimation of incomplete count data well. Finally, some examples of simulated and real data are given.     

Priors and component structures in autoregressive time series models

New approaches to prior specification and structuring in autoregressive time series models are introduced and developed. We focus on defining classes of prior distributions for parameters and latent variables related to latent components of an autoregressive model for an observed time series. These new priors naturally permit the incorporation of both qualitative and quantitative prior information about the number and relative importance of physically meaningful components that represent low frequency trends, quasi-periodic subprocesses and high frequency residual noise components of observed series. The class of priors also naturally incorporates uncertainty about model order and hence leads in posterior analysis to model order assessment and resulting posterior and predictive inferences that incorporate full uncertainties about model order as well as model parameters. Analysis also formally incorporates uncertainty and leads to inferences about unknown initial values of the time series, as it does for predictions of future values. Posterior analysis involves easily implemented iterative simulation methods, developed and described here. One motivating field of application is climatology, where the evaluation of latent structure, especially quasi-periodic structure, is of critical importance in connection with issues of global climatic variability. We explore the analysis of data from the southern oscillation index, one of several series that has been central in recent high profile debates in the atmospheric sciences about recent apparent trends in climatic indicators.     

An information-theoretic approach to incorporating prior information in binomial sampling

The incorporation of prior information about θ, where θ is the success probability in a binomial sampling model, is an essential feature of Bayesian statistics. Methodology based on information-theoretic concepts is introduced which (a) quantifies the amount of information provided by the sample data relative to that provided by the prior distribution and (b) allows for a ranking of prior distributions with respect to conservativeness, where conservatism refers to restraint of extraneous information about θ which is embedded in any prior distribution. In effect, the most conservative prior distribution from a specified class (each member o f which carries the available prior information about θ) is that prior distribution within the class over which the likelihood function has the greatest average domination. The most conservative prior distributions from five different families of prior distributions over the interval (0,1) including the beta distribution are determined and compared for three situations: (1) no prior estimate of θ is available, (2) a prior point estimate or θ is available, and (3) a prior interval estimate of θ is available. The results of the comparisons not only advocate the use of the beta prior distribution in binomial sampling but also indicate which particular one to use in the three aforementioned situations.     

On the Bayesian analysis of categorical data: the problem of nonresponse

It is demonstrated how a suitably chosen prior for the frequency parameters can streamline the Bayesian analysis of categorical data with missing entries due to nonresponse or other causes. The two cases where the data follow the Multinomial or the Hypergeometric model are treated separately. In the first case it is adequate to restrict the prior (for the cell probabilities) to the class of Dirichlet distributions. In the case of the Hypergeometric model it is convenient to select a prior from the class of Dirichlet-Multinomial (DM) distributions. The DM distributions are studied in some details.     

Infinitesimal sensitivity of posterior distributions

We measure the local sensitivity of a posterior expectation with respect to the prior by computing the norm of the Fréchet derivative of the posterior with respect to the prior over several different classes of measures. We compute the derivative of the posterior upper expectation when the prior varies in a restricted ?-contamination class. A bound on the global sensitivity of a class of priors is obtained. As an application, we show that of all sets with posterior probability 1 — α, the likelihood region minimizes the norm of the Fréchet derivative over the ?-contamination class and so is, in some sense, the most robust region with this posterior probability. But there exist counterexamples to this result for other classes of priors.     

Frequentist and Bayesian Interval Estimators for the Normal Mean

It is well known that a Bayesian credible interval for a parameter of interest is derived from a prior distribution that appropriately describes the prior information. However, it is less well known that there exists a frequentist approach developed by Pratt (1961 Pratt , J. W. ( 1961 ). Length of confidence intervals . J. Amer. Statist. Assoc. 56 : 549 – 657 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) that also utilizes prior information in the construction of frequentist confidence intervals. This frequentist approach produces confidence intervals that have minimum weighted average expected length, averaged according to some weight function that appropriately describes the prior information. We begin with a simple model as a starting point in comparing these two distinct procedures in interval estimation. Consider X ₁,…, X _n that are independent and identically N(μ, σ²) distributed random variables, where σ² is known, and the parameter of interest is μ. Suppose also that previous experience with similar data sets and/or specific background and expert opinion suggest that μ = 0. Our aim is to: (a) develop two types of Bayesian 1 ? α credible intervals for μ, derived from an appropriate prior cumulative distribution function F(μ) more importantly; (b) compare these Bayesian 1 ? α credible intervals for μ to the frequentist 1 ? α confidence interval for μ derived from Pratt's frequentist approach, in which the weight function corresponds to the prior cumulative distribution function F(μ). We show that the endpoints of the Bayesian 1 ? α credible intervals for μ are very different to the endpoints of the frequentist 1 ? α confidence interval for μ, when the prior information strongly suggests that μ = 0 and the data supports the uncertain prior information about μ. In addition, we assess the performance of these intervals by analyzing their coverage probability properties and expected lengths.     

A procedure to select a ML-II prior in a multivariate normal case

Berger (1985) derived a procedure to select a maximum likelihood II prior distribution. In this paper a method is suggested to construct such a prior distribution from a multivariate ε-contamination class of distributions. The method is illustrated by the conetruction of a ML-II prior in the multivariate normal case.     

A note on coherent invariant distributions as non-informative priors for exponential and location-scale families

The problem of finding a non-informative prior distribution for a parameter is approached using the notion of context-invariance. This concept is revisited and discussed with the aim of applying it to finding context-invariant non-informative priors for the one-parameter exponential family (suitably redefined) and the location-scale family. Our approach, carried-out in a finitely-additive framework, generally leads to a class of non-informative priors with respect to any given problem. For most common statistical models such a class does not always contain the corresponding Jeffreys' prior, but does contain the so-called ALI prior by Hartigan.     

Robust bayesian analysis given a lower bound on the probability of a set

Suppose that just the lower bound of the probability of a measurable subset K in the parameter space Ω is a priori known, when inferences are to be made about measurable subsets A in Ω. Instead of eliciting a unique prior distribution, consider the class Г of all the distributions compatible with such bound. Under mild regularity conditions about the likelihood function, the range of the posterior probability of any A is found, as the prior distribution varies in Г. Such ranges are analysed according to the robust Bayesian viewpoint. Furthermore, some characterising properties of the extended likelihood sets are proved. The prior distributions in Г are then considered as a neighbour class of an elicited prior, comparing likelihood sets and HPD in terms of robustness.     

Robust bayesian analysis given bounds on the probability of a set

Suppose that just the lower and the upper bounds on the probability of a measurable subset K in the parameter space ω are a priori known. Instead of eliciting a unique prior probability measure, consider the class Γ of all the probability measures compatible with such bounds. Under mild regularity conditions about the likelihood function, both prior and posterior bounds on the expected value of any function of the unknown parameter ω are computed, as the prior measure varies in Γ. Such bounds are analysed according to the robust Bayesian viewpoint. Furthermore, lower and upper bounds on the Bayes factor are corisidered. Finally, the local sensitivity analysis is performed, considering the class Γ as a aeighbourhood of an elicited prior     

An empirical Bayes procedure for the selection of Gaussian graphical models

A new methodology for model determination in decomposable graphical Gaussian models (Dawid and Lauritzen in Ann. Stat. 21(3), 1272?C1317, 1993) is developed. The Bayesian paradigm is used and, for each given graph, a hyper-inverse Wishart prior distribution on the covariance matrix is considered. This prior distribution depends on hyper-parameters. It is well-known that the models??s posterior distribution is sensitive to the specification of these hyper-parameters and no completely satisfactory method is registered. In order to avoid this problem, we suggest adopting an empirical Bayes strategy, that is a strategy for which the values of the hyper-parameters are determined using the data. Typically, the hyper-parameters are fixed to their maximum likelihood estimations. In order to calculate these maximum likelihood estimations, we suggest a Markov chain Monte Carlo version of the Stochastic Approximation EM algorithm. Moreover, we introduce a new sampling scheme in the space of graphs that improves the add and delete proposal of Armstrong et al. (Stat. Comput. 19(3), 303?C316, 2009). We illustrate the efficiency of this new scheme on simulated and real datasets.     

Bayes and robust Bayes predictions in a subfamily of scale parameters under a precautionary loss function

ABSTRACT

This paper deals with Bayes, robust Bayes, and minimax predictions in a subfamily of scale parameters under an asymmetric precautionary loss function. In Bayesian statistical inference, the goal is to obtain optimal rules under a specified loss function and an explicit prior distribution over the parameter space. However, in practice, we are not able to specify the prior totally or when a problem must be solved by two statisticians, they may agree on the choice of the prior but not the values of the hyperparameters. A common approach to the prior uncertainty in Bayesian analysis is to choose a class of prior distributions and compute some functional quantity. This is known as Robust Bayesian analysis which provides a way to consider the prior knowledge in terms of a class of priors Γ for global prevention against bad choices of hyperparameters. Under a scale invariant precautionary loss function, we deal with robust Bayes predictions of Y based on X. We carried out a simulation study and a real data analysis to illustrate the practical utility of the prediction procedure.     

Bayesian analysis of correlated mixed categorical data by incorporating historical prior information

In this article, we develop statistical models for analysis of correlated mixed categorical (binary and ordinal) response data arising in medical and epidemi-ologic studies. There is evidence in the literature to suggest that models including correlation structure can lead to substantial improvement in precision of estimation or are more appropriate (accurate). We use a very rich class of scale mixture of multivariate normal (SMMVN) iink functions to accommodate heavy tailed distributions. In order to incorporate available historical information, we propose a unified prior elicitation scheme based on SMMVN-link models. Further, simulation-based techniques are developed to assess model adequacy. Finally, a real data example from prostate cancer studies is used to illustrate the proposed methodologies.     

Empirical Bayes approach to wavelet regression using ϵ-contaminated priors

We consider an empirical Bayes approach to standard nonparametric regression estimation using a nonlinear wavelet methodology. Instead of specifying a single prior distribution on the parameter space of wavelet coefficients, which is usually the case in the existing literature, we elicit the ?-contamination class of prior distributions that is particularly attractive to work with when one seeks robust priors in Bayesian analysis. The type II maximum likelihood approach to prior selection is used by maximizing the predictive distribution for the data in the wavelet domain over a suitable subclass of the ?-contamination class of prior distributions. For the prior selected, the posterior mean yields a thresholding procedure which depends on one free prior parameter and it is level- and amplitude-dependent, thus allowing better adaptation in function estimation. We consider an automatic choice of the free prior parameter, guided by considerations on an exact risk analysis and on the shape of the thresholding rule, enabling the resulting estimator to be fully automated in practice. We also compute pointwise Bayesian credible intervals for the resulting function estimate using a simulation-based approach. We use several simulated examples to illustrate the performance of the proposed empirical Bayes term-by-term wavelet scheme, and we make comparisons with other classical and empirical Bayes term-by-term wavelet schemes. As a practical illustration, we present an application to a real-life data set that was collected in an atomic force microscopy study.     

Bayesian Kernel Mixtures for Counts

Although Bayesian nonparametric mixture models for continuous data are well developed, there is a limited literature on related approaches for count data. A common strategy is to use a mixture of Poissons, which unfortunately is quite restrictive in not accounting for distributions having variance less than the mean. Other approaches include mixing multinomials, which requires finite support, and using a Dirichlet process prior with a Poisson base measure, which does not allow smooth deviations from the Poisson. As a broad class of alternative models, we propose to use nonparametric mixtures of rounded continuous kernels. An efficient Gibbs sampler is developed for posterior computation, and a simulation study is performed to assess performance. Focusing on the rounded Gaussian case, we generalize the modeling framework to account for multivariate count data, joint modeling with continuous and categorical variables, and other complications. The methods are illustrated through applications to a developmental toxicity study and marketing data. This article has supplementary material online.     

The estimation of exchangeable binomial parameters

A general class of Bayesian estimators for several Binomial parameters is proposed. The estimati on procedures are based on a logistic transformation of the parameters and a class of symmetric prior distributions.