Experience with a bayesian bootstrap method incorporating proper prior information

Although bootstrapping has become widely used in statistical analysis, there has been little reported concerning bootstrapped Bayesian analyses, especially when there is proper prior informa-tion concerning the parameter of interest. In this paper, we first propose an operationally implementable definition of a Bayesian bootstrap. Thereafter, in simulated studies of the estimation of means and variances, this Bayesian bootstrap is compared to various parametric procedures. It turns out that little information is lost in using the Bayesian bootstrap even when the sampling distribution is known. On the other hand, the parametric procedures are at times very sensitive to incorrectly specified sampling distributions, implying that the Bayesian bootstrap is a very robust procedure for determining the posterior distribution of the parameter.     

Pooling means under uncertain prior information with application to discrete distributions

The problem of pooling means is considered based on two samples in presence of the uncertain prior information that these samples are taken from possibly identical populations. Two discrete models, Poisson and binomial are considered in particular. Three estimators, i.e. the unrestricted estimator, shrinkage restricted estimator and estimators based on preliminary test are proposed. Their asymptotic mean squared errors are derived and compared. It is demonstrated via asymptotic results that the range of the parameter space in which shrinkage preliminary test estimator dominates the unrestricted estimator is wider than that of the usual preliminary test estimator. A Monte Carlo study for Poisson model is presented to compare the performance of the estimators for small samples.     

Eliciting prior information to enhance the predictive performance of bayesian graphical models

Both knowledge-based systems and statistical models are typically concerned with making predictions about future observables. Here we focus on assessment of predictive performance and provide two techniques for improving the predictive performance of Bayesian graphical models. First, we present Bayesian model averaging, a technique for accounting for model uncertainty.

Second, we describe a technique for eliciting a prior distribution for competing models from domain experts. We explore the predictive performance of both techniques in the context of a urological diagnostic problem.     

Value of the beta prior information

Information gain or loss is studied here by considering changes in the reciprocal of the expected posterior variance. For two beta distributions, the ratio of the expected values of their posterior variances provides a convenient criterion which is consistent with several results of the bayesian set-up and also permits the determination of the least informative prior beta distribution. Relations with results obtained by other authors are also discussed.     

Single-parameter inference based on partial prior information

Partial specification of a prior distribution can be appealing to an analyst, but there is no conventional way to update a partial prior. In this paper, we show how a framework for Bayesian updating with data can be based on the Dirichlet(a) process. Within this framework, partial information predictors generalize standard minimax predictors and have interesting multiple-point shrinkage properties. Approximations to partial-information estimators for squared error loss are defined straightforwardly, and an estimate of the mean shrinks the sample mean. The proposed updating of the partial prior is a consequence of four natural requirements when the Dirichlet parameter a is continuous. Namely, the updated partial posterior should be calculable from knowledge of only the data and partial prior, it should be faithful to the full posterior distribution, it should assign positive probability to every observed event {X,}, and it should not assign probability to unobserved events not included in the partial prior specification.     

A bayesian analysis of interrelated bernoulli processes with an application for clinical trials

Prior information regarding the interrelation of two Bernoulli processes may justify a clinical trial designed to corroborate this information. Antelman (1973) has studied the Dirichlet-beta which permits the expression of the prior knowledge of such interrelation. However, use of this prior distribution leads to complicated and intractable analyses. Alternately, such prior information regarding the interrelation of the processes may be adequately summarized by a simple Dirichlet distribution. Procedures for testing hypotheses regarding a priori interrelations of the success probabilities of the processes are given. Exact expressions for the posterior probabi1ities of these hypotheses are shown to be approximately equal to weighted p-values or 1ikelihood ratios.     

Analysis of incomplete data under non-random mechanisms: bayesian inference

Inferences are made concerning population proportions when data are not missing at random.Both one sample and two sample situations are considered with examples in clinical trials.The one samplesituation involves the existence of response related incomplete data in a study conducted to make inferences involving the proportion. The two sample problem involves comparing two treatments in clinical trials when there exists dropouts due to both the treatment and the response to the treatment.Bayes procedures are used in estimating parameters of interest and testing hypotheses of interest in these two situations. An ad-hoc approach to the classical inference is presented for each ofthe two situations and compared with the Bayesian approach discussed. To illustrate the theory developed, data from clinical trials of severe head trauma patients at the Medical College of Virginia Head Injury Center from 1984 to 1987 is considered     

Image reconstruction by adaptive bayesian classification with a locally proportional prior

This paper concerns the problem of reconstructing images from noisy data by means of Bayesian classification methods. In Klein and Press, 1992, the authors presented a method for reconstructing images called Adaptive Bayesian Classification (ABC). The ABC procedure was shown to preform very well in simulation experiments. The ABC procedure was multistaged; moreover, it involved selecting a prior at Stage n that was the posterior at Stage n - 1. In this paper the authors show that we can improve upon ABC for some problems by modifying the way we take the prior at each stage. The new proposal is to take the prior for the pixel label at each stage as proportional to the number of pixels with that label in a small neighborhood of the pixel. The ABC procedure with a locally proportional prior (ABC/LPP) tends to improve upon the ABC procedure for some problems because the prior in the iterative portion of ABC/LPP is contextual, while that in ABC in non- contextual.     

Hierarchical modeling with gaussian processes

We formulate a hierarchical version of the Gaussian Process model. In particular, we assume there to be data on several units randomly drawn from the same population. For each unit, several responses are available that arise from a Gaussian Process model. The parameters characterizing the Gaussian Process model for the units are modeled to arise from normal or gamma distributions. Results for two simulations are given that compare the performance of the hierarchical and non-hierarchical models.     

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.     

The biases associated with maximum likelihood methods of estimation of the multivariate logistic risk function

The posterior probability of an object belonging to one of two populations can be estimated using multivariate logistic regression. The bias associated with this procedure is derived In the context of normal populations with different mean vectors and a common covariance matrix and is compared with the bias of the classical method based on this normality assumption, -It Is found that the bias of the more robust procedure of logistic regression is of a lower order than that of the normality based method.     

Importance of the prior mass for agreement between frequentist and Bayesian approaches in the two-sided test

We study distributional properties of generalized order statistics (gos) related by a random shift or scaling scheme in the continuous and discrete case, respectively. In the continuous case, we obtain new characterizations of distributions relating non-neighbouring gos extending some results given in the literature for the neighbouring cases. On the other hand, in the discrete case, we investigate the existence and uniqueness of a discrete parent distribution supported on the integers whose gos are related by a random translation.     

Hierarchical Bayesian modeling of marked non-homogeneous Poisson processes with finite mixtures and inclusion of covariate information

We investigate marked non-homogeneous Poisson processes using finite mixtures of bivariate normal components to model the spatial intensity function. We employ a Bayesian hierarchical framework for estimation of the parameters in the model, and propose an approach for including covariate information in this context. The methodology is exemplified through an application involving modeling of and inference for tornado occurrences.     

Simultaneous Credible Bands for Latent Gaussian Models

Abstract. Deterministic Bayesian inference for latent Gaussian models has recently become available using integrated nested Laplace approximations (INLA). Applying the INLA‐methodology, marginal estimates for elements of the latent field can be computed efficiently, providing relevant summary statistics like posterior means, variances and pointwise credible intervals. In this article, we extend the use of INLA to joint inference and present an algorithm to derive analytical simultaneous credible bands for subsets of the latent field. The algorithm is based on approximating the joint distribution of the subsets by multivariate Gaussian mixtures. Additionally, we present a saddlepoint approximation to compute Bayesian contour probabilities, representing the posterior support of fixed parameter vectors of interest. We perform a simulation study and apply the given methods to two real examples.     

Discriminant analysis with stratified prior probabilities

This study investigates the use of stratification to improve discrimination when prior probabilities vary across strata of a population of interest. Sources of heterogeneity in prior probabilities include differences in geographic locale, age differences in the population studied, or differences in the time component of the data collected. The article suggests using logistic regression both to identify the underlying stratification and to estimate prior probabilities. A simulation study compares misclassification rates under two alternative stratification schemes with the traditional discriminant approach that ignores stratification in favor of pooled prior estimates. The simulations show that large asymptotic gains can be realized by stratification, and that these gains can be realized in finite samples, given moderate differences in prior probabilities.     

A simple hidden markov model for bayesian modeling with time dependent data

Consider a set of real valued observations collected over time. We pro¬pose a simple hidden Markow model for these realizations in which the the predicted distribution of the next future observation given the past is easily computed. The hidden or unobservable set of parameters is assumed to have a Markov structure of a special type. The model is quite flexible and can be used to incorporate different types of prior information in straightforward and sensible ways.     

Comparison of proportions with asymmetric prior information

In this paper a Bayesian model is developed for comparing two binomial proportions. A two stage hierarchical prior distribution is used to represent prior dependence. Prior exchangeability and independence are shown to be but special cases. The relevant distributions have to be computed numerically and some examples are presented.     

Inference with three-level prior distributions in quantile regression problems

In this paper, we propose a three level hierarchical Bayesian model for variable selection and estimation in quantile regression problems. Specifically, at the first level we consider a zero mean normal priors for the coefficients with unknown variance parameters. At the second level, we specify two different priors for the unknown variance parameters which introduce two different models producing different levels of sparsity. Then, at the third level we suggest joint improper priors for the unknown hyperparameters assuming they are independent. Simulations and Boston Housing data are utilized to compare the performance of our models with six existing models. The results indicate that our models perform good in the simulations and Boston Housing data.     

Practicing safe statistics with the mid-p

The mid-p-value is the standard p-value for a test minus half the difference between it and the nearest lower possible value. Its smaller size lends it an obvious appeal to users — it provides a more significant-looking summary of the evidence against the null hypothesis. This paper examines the possibility that the user might overstate the significance of the evidence by using the smaller mid-p in place of the standard p-value. Routine use of the mid-p is shown to control a quantity related to the Type I error rate. This related quantity is appropriate to consider when the decision to accept or reject the null hypothesis is not always firm. The natural, subjective interpretation of a p-value as the probability that the null hypothesis is true is also examined. The usual asymptotic correspondence between these two probabilities for one-sided hypotheses is shown to be strengthened when the standard p-value is replaced by the mid-p.     

Bayesian Poisson models for the graphical combination of dependent expert information

A supra-Bayesian (SB) wants to combine the information from a group of k experts to produce her distribution of a probability θ. Each expert gives his counts of what he thinks are the numbers of successes and failures in a sequence of independent trials, each with probability θ of success. These counts, used as a surrogate for each expert's own individual probability assessment (together with his associated level of confidence in his estimate), allow the SB to build various plausible conjugate models. Such models reflect her beliefs about the reliability of different experts and take account of different possible patterns of overlap of information between them. Corresponding combination rules are then obtained and compared with other more established rules and their properties examined.