Class Of Prior Distributions Research Articles

A Bayesian approach to sample size selection in a multiple hypotheses test for the Bernoulli model

The situation where a certain type of seed need to be classified into one of three different categories, according to its germination level, can be formulated as the simultaneous test of three statistical hypotheses. In this paper, the problem of assessing an optimal sample size for the simultaneous test of several hypotheses on a Bernoulli process is studied within the Bayesian framework, and a solution is obtained for a wide class of prior distributions and a logarithmic loss function.

Global prior distributions for the analysis of discrete graphical models

We propose a new class of prior distributions for the analysis of discrete graphical models. Such a class, obtained following a conditional approach, generalizes the hyper Dirichlet distributions of Dawid and Lauritzen (1993), since it can be extended to non decomposable graphical models. The two classes are compared in terms of model selection, with an application to a medical data-set illustrating the performance of the two resulting procedures. The proposed class turns out to select simpler, more par-simonious structures.

Classes of bidimensional priors specified on a collection of set: Bayesian robustness

When the parameter space is multidimensional, to elicit the joint prior distribution is a very difficult task. An accessible prior information might then be the class of prior distributions with given one-dimensional marginals. Unfortunately, even in bidimensional parameter spaces, the variational problems encountered in the Bayesian analysis of this class have not yet been solved. This paper is devoted to studying two approximations to these variational problem based on the observation that the above class of priors can be seen as the class of priors with specified probabilities for a given (non-countable) collection of sets. Illustrations, including a recent clinical trial (ECMO), are given.

Robust bayesian methods for monitoring clinical trials

Bayesian methods for the analysis of clinical trials data have received increasing attention recently as they offer an approach for dealing with difficult problems that arise in practice. A major criticism of the Bayesian approach, however, has focused on the need to specify a single, often subjective, prior distribution for the parameters of interest. In an attempt to address this criticism, we describe methods for assessing the robustness of the posterior distribution to the specification of the prior. The robust Bayesian approach to data analysis replaces the prior distribution with a class of prior distributions and investigates how the inferences might change as the prior varies over this class. The purpose of this paper is to illustrate the application of robust Bayesian methods to the analysis of clinical trials data. Using two examples of clinical trials taken from the literature, we illustrate how to use these methods to help a data monitoring committee decide whether or not to stop a trial early.

Bayesian inference for contingency tables with given marginals

In this paper we introduce a class of prior distributions for contingency tables with given marginals. We are interested in the structrre of concordance/discordance of such tables. There is actually a minor limitation in that the marginals are required to assume only rational values. We do argue, though, that this is not a serious drawback for all applicatory purposes. The posterior and predictive distributions given anM-sample are computed. Examples of Bayesian estimates of some classical indices of concordance are also given. Moreover, we show how to use simulation in order to overcome some difficulties which arise in the computation of the posterior distribution.

Sensitivity to prior independence via farlie-gumbel-morgenstern model

When the parameter spare is multidimensional, the prior marginals are sometimes simple to specify, whereas the joint distribution may be extremely hard to elicit; as a consequence of this, prior independence is often assumed. When this is the case, it seems convenient to perform a sensitivity analysis, i.e., to consider a class of prior distributions (containing the independence as a partieular case) and to compute the range of variation of the posterior quantity of interest, when the prior moves in that class. In this paper, the well-known Farlie-Gumbel- Morgens tern system (or a generalization of it) is proposed as the class of priors. We show that the robustness analysis is easy to carry out and, although this family may sometimes seem too narrow, it could be used as a first step in a robustness study.

Bayesian estimation of a normal mean parameter using the Linex loss function and robustness considerations

This paper presents a derivation of an explicit analytical form for the Bayes estimator of the normal location parameter using the Linex loss function with a general class of prior distributions. Exact and approximate results based on Pericchi and Smith’s paper (1992) are given, where the priors are double-exponential and Studentt, respectively. The results of this paper provide a link between the robust Bayesian analysis for the normal location parameter when adopting either the Linex loss function or the squared error loss function.

Bayesian robustness for multiparameter problems

Abstract Bayesian robustness is studied for e-contamination classes of prior distributions. Nonparametric classes of contsminations such as the class of all unimodal spherically symmetric densities are considered here. Posterior φ-divergence and its curvature are used to measure the sensitivity of priors on the resulting posterior densities. Examples are provided to illustrate our results.

Robust Bayesian hypothesis testing in the presence of nuisance parameters

Abstract Robust Bayesian testing of point null hypotheses is considered for problems involving the presence of nuisance parameters. The robust Bayesian approach seeks answers that hold for a range of prior distributions. Three techniques for handling the nuisance parameter are studied and compared. They are (i) utilize a noninformative prior to integrate out the nuisance parameter; (ii) utilize a test statistic whose distribution does not depend on the nuisance parameter; and (iii) use a class of prior distributions for the nuisance parameter. These approaches are studied in two examples, the univariate normal model with unknown mean and variance, and a multivariate normal example.

Bayesian Robustness with Mixture Classes of Priors

Uncertainty in specification of the prior distribution is a common concern with Bayesian analysis. The robust Bayesian approach is to work with a class of prior distributions, which model uncertainty about the prior, instead of a single distribution. One is interested in the range of the posterior expectations of certain parametric functions as the prior varies over the class being considered--if this range is small, the analysis is robust to misspecification of the prior. Relatively little research has dealt with robustness with respect to priors on several parameters, especially the problem of imposing shape and smoothness constraints on the priors in the class. To address this problem, we consider neighborhood classes of mixture priors. Results are presented for two kinds of "mixture classes," which yield different types of neighborhoods. The problem of finding suprema and infima of posterior expectations of parametric functions is seen to reduce to numerical maximization and minimization. In the applications we consider mixtures of uniform densities on variously shaped sets. This allows one to model symmetry and unimodality of different types in more than one dimension. Numerical examples are provided.

A Bayesian Approach to Estimation of GPAs of University of Iowa Freshmen Under Order Restrictions

The problem of interest is to estimate a two-way table of means of grade point averages (GPAs) of University of Iowa freshmen grouped by levels of ACT scores and high school percentiles. Suppose one believes a priori that the table means satisfy a specific partial order. The use of two different classes of prior distributions is considered in modeling this prior information. The use of simulation and, in particular, the Gibbs sampler are outlined to summarize various posterior distributions of interest. The posterior distribution is used to predict the grade point averages of future Iowa freshmen.

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

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.

Interpreting the Stars in Precise Hypothesis Testing

The problem of testing a precise null hypothesis is considered when available information is limited to knowledge of either the P-value or that the P-value is in some interval (e.g., the classical one or two 'stars'). Because of the recognized conflict between classical and Bayesian measures of evidence in testing a precise null hypothesis, the interpretation of a P-value or of 'stars', from a Bayesian perspective, is explored. This is done by treating the P-value or the 'stars' as the data, and computing corresponding posterior probabilities or Bayes factors. Of particular interest are lower bounds on these measures over wide classes of prior distributions. Comparisons are also made between classical meta-analysis techniques for combining many tests of statistical significance and lower bounds on Bayes factors and posterior probabilities.

Bayesian Diagnostics for Heterogeneity

In this paper we examine the problem of testing for hetero geneity and heterosckedasticity in a Bayesian framework. We first show that a model with random coefficients is identical to a model with heteroskedastic residuals. We then consider two approaches for testing. The first one is concer ned with the point of view of misspecification. The original model is homoskedastic. One is willing to detect any departure from homoskedasticity. We propose diagnostics based on the examination of the Bayesian residuals, after stressing the differences between classical and Bayesian residuals. In the second approach, the starting point is a precise form of the alternative hypothesis, and the model for inference is heteroskedastic. A test for homoskedasticity is then a test for a parameter restriction. This can be done by looking at highest posterior probability regions or by the use of the posterior odds ratio. As a joint product, we develop the posterior analysis of a heteroskedastic regression model for several classes of prior distributions.

Very Weak Expansions for Sequentially Designed Experiments: Linear Models

In sequentially designed experiments with linear models, each design variable may depend on previous responses. The use of such sequential designs does not affect the likelihood function or the functional form of the maximum likelihood estimator, but it may affect sampling distributions. In this paper, asymptotic expansions for sampling distributions are obtained. The expansions are very weak ones in which a confidence curve (a function of the unknown parameters) is replaced by a confidence functional defined on a class of prior distributions. The proofs use a version of Stein's identity.

Ranges of Posterior Probabilities for Quasiunimodal Priors with Specified Quantiles

Abstract Suppose several quantiles of the prior distribution for θ are specified or, equivalently, the prior probabilities of a partitioning collection of intervals {Ii } are given. In addition, suppose that the prior distribution is assumed to be unimodal. Rather than selecting a single prior distribution to perform a Bayesian analysis, it is of interest to consider the class of all prior distributions compatible with these inputs. For this class and unimodal likelihood functions, the ranges of the posterior probabilities of the Ii and the ranges of the posterior cdf at the specified prior quantiles were determined in Berger and O'Hagan (in press). Unfortunately, calculations with this class can be difficult. Here a similar, much more easily analyzed class of quasiunimodal prior distributions is considered and compared with other classes.

Reconciling Bayesian and Frequentist Evidence in the One-Sided Testing Problem

Abstract For the one-sided hypothesis testing problem it is shown that it is possible to reconcile Bayesian evidence against H 0, expressed in terms of the posterior probability that H 0 is true, with frequentist evidence against H 0, expressed in terms of the p value. In fact, for many classes of prior distributions it is shown that the infimum of the Bayesian posterior probability of H 0 is equal to the p value; in other cases the infimum is less than the p value. The results are in contrast to recent work of Berger and Sellke (1987) in the two-sided (point null) case, where it was found that the p value is much smaller than the Bayesian infimum. Some comments on the point null problem are also given.

Reconciling Bayesian and Frequentist Evidence in the One-Sided Testing Problem

Abstract For the one-sided hypothesis testing problem it is shown that it is possible to reconcile Bayesian evidence against H 0, expressed in terms of the posterior probability that H 0 is true, with frequentist evidence against H 0, expressed in terms of the p value. In fact, for many classes of prior distributions it is shown that the infimum of the Bayesian posterior probability of H 0 is equal to the p value; in other cases the infimum is less than the p value. The results are in contrast to recent work of Berger and Sellke (1987) in the two-sided (point null) case, where it was found that the p value is much smaller than the Bayesian infimum. Some comments on the point null problem are also given.

Simultaneous control and estimation of linear stochastic systems with unknown parameters of disturbances

In the present paper methods of the decision theory are applied to determine minimax policies of simultaneous control and estimation for stochastic system (1). There are solved the following cases: a)when disturbances of the system have the binomial distribution with unknown parameter b)when disturbances of the system have distribution belonging to an exponential family dependent on natural unknown parameter and the class of prior distributions of parameter is restricted by fixing the second moment In both cases open analytical forms of minimax policies are given

Empirical bayes estimation of a set of binomial probabilities

A class of prior distributions is defined to reflect exchangeability of a set of binomial probabilities. The class is indexed by the hyperparameter K, which indicates the precision of the user's prior belief about the similarity of the probabilities. By estimating the unknown value of K from the marginal distribution, simple new point and interval estimators are proposed.