Comparisons of Bayesian estimation procedures for two-way contingency tables without interaction

Bayesian and empirical Bayesian estimation methods are reviewed and proposed for the row and column parameters in two-way Contingency tables without interaction. Rasch's multiplicative Poisson model for misreadings is discussed in an example. The case is treated where assumptions of exchangeability are reasonable a priori for the unknown parameters. Two different types of prior distributions are compared, It appears that gamma priors yield more tractable results than lognormal priors.     

Comparison of Hypothesis Testing and Bayesian Model Selection

The main goal of both Bayesian model selection and classical hypotheses testing is to make inferences with respect to the state of affairs in a population of interest. The main differences between both approaches are the explicit use of prior information by Bayesians, and the explicit use of null distributions by the classicists. Formalization of prior information in prior distributions is often difficult. In this paper two practical approaches (encompassing priors and training data) to specify prior distributions will be presented. The computation of null distributions is relatively easy. However, as will be illustrated, a straightforward interpretation of the resulting p-values is not always easy. Bayesian model selection can be used to compute posterior probabilities for each of a number of competing models. This provides an alternative for the currently prevalent testing of hypotheses using p-values. Both approaches will be compared and illustrated using case studies. Each case study fits in the framework of the normal linear model, that is, analysis of variance and multiple regression.     

On the lack of invariance of some Bayesian outlier models

The property of non-invariance to the scale of the data for the Abraham and Box (1978) Bayesian outlier model, and the model that generalizes the Guttman, Dutter and Freeman (1978) outlier analysis is discussed. This drawback is due to the non-informative prior taken for the parameters. Freeman (1980) expected that most posterior weight would be put on the model with the most outliers if the improper prior is used. We show that this may not be correct. An illustrative example is given.     

Limit theorems for multivariate discrete distributions

According to the usual law of small numbers a multivariate Poisson distribution is derived by defining an appropriate model for multivariate Binomial distributions and examining their behaviour for large numbers of trials and small probabilities of marginal and simultaneous successes. The weak limit law is a generalization of Poisson's distribution to larger finite dimensions with arbitrary dependence structure. Compounding this multivariate Poisson distribution by a Gamma distribution results in a multivariate Pascal distribution which is again asymptotically multivariate Poisson. These Pascal distributions contain a class of multivariate geometric distributions. Finally the bivariate Binomial distribution is shown to be the limit law of appropriate bivariate hypergeometric distributions. Proving the limit theorems mentioned here as well as understanding the corresponding limit distributions becomes feasible by using probability generating functions.     

Bayesian model selection using encompassing priors

This paper deals with Bayesian selection of models that can be specified using inequality constraints among the model parameters. The concept of encompassing priors is introduced, that is, a prior distribution for an unconstrained model from which the prior distributions of the constrained models can be derived. It is shown that the Bayes factor for the encompassing and a constrained model has a very nice interpretation: it is the ratio of the proportion of the prior and posterior distribution of the encompassing model in agreement with the constrained model. It is also shown that, for a specific class of models, selection based on encompassing priors will render a virtually objective selection procedure. The paper concludes with three illustrative examples: an analysis of variance with ordered means; a contingency table analysis with ordered odds-ratios; and a multilevel model with ordered slopes.     

Generalized P‐values and Bayesian evidence in the one‐sided testing problems under exponential distributions

The problem of comparing the frequentist evidence and the Bayesian evidence in the one‐sided testing problems has been widely treated and many researches revealed that these two methods can reach an agreement approximately. However, most of the previous work dealt mainly with situations without nuisance parameters. Since the presence of nuisance parameters is very common in practice, whether these two kinds of evidence still reach an agreement is a problem worthy of study. In this article, we establish in a systematic way under the exponential distributions the agreement of the Bayesian evidence and the generalized frequentist evidence (the generalized P‐value) for a variety of one‐sided testing problems where the nuisance parameters are involved.     

On truncated bivariate discrete distributions: A unified treatment

A unified treatment of three types of zero class truncation for bivariate discrete distributions is presented. Using the probability generating function approach, various properties of the truncated distributions are examined in association with the corresponding properties of the initial complete form of the distribution. Expressions for moments and conditional distributions are also obtained. Bivariate versions of the Thomas and the Intervened Poisson distributions are introduced and used as illustrative examples. Received November 2000/Revised March 2002     

Some properties of the negative hypergeometric distribution and its limit distributions

Summary This paper deals with a class of frequency distributions consisting of the negative hypergeometric distribution and its limit cases, namely, the negative binomial, binomial, Poisson, gamma, beta and normal distributions. In view of the relationships existing between the various members of this class, it is found convenient to discuss jointly for all the distributions certain topics, including additivity properties, a property related to the information measure in point estimation, as well as a comparison of frequency theory and Bayesian theory interpretations of interval estimation.     

Highest posterior mass prediction intervals for binomial and poisson distributions

The problems of constructing prediction intervals (PIs) for the binomial and Poisson distributions are considered. New highest posterior mass (HPM) PIs based on fiducial approach are proposed. Other fiducial PIs, an exact PI and approximate PIs are reviewed and compared with the HPM-PIs. Exact coverage studies and expected widths of prediction intervals show that the new prediction intervals are less conservative than other fiducial PIs and comparable with the approximate one based on the joint sampling approach for the binomial case. For the Poisson case, the HPM-PIs are better than the other PIs in terms of coverage probabilities and precision. The methods are illustrated using some practical examples.     

Stepwise Bayes distributional inference for a finite population

Stepwise Bayes arguments can be used to derive various decision rules. Admissibility of such rules follows if additional conditions are satisfied. We show that in complete generality almost admissibility is in place. A uniform distribution example is used to demonstrate how stepwise Bayes arguments can be used when the support of the observation distribution depends on the unknown parameter. We then discuss distributional inference and show that weighted Polya posterior distributions provide admissible distributional inference for finite population problems when strictly proper loss functions are used.     

What Does Objective Mean in a Dirichlet‐multinomial Process?

The Dirichlet‐multinomial process can be seen as the generalisation of the binomial model with beta prior distribution when the number of categories is larger than two. In such a scenario, setting informative prior distributions when the number of categories is great becomes difficult, so the need for an objective approach arises. However, what does objective mean in the Dirichlet‐multinomial process? To deal with this question, we study the sensitivity of the posterior distribution to the choice of an objective Dirichlet prior from those presented in the available literature. We illustrate the impact of the selection of the prior distribution in several scenarios and discuss the most sensible ones.     

Inference of Dynamic Generalized Linear Models: On-Line Computation and Appraisal

The purpose of this paper is to provide a critical discussion on real-time estimation of dynamic generalized linear models. We describe and contrast three estimation schemes, the first of which is based on conjugate analysis and linear Bayes methods, the second based on posterior mode estimation, and the third based on sequential Monte Carlo sampling methods, also known as particle filters. For the first scheme, we give a summary of inference components, such as prior/posterior and forecast densities, for the most common response distributions. Considering data of arrivals of tourists in Cyprus, we illustrate the Poisson model, providing a comparative analysis of the above three schemes.     

On model selection in Bayesian regression

We discuss the problem of constructing a suitable regression model from a nonparametric Bayesian viewpoint. For this purpose, we consider the case when the error terms have symmetric and unimodal densities. By the Khintchine and Shepp theorem, the density of response variable can be written as a scale mixture of uniform densities. The mixing distribution is assumed to have a Dirichlet process prior. We further consider appropriate prior distributions for other parameters as the components of the predictive device. Among the possible submodels, we select the one which has the highest posterior probability. An example is given to illustrate the approach.     

Posterior Inference in Bayesian Quantile Regression with Asymmetric Laplace Likelihood

The paper discusses the asymptotic validity of posterior inference of pseudo‐Bayesian quantile regression methods with complete or censored data when an asymmetric Laplace likelihood is used. The asymmetric Laplace likelihood has a special place in the Bayesian quantile regression framework because the usual quantile regression estimator can be derived as the maximum likelihood estimator under such a model, and this working likelihood enables highly efficient Markov chain Monte Carlo algorithms for posterior sampling. However, it seems to be under‐recognised that the stationary distribution for the resulting posterior does not provide valid posterior inference directly. We demonstrate that a simple adjustment to the covariance matrix of the posterior chain leads to asymptotically valid posterior inference. Our simulation results confirm that the posterior inference, when appropriately adjusted, is an attractive alternative to other asymptotic approximations in quantile regression, especially in the presence of censored data.     

Some results on discrete a–monotonicity

Some characterizations and preservation properties of a–monotonicity are investigated. In particular it is shown that a mixture of binomial or Poisson distributions preserves the a–monotonicity of the mixing distribution. Also, a characterization of a class of infinitely divisible distributions, within the discrete α–unimodal class, is given. This characterization is the discrete version of that of ALAMATSAZ (1985) and KOTZ and STEUTEL (1988) in the continuous case.     

Bayesian semiparametric stochastic volatility modeling

This paper extends the existing fully parametric Bayesian literature on stochastic volatility to allow for more general return distributions. Instead of specifying a particular distribution for the return innovation, nonparametric Bayesian methods are used to flexibly model the skewness and kurtosis of the distribution while the dynamics of volatility continue to be modeled with a parametric structure. Our semiparametric Bayesian approach provides a full characterization of parametric and distributional uncertainty. A Markov chain Monte Carlo sampling approach to estimation is presented with theoretical and computational issues for simulation from the posterior predictive distributions. An empirical example compares the new model to standard parametric stochastic volatility models.     

Minnesota-type adaptive hierarchical priors for large Bayesian VARs

Large Bayesian VARs with stochastic volatility are increasingly used in empirical macroeconomics. The key to making these highly parameterized VARs useful is the use of shrinkage priors. We develop a family of priors that captures the best features of two prominent classes of shrinkage priors: adaptive hierarchical priors and Minnesota priors. Like adaptive hierarchical priors, these new priors ensure that only ‘small’ coefficients are strongly shrunk to zero, while ‘large’ coefficients remain intact. At the same time, these new priors can also incorporate many useful features of the Minnesota priors such as cross-variable shrinkage and shrinking coefficients on higher lags more aggressively. We introduce a fast posterior sampler to estimate BVARs with this family of priors—for a BVAR with 25 variables and 4 lags, obtaining 10,000 posterior draws takes about 3 min on a standard desktop computer. In a forecasting exercise, we show that these new priors outperform both adaptive hierarchical priors and Minnesota priors.     

Uncertainty Estimation for Pseudo‐Bayesian Inference Under Complex Sampling

Social and economic studies are often implemented as complex survey designs. For example, multistage, unequal probability sampling designs utilised by federal statistical agencies are typically constructed to maximise the efficiency of the target domain level estimator (e.g. indexed by geographic area) within cost constraints for survey administration. Such designs may induce dependence between the sampled units; for example, with employment of a sampling step that selects geographically indexed clusters of units. A sampling‐weighted pseudo‐posterior distribution may be used to estimate the population model on the observed sample. The dependence induced between coclustered units inflates the scale of the resulting pseudo‐posterior covariance matrix that has been shown to induce under coverage of the credibility sets. By bridging results across Bayesian model misspecification and survey sampling, we demonstrate that the scale and shape of the asymptotic distributions are different between each of the pseudo‐maximum likelihood estimate (MLE), the pseudo‐posterior and the MLE under simple random sampling. Through insights from survey‐sampling variance estimation and recent advances in computational methods, we devise a correction applied as a simple and fast postprocessing step to Markov chain Monte Carlo draws of the pseudo‐posterior distribution. This adjustment projects the pseudo‐posterior covariance matrix such that the nominal coverage is approximately achieved. We make an application to the National Survey on Drug Use and Health as a motivating example and we demonstrate the efficacy of our scale and shape projection procedure on synthetic data on several common archetypes of survey designs.     

Models for truncated counts

This paper examines a class of maximum-likelihood regression estimators for count data from truncated samples. Estimators for the truncated Poisson and negative binomial distributions are illustrated. Simulation results are given to illustrate the magnitude of the bias that may result from the failure to account for overdispersion in truncated samples. An empirical application based upon the number of recreational fishing trips taken by a sample of Alaskan fishermen is provided.     

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