Reference priors for constrained rate models of count data

We derive reference priors for constrained rate models of count data using the sequential algorithm of Berger and Bernardo (1992b). The event counts for various groups of subjects are modeled as discrete random variables (Poisson, binomial, or negative binomial) with group specific rates. We consider situations in which the groups can be completely ordered according to one covariate. The priors enforce monotonicity (or monotonicity and convexity) of the rates with respect to the ordering. We use the priors to model a data set on mortality rates for men in different age groups assuming that the mortality rates increase with respect to age. We also consider the situation in which the parameter space is augmented to include rates corresponding to unobserved age groups, and the case of a random upper bound on the mortality rates. In addition, we provide an evaluation of the out-of-sample predictive performance of the proposed methods.     

Bayesian model selection in linear mixed effects models with autoregressive(p) errors using mixture priors

In this article, we apply the Bayesian approach to the linear mixed effect models with autoregressive(p) random errors under mixture priors obtained with the Markov chain Monte Carlo (MCMC) method. The mixture structure of a point mass and continuous distribution can help to select the variables in fixed and random effects models from the posterior sample generated using the MCMC method. Bayesian prediction of future observations is also one of the major concerns. To get the best model, we consider the commonly used highest posterior probability model and the median posterior probability model. As a result, both criteria tend to be needed to choose the best model from the entire simulation study. In terms of predictive accuracy, a real example confirms that the proposed method provides accurate results.     

Stylometric analyses using Dirichlet process mixture models

Stylometry refers to the statistical analysis of literary style of authors based on the characteristics of expression in their writings. We propose an approach to stylometry based on a Bayesian Dirichlet process mixture model using multinomial word frequency data. The parameters of the multinomial distribution of word frequency data are the “word prints” of the author. Our approach is based on model-based clustering of the vectors of probability values of the multinomial distribution. The resultant clusters identify different writing styles that assist in author attribution for disputed works in a corpus. As a test case, the methodology is applied to the problem of authorship attribution involving the Federalist papers. Our results are consistent with previous stylometric analyses of these papers.     

Noninformative priors for the normal variance ratio

In this paper, we consider noninformative priors for the ratio of variances in two normal populations. We develop first and second order matching priors. We find that the second order matching prior matches alternative coverage probabilities up to the second order and is also a HPD matching prior. It turns out that among the reference priors, only one-at-a-time reference prior satisfies a second order matching criterion. Our simulation study indicates that the one-at-a-time reference prior performs better than other reference priors in terms of matching the target coverage probabilities in a frequentist sense. This work is supported by Korea Research Foundation Grant (KRF-2004-002-C00041).     

Bayesian analysis for a stress-strength system under noninformative priors

Noninformative priors are used for estimating the reliability of a stress-strength system. Several reference priors (cf. Berger and Bernardo 1989, 1992) are derived. A class of priors is found by matching the coverage probabilities of one-sided Bayesian credible intervals with the corresponding frequentist coverage probabilities. It turns out that none of the reference priors is a matching prior. Sufficient conditions for propriety of posteriors under reference priors and matching priors are provided. A simple matching prior is compared with three reference priors when sample sizes are small. The study shows that the matching prior performs better than Jeffreys's prior and reference priors in meeting the target coverage probabilities.     

Discretisation for inference on normal mixture models

The problem of inference in Bayesian Normal mixture models is known to be difficult. In particular, direct Bayesian inference (via quadrature) suffers from a combinatorial explosion in having to consider every possible partition of n observations into k mixture components, resulting in a computation time which is O(k ⁿ). This paper explores the use of discretised parameters and shows that for equal-variance mixture models, direct computation time can be reduced to O(D ^k n ^k), where relevant continuous parameters are each divided into D regions. As a consequence, direct inference is now possible on genuine data sets for small k, where the quality of approximation is determined by the level of discretisation. For large problems, where the computational complexity is still too great in O(D ^k n ^k) time, discretisation can provide a convergence diagnostic for a Markov chain Monte Carlo analysis.     

Subjective priors for the dirichlet process

This paper reviews difficulties with the interpretation and use of the prior parameter u required in the Dirichlet approach to nonpararnetric Bayesian statistics. Two subjective prior distributions are introduced and studied. These priors are obtained computationally by requiring that the experimenter specify certain constraints.     

Generating pseudo-random variables from mixture models by exemplary sampling

Generating samples from a two-stage distribution is an important part of the study of mixture models. These samples are used to examine estimation procedures, and other properties of the mixture model. In this paper we present an exemplary sampling method for generating data from the mixed distribution. This method uses the order statistic spacings of the mixing distribution and random sampling from the distribution conditional on the mixing variable to produce samples from the mixed distribution. We show that this exemplary procedure often produces data with an empirical distribution function closer to the mixed distribution than the Method of Composition. We illustrate the method with an example.     

Noninformative priors for linear combinations of the normal means

In this paper, we develop noninformative priors for linear combinations of the means under the normal populations. It turns out that among the reference priors the one-at-a-time reference prior satisfies a second order probability matching criterion. Moreover, the second order probability matching priors match alternative coverage probabilities up to the second order and are also HPD matching priors. Our simulation study indicates that the one-at-a-time reference prior performs better than the other reference priors in terms of matching the target coverage probabilities in a frequentist sense.     

Improper priors for translation families

We consider the problem of inference when sampling from a translation family with an improper prior. Properties of the formal Bayes inference will be studied. We give conditions (on the prior and/or the family) guaranteeing the HS-coherence (see Heath and Sudderth, Ann. Statist. 6 (1978), 333–345) of the formal Bayes posterior. Since HS-coherence is equivalent to being a posterior of a finitely additive prior, all coherence results imply the existence of a finitely additive prior which has the formal Bayes inference as a posterior.     

Nonparametric Bayesian inference for multivariate density functions using Feller priors

Multivariate density estimation plays an important role in investigating the mechanism of high-dimensional data. This article describes a nonparametric Bayesian approach to the estimation of multivariate densities. A general procedure is proposed for constructing Feller priors for multivariate densities and their theoretical properties as nonparametric priors are established. A blocked Gibbs sampling algorithm is devised to sample from the posterior of the multivariate density. A simulation study is conducted to evaluate the performance of the procedure.     

Empirical Bayes estimators in hierarchical models with mixture priors

We consider subgroup analyses within the framework of hierarchical modeling and empirical Bayes (EB) methodology for general priors, thereby generalizing the normal–normal model. By doing this one obtains greater flexibility in modeling. We focus on mixture priors, that is, on the situation where group effects are exchangeable within clusters of subgroups only. We establish theoretical results on accuracy, precision, shrinkage and selection bias of EB estimators under the general priors. The impact of model misspecification is investigated and the applicability of the methodology is illustrated with datasets from the (medical) literature.     

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.     

Hybrid Dirichlet mixture models for functional data

Summary.  In functional data analysis, curves or surfaces are observed, up to measurement error, at a finite set of locations, for, say, a sample of n individuals. Often, the curves are homogeneous, except perhaps for individual-specific regions that provide heterogeneous behaviour (e.g. 'damaged' areas of irregular shape on an otherwise smooth surface). Motivated by applications with functional data of this nature, we propose a Bayesian mixture model, with the aim of dimension reduction, by representing the sample of n curves through a smaller set of canonical curves. We propose a novel prior on the space of probability measures for a random curve which extends the popular Dirichlet priors by allowing local clustering: non-homogeneous portions of a curve can be allocated to different clusters and the n individual curves can be represented as recombinations (hybrids) of a few canonical curves. More precisely, the prior proposed envisions a conceptual hidden factor with k -levels that acts locally on each curve. We discuss several models incorporating this prior and illustrate its performance with simulated and real data sets. We examine theoretical properties of the proposed finite hybrid Dirichlet mixtures, specifically, their behaviour as the number of the mixture components goes to ∞ and their connection with Dirichlet process mixtures.     

Almost nonparametric inference for repeated measures in mixture models

We consider ways to estimate the mixing proportions in a finite mixture distribution or to estimate the number of components of the mixture distribution without making parametric assumptions about the component distributions. We require a vector of observations on each subject. This vector is mapped into a vector of 0s and 1s and summed. The resulting distribution of sums can be modelled as a mixture of binomials. We then work with the binomial mixture. The efficiency and robustness of this method are compared with the strategy of assuming multivariate normal mixtures when, typically, the true underlying mixture distribution is different. It is shown that in many cases the approach based on simple binomial mixtures is superior.     

Bayesian inference for generalized additive mixed models based on Markov random field priors

Most regression problems in practice require flexible semiparametric forms of the predictor for modelling the dependence of responses on covariates. Moreover, it is often necessary to add random effects accounting for overdispersion caused by unobserved heterogeneity or for correlation in longitudinal or spatial data. We present a unified approach for Bayesian inference via Markov chain Monte Carlo simulation in generalized additive and semiparametric mixed models. Different types of covariates, such as the usual covariates with fixed effects, metrical covariates with non-linear effects, unstructured random effects, trend and seasonal components in longitudinal data and spatial covariates, are all treated within the same general framework by assigning appropriate Markov random field priors with different forms and degrees of smoothness. We applied the approach in several case-studies and consulting cases, showing that the methods are also computationally feasible in problems with many covariates and large data sets. In this paper, we choose two typical applications.     

Bayesian estimation of the generalized lognormal distribution using objective priors

The generalized lognormal distribution plays an important role in analysing data from different life testing experiments. In this paper, we consider Bayesian analysis of this distribution using various objective priors for the model parameters. Specifically, we derive expressions for the Jeffreys-type priors, the reference priors with different group orderings of the parameters, and the first-order matching priors. We also study the properties of the posterior distributions of the parameters under these improper priors. It is shown that only two of them result in proper posterior distributions. Numerical simulation studies are conducted to compare the performances of the Bayesian estimators under the considered priors and the maximum likelihood estimates. Finally, a real-data application is also provided for illustrative purposes.     

Bayesian analysis of the amended Davidson model for paired comparison using noninformative and informative priors

In recent years, numerous statisticians have focused their attention on the Bayesian analysis of different paired comparison models. While studying paired comparison techniques, the Davidson model is considered to be one of the famous paired comparison models in the available literature. In this article, we have introduced an amendment in the Davidson model which has been commenced to accommodate the option of not distinguishing the effects of two treatments when they are compared pairwise. Having made this amendment, the Bayesian analysis of the Amended Davidson model is performed using the noninformative (uniform and Jeffreys’) and informative (Dirichlet–gamma–gamma) priors. To study the model and to perform the Bayesian analysis with the help of an example, we have obtained the joint and marginal posterior distributions of the parameters, their posterior estimates, graphical presentations of the marginal densities, preference and predictive probabilities and the posterior probabilities to compare the treatment parameters.     

Empirical likelihood‐based inference for genetic mixture models

The authors show how the genetic effect of a quantitative trait locus can be estimated by a nonparametric empirical likelihood method when the phenotype distributions are completely unspecified. They use an empirical likelihood ratio statistic for testing the genetic effect and obtaining confidence intervals. In addition to studying the asymptotic properties of these procedures, the authors present simulation results and illustrate their approach with a study on breast cancer resistance genes.     

Reference priors for a product of normal means when variances are unknown

The reference priors of Berger and Bernardo (1992) are derived for normal populations with unknown variances when the product of means is of interest. The priors are also shown to be Tibshirani's (1989) matching priors.