characterization of the dunkl classical symmetric orthogonal polynomials in this paper we introduce the notion of dunkl classical orthogonal polynomials then we show that generalized hermite and generalized gegenbauer polynomials are the only dunkl classical symmetric orthogonal polynomials by solving a suitable differential difference equation

bayesian isotonic regression and trend analysis in many applications the mean of a response variable can be assumed to be a nondecreasing function of a continuous predictor controlling for covariates in such cases interest often focuses on estimating the regression function while also assessing evidence of an association this article proposes a new framework for bayesian isotonic regression and order restricted inference approximating the regression function with a high dimensional piecewise linear model the nondecreasing constraint is incorporated through a prior distribution for the slopes consisting of a product mixture of point masses accounting for flat regions and truncated normal densities to borrow information across the intervals and smooth the curve the prior is formulated as a latent autoregressive normal process this structure facilitates efficient posterior computation since the full conditional distributions of the parameters have simple conjugate forms point and interval estimates of the regression function and posterior probabilities of an association for different regions of the predictor can be estimated from a single mcmc run generalizations to categorical outcomes and multiple predictors are described and the approach is applied to an epidemiology application

wilcoxon type generalized bayesian information criterion we develop a generalized bayesian information criterion for regression model selection the new criterion relaxes the usually strong distributional assumption associated with schwarz s bic by adopting a wilcoxon type dispersion function and appropriately adjusting the penalty term we establish that the wilcoxon type generalized bic preserves the consistency of schwarz s bic without the need to assume a parametric likelihood we also show that it outperforms schwarz s bic with heavier tailed data in the sense that asymptotically it can yield substantially smaller l 2 risk on the other hand when the data are normally distributed both criteria have similar l 2 risk the new criterion function is convex and can be conveniently computed using existing statistical software our proposal provides a flexible yet highly efficient alternative to schwarz s bic at the same time it broadens the scope of wilcoxon inference which has played a fundamental role in classical nonparametric analysis

when should epidemiologic regressions use random coefficients summary regression models with random coefficients arise naturally in both frequentist and bayesian approaches to estimation problems they are becoming widely available in standard computer packages under the headings of generalized linear mixed models hierarchical models and multilevel models i here argue that such models offer a more scientifically defensible framework for epidemiologic analysis than the fixed effects models now prevalent in epidemiology the argument invokes an antiparsimony principle attributed to l j savage which is that models should be rich enough to reflect the complexity of the relations under study it also invokes the countervailing principle that you cannot estimate anything if you try to estimate everything often used to justify parsimony regression with random coefficients offers a rational compromise between these principles as well as an alternative to analyses based on standard variable selection algorithms and their attendant distortion of uncertainty assessments these points are illustrated with an analysis of data on diet nutrition and breast cancer

approximate bayes factors and accounting for model uncertainty in generalised linear models summary ways of obtaining approximate bayes factors for generalised linear models are described based on the laplace method for integrals we propose a new approximation which uses only the output of standard computer programs for estimating generalised linear models this appears to be quite accurate a reference set of proper priors is suggested both to represent the situation where there is not much prior information and to assess the sensitivity of the results to the prior distribution the methods can be used when the dispersion parameter is unknown when there is overdispersion to compare link functions and to compare error distributions and variance functions the methods can be used to implement the bayesian approach to accounting for model uncertainty we describe an application to inference about relative risks in the presence of control factors where model uncertainty is large and important software to implement the methods is available at no cost from statlib

a missing family of classical orthogonal polynomials we study a family of classical orthogonal polynomials which satisfy apart from a three term recurrence relation an eigenvalue problem with a differential operator of dunkl type these polynomials can be obtained from the little q jacobi polynomials in the limit q 1 we also show that these polynomials provide a nontrivial realization of the askey wilson algebra for q 1

bayesian covariance selection in generalized linear mixed models repeated measures and longitudinal data are commonly collected for analysis in epidemiology clinical trials biology sociology and economic sciences in such studies a response is measured repeatedly over time for each subject under study and the number and timing of the observations often varies among subjects in contrast to cross sectional studies that collect a single measurement per subject longitudinal studies have the extra complication of within subject dependence in the repeated measures such dependence can be thought to arise due to the impact of unmeasured predictors main effects of unmeasured predictors lead to variation in the average level of response among subjects while interactions with measured predictors lead to heterogeneity in the regression coefficients this justification has motivated random effects models which allow the intercept and slopes in a regression model to be subject specific random effects models are broadly useful for modeling of dependence not only for longitudinal data but also in multicenter studies meta analysis and functional data analysis

a rodrigues formula approach to determining closed form solutions to the schrodinger equation for symmetric anharmonic oscillators a natural generalization to the rodrigues formula for the hermite polynomials leads to the definition of sets of polynomials under certain constraints they are used to explain the existence of closed form solutions to the schrodinger equation for symmetric anharmonic potential wells and to obtain further closed form solutions

bayes factors for non homogeneous poisson processes with vague prior information on developpe une approche bayesienne au probleme de la comparaison de modeles pour des processus de poisson non homogenes

inverse groundwater modeling for hydraulic conductivity estimation using bayesian model averaging and variance window 1 this study proposes a bayesian model averaging bma method to address parameter estimation uncertainty arising from nonuniqueness in parameterization methods bma is able to incorporate multiple parameterization methods for prediction through the law of total probability and to obtain an ensemble average of hydraulic conductivity estimates two major issues in applying bma to hydraulic conductivity estimation are discussed the first problem is using occam s window in usual bma applications to measure approximated posterior model probabilities occam s window only accepts models in a very narrow range tending to single out the best method and discard other good methods we propose a variance window to replace occam s window to cope with this problem the second problem is the kashyap information criterion kic in the approximated posterior model probabilities which tends to prefer highly uncertain parameterization methods by considering the fisher information matrix with sufficient amounts of observation data the bayesian information criterion bic is a good approximation and is able to avoid controversial results from using kic this study adopts multiple generalized parameterization gp methods such as the bma models to estimate spatially correlated hydraulic conductivity numerical examples illustrate the issues of using kic and occam s window and show the advantages of using bic and the variance window in bma application finally we apply bma to the hydraulic conductivity estimation of the 1500 foot sand in east baton rouge parish louisiana

bayesian model averaging for linear regression models abstract we consider the problem of accounting for model uncertainty in linear regression models conditioning on a single selected model ignores model uncertainty and thus leads to the underestimation of uncertainty when making inferences about quantities of interest a bayesian solution to this problem involves averaging over all possible models i e combinations of predictors when making inferences about quantities of interest this approach is often not practical in this article we offer two alternative approaches first we describe an ad hoc procedure occam s window which indicates a small set of models over which a model average can be computed second we describe a markov chain monte carlo approach that directly approximates the exact solution in the presence of model uncertainty both of these model averaging procedures provide better predictive performance than any single model that might reasonably have been selected in the extreme case where there are many candidate predictors but

estimating bayes factors via posterior simulation with the laplace metropolis estimator abstract the key quantity needed for bayesian hypothesis testing and model selection is the integrated or marginal likelihood of a model we describe a way to use posterior simulation output to estimate integrated likelihoods we describe the basic laplace metropolis estimator for models without random effects for models with random effects we introduce the compound laplace metropolis estimator we apply this estimator to data from the world fertility survey and show it to give accurate results batching of simulation output is used to assess the uncertainty involved in using the compound laplace metropolis estimator the method allows us to test for the effects of independent variables in a random effects model and also to test for the presence of the random effects

generalized gegenbauer orthogonal polynomials abstract in this paper we explore a specific semi classical orthogonal sequence namely the generalized gegenbauer orthogonal polynomials gg which appear in many applications such as the weighted l p mean convergence of hermite fejer interpolation or the chain of harmonic oscillators in the absence of externally applied forces first we trace back the genesis of gg underlining its links with the jacobi orthogonal polynomials second we establish a differential difference relation and the second order differential equation satisfied by this sequence we end by giving the fourth order differential equation satisfied by the association of arbitrary order of the gg

a reference bayesian test for nested hypotheses and its relationship to the schwarz criterion abstract to compute a bayes factor for testing h 0 0 in the presence of a nuisance parameter priors under the null and alternative hypotheses must be chosen as in bayesian estimation an important problem has been to define automatic or reference methods for determining priors based only on the structure of the model in this article we apply the heuristic device of taking the amount of information in the prior on equal to the amount of information in a single observation then after transforming to be null orthogonal to we take the marginal priors on to be equal under the null and alternative hypotheses doing so and taking the prior on to be normal we find that the log of the bayes factor may be approximated by the schwarz criterion with an error of order o p n rather than the usual error of order o p 1 this result suggests the schwarz criterion should provide sensible approximate solutions to bayesian testing problems at least when the hypotheses are nested when

testing a point null hypothesis the irreconcilability of p values and evidence abstract the problem of testing a point null hypothesis or a small interval null hypothesis is considered of interest is the relationship between the p value or observed significance level and conditional and bayesian measures of evidence against the null hypothesis although one might presume that a small p value indicates the presence of strong evidence against the null such is not necessarily the case expanding on earlier work especially edwards lindman and savage 1963 and dickey 1977 it is shown that actual evidence against a null as measured say by posterior probability or comparative likelihood can differ by an order of magnitude from the p value for instance data that yield a p value of 05 when testing a normal mean result in a posterior probability of the null of at least 30 for any objective prior distribution objective here means that equal prior weight is given the two hypotheses and that the prior is symmetric and nonincreasing away from the null other definiti

bayesian analysis of autocorrelated ordered categorical data for industrial quality monitoring presently available methods to analyze the link between explanatory variables and an ordered categorical response implicitly assume independence this hypothesis is no longer valid when data are collected over time we assume temporal dependence and introduce autocorrelation using a latent variable formulation due to intractable distributions we resort to gibbs sampling for statistical inference within the bayesian paradigm variable selection is also addressed and appears as a straightforward byproduct of this framework we illustrate the method by analyzing on line quality data that possess such autocorrelation

benchmark priors for bayesian model averaging in contrast to a posterior analysis given a particular sampling model posterior model probabilities in the context of model uncertainty are typically rather sensitive to the speciflcation of the prior in particular difiuse priors on model speciflc parameters can lead to quite unexpected consequences here we focus on the practically relevant situation where we need to entertain a large number of sampling models and we have or wish to use little or no subjective prior information we aim at providing an automatic or benchmark prior structure that can be used in such cases we focus on the normal linear regression model with uncertainty in the choice of re gressors we propose a partly noninformative prior structure related to a natural conjugate g prior speciflcation where the amount of subjective information requested from the user is limited to the choice of a single scalar hyperparameter g0j the consequences of difierent choices for g0j are examined we investigate theoretical properties such as consistency of the implied bayesian procedure links with classical information criteria are provided in addition we examine the flnite sample implications of several choices of g0j in a simulation study the use of the mc 3 algorithm of madigan and york 1995 combined with e cient coding in fortran makes it feasible to conduct large simulations in addition to posterior criteria we shall also compare the predictive performance of difierent priors a classic exam ple concerning the economics of crime will also be provided and contrasted with results in the literature the main flndings of the paper will lead us to propose a benchmark prior speciflcation in a linear regression context with model uncertainty

model determination for categorical data with factor level merging we deal with contingency table data that are used to examine the relationships between a set of categorical variables or factors we assume that such relationships can be adequately described by the conditional independence structure that is imposed by an undirected graphical model if the contingency table is large a desirable simplified interpretation can be achieved by combining some categories or levels of the factors we introduce conditions under which such an operation does not alter the markov properties of the graph implementation of these conditions leads to bayesian model uncertainty procedures based on reversible jump markov chain monte carlo methods the methodology is illustrated on a 2 x 3 x 4 and up to a 4 x 5 x 5 x 2 x 2 contingency table

a bayesian model averaging approach for estimating the relative risk of mortality associated with heat waves in 105 u s cities estimating the risks heat waves pose to human health is a critical part of assessing the future impact of climate change in this article we propose a flexible class of time series models to estimate the relative risk of mortality associated with heat waves and conduct bayesian model averaging bma to account for the multiplicity of potential models applying these methods to data from 105 u s cities for the period 1987 2005 we identify those cities having a high posterior probability of increased mortality risk during heat waves examine the heterogeneity of the posterior distributions of mortality risk across cities assess sensitivity of the results to the selection of prior distributions and compare our bma results to a model selection approach our results show that no single model best predicts risk across the majority of cities and that for some cities heat wave risk estimation is sensitive to model choice although model averaging leads to posterior distributions with increased variance as compared to statistical inference conditional on a model obtained through model selection we find that the posterior mean of heat wave mortality risk is robust to accounting for model uncertainty over a broad class of models

fractional bayes factors for model comparison bayesian comparison of models is achieved simply by calculation of posterior probabilities of the models themselves however there are difficulties with this approach when prior information about the parameters of the various models is weak partial bayes factors offer a resoiution of the problem by setting aside part of the data as a training sampie the training sampie is used to obtain an initiai informative posterior distribution of the parameters in each model model comparison is then based on a bayes factor calculated from the remaining data properties of partial bayes factors are discussed particularly in the context of weak prior information and they are found to have advantages over other proposed methods of model comparison a new variant of the partial bayes factor the fractional bayes factor is advocated on grounds of consistency simplicity robustness and coherence

the selection of prior distributions by formal rules abstract subjectivism has become the dominant philosophical foundation for bayesian inference yet in practice most bayesian analyses are performed with so called noninformative priors that is priors constructed by some formal rule we review the plethora of techniques for constructing such priors and discuss some of the practical and philosophical issues that arise when they are used we give special emphasis to jeffreys s rules and discuss the evolution of his viewpoint about the interpretation of priors away from unique representation of ignorance toward the notion that they should be chosen by convention we conclude that the problems raised by the research on priors chosen by formal rules are serious and may not be dismissed lightly when sample sizes are small relative to the number of parameters being estimated it is dangerous to put faith in any default solution but when asymptotics take over jeffreys s rules and their variants remain reasonable choices we also provide an annotated b

a bayesian approach to finite mixture models in bioassay via data augmentation and gibbs sampling and its application to insecticide resistance after continued treatment with an insecticide within the population of the susceptible insects resistant strains will occur it is important to know whether there are any resistant strains what the proportions are and what the median lethal doses are for the insecticide lwin and martin 1989 biometrics 45 721 732 propose a probit mixture model and use the em algorithm to obtain the maximum likelihood estimates for the parameters this approach has difficulties in estimating the confidence intervals and in testing the number of components we propose a bayesian approach to obtaining the credible intervals for the location and scale of the tolerances in each component and for the mixture proportions by using data augmentation and gibbs sampler we use bayes factor for model selection and determining the number of components we illustrate the method with data published in lwin and martin 1989

scale parameters a bayesian treatment a definition of a scale parameter is given which includes the usual definition of a scale parameter in the exponential family some suggestions for the estimation of and approximations to the marginal posterior distribution of a vector valued parameter of primary interest are made in particular the possibility of a 2 approximation to the posterior distribution of leading to h p d regions for based on the f distribution is explored two examples are discussed where observations are from a normal and gamma distribution

model selection and accounting for model uncertainty in graphical models using occam s window abstract we consider the problem of model selection and accounting for model uncertainty in high dimensional contingency tables motivated by expert system applications the approach most used currently is a stepwise strategy guided by tests based on approximate asymptotic p values leading to the selection of a single model inference is then conditional on the selected model the sampling properties of such a strategy are complex and the failure to take account of model uncertainty leads to underestimation of uncertainty about quantities of interest in principle a panacea is provided by the standard bayesian formalism that averages the posterior distributions of the quantity of interest under each of the models weighted by their posterior model probabilities furthermore this approach is optimal in the sense of maximizing predictive ability but this has not been used in practice because computing the posterior model probabilities is hard and the number of models is very large often greater than 1

accurate approximations for posterior moments and marginal densities abstract this article describes approximations to the posterior means and variances of positive functions of a real or vector valued parameter and to the marginal posterior densities of arbitrary i e not necessarily positive parameters these approximations can also be used to compute approximate predictive densities to apply the proposed method one only needs to be able to maximize slightly modified likelihood functions and to evaluate the observed information at the maxima nevertheless the resulting approximations are generally as accurate and in some cases more accurate than approximations based on third order expansions of the likelihood and requiring the evaluation of third derivatives the approximate marginal posterior densities behave very much like saddle point approximations for sampling distributions the principal regularity condition required is that the likelihood times prior be unimodal

mixtures of g priors in generalized linear models abstractmixtures of zellner s g priors have been studied extensively in linear models and have been shown to have numerous desirable properties for bayesian variable selection and model averaging

bayesian parameter inference for dynamic infectious disease modelling rotavirus in germany understanding infectious disease dynamics using epidemic models based on ordinary differential equations requires the calibration of model parameters from data a commonly used approach in practice to simplify this task is to fix many parameters on the basis of expert or literature information however this not only leaves the corresponding uncertainty unexamined but often also leads to biased inference for the remaining parameters because of dependence structures inherent in any given model in the present work we develop a bayesian inference framework that lessens the reliance on such external parameter quantifications by pursuing a more data driven calibration approach this includes a novel focus on residual autocorrelation combined with model averaging techniques in order to reduce these estimates dependence on the underlying model structure we applied our methods to the modelling of age stratified weekly rotavirus incidence data in germany from 2001 to 2008 using a complex susceptible infectious susceptible type model complemented by the stochastic reporting of new cases as a result we found the detection rate in the eastern federal states to be more than four times higher compared with that of the western federal states 19 0 vs 4 3 and also the infectiousness of symptomatically infected individuals was estimated to be more than 10 times higher than that of asymptomatically infected individuals 95 credibility interval 8 1 19 6 not only do these findings give valuable epidemiological insight into the transmission processes we were also able to examine the considerable impact on the model predicted transmission dynamics when fixing parameters beforehand

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12