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

hyper markov laws in the statistical analysis of decomposable graphical models this paper introduces and investigates the notion of a hyper markov law which is a probability distribution over the set of probability measures on a multivariate space that i is concentrated on the set of markov probabilities over some decomposable graph and ii satisfies certain conditional independence restrictions related to that graph a stronger version of this hyper markov property is also studied our analysis starts by reconsidering the properties of markov probabilities using an abstract approach which thereafter proves equally applicable to the hyper markov case next it is shown constructively that hyper markov laws exist that they appear as sampling distributions of maximum likelihood estimators in decomposable graphical models and also that they form natural conjugate prior distributions for a bayesian analysis of these models as examples we construct a range of specific hyper markov laws including the hyper multinomial hyper dirichlet and the hyper wishart and inverse wishart laws these laws occur naturally in connection with the analysis of decomposable log linear and covariance selection models

asymptotic normality with small relative errors of posterior probabilities of half spaces some examples of half spaces of interest in parameter spaces are in a clinical trial of a treatment versus placebo the half spaces where the treatment is a helpful or b harmful thus one may not only want to test the hypothesis that the treatment c makes no difference but to assign posterior probabilities to a b and c under conditions as unrestrictive as possible on the choice of prior probabilities e g dudley and haughton 2001 more generally we have in mind applications to model selections as in the bic criterion of schwarz 1978 and its extensions poskitt 1987 haughton 1988 specifically to one sided models and multiple data sets dudley and haughton 1997

model comparison of coordinate free multivariate skewed distributions with an application to stochastic frontiers we consider classes of multivariate distributions which can model skewness and are closed under orthogonal transformations we review two classes of such distributions proposed in the literature and focus our attention on a particular yet quite flexible subclass of one of these classes members of this subclass are defined by affine transformations of univariate skewed distributions that ensure the existence of a set of coordinate axes along which there is independence and the marginals are known analytically the choice of an appropriate m dimensional skewed distribution is then restricted to the simpler problem of choosing m univariate skewed distributions we introduce a bayesian model comparison setup for selection of these univariate skewed distributions the analysis does not rely on the existence of moments allowing for any tail behaviour and uses equivalent priors on the common characteristics of the different models finally we apply this framework to multi output stochastic frontiers using data from dutch dairy farms

bayesian hypothesis testing in latent variable models hypothesis testing using bayes factors bfs is known not to be well defined under the improper prior in the context of latent variable models an additional problem with bfs is that they are difficult to compute in this paper a new bayesian method based on the decision theory and the em algorithm is introduced to test a point hypothesis in latent variable models the new statistic is a by product of the bayesian mcmc output and hence easy to compute it is shown that the new statistic is appropriately defined under improper priors because the method employs a continuous loss function in addition it is easy to interpret the method is illustrated using a one factor asset pricing model and a stochastic volatility model with jumps

generalization of jeffreys divergence based priors for bayesian hypothesis testing summary we introduce objective proper prior distributions for hypothesis testing and model selection based on measures of divergence between the competing models we call them diver gence based db priors db priors have simple forms and desirable properties like information finite sample consistency and are often similar to other existing proposals like intrinsic priors moreover in normal linear model scenarios they reproduce the jeffreys zellner siow priors exactly most importantly in challenging scenarios such as irregular models and mixture mod els db priors are well defined and very reasonable whereas alternative proposals are not we derive approximations to the db priors as well as markov chain monte carlo and asymptotic expressions for the associated bayes factors

a flexible approach to parametric inference in nonlinear and time varying time series models many structural break and regime switching models have been used with macroeconomic and financial data in this paper we develop an extremely flexible modeling approach which can accommodate virtually any of these specifications we build on earlier work showing the relationship between flexible functional forms and random variation in parameters our contribution is based around the use of priors on the time variation that is developed from considering a hypothetical reordering of the data and distance between neighboring reordered observations the range of priors produced in this way can accommodate a wide variety of nonlinear time series models including those with regime switching and structural breaks by allowing the amount of random variation in parameters to depend on the distance between reordered observations the parameters can evolve in a wide variety of ways allowing for everything from models exhibiting abrupt change e g threshold autoregressive models or standard structural break models to those which allow for a gradual evolution of parameters e g smooth transition autoregressive models or time varying parameter models bayesian econometric methods for inference are developed for estimating the distance function and types of hypothetical reordering conditional on a hypothetical reordering and distance function a simple reordering of the actual data allows us to estimate our models with standard state space methods by a simple adjustment to the measurement equation we use artificial data to show the advantages of our approach before providing two empirical illustrations involving the modeling of real gdp growth

objective bayesian comparison of constrained analysis of variance models in the social sciences we are often interested in comparing models specified by parametric equality or inequality constraints for instance when examining three group means mu 1 mu 2 mu 3 through an analysis of variance anova a model may specify that mu 1 mu 2 mu 3 while another one may state that mu 1 mu 3 mu 2 and finally a third model may instead suggest that all means are unrestricted this is a challenging problem because it involves a combination of nonnested models as well as nested models having the same dimension we adopt an objective bayesian approach requiring no prior specification from the user and derive the posterior probability of each model under consideration our method is based on the intrinsic prior methodology suitably modified to accommodate equality and inequality constraints focussing on normal anova models a comparative assessment is carried out through simulation studies we also present an application to real data collected in a psychological experiment

posterior predictive p values extending work of rubin this paper explores a bayesian counterpart of the classical p value namely a tail area probability of a test statistic under a null hypothesis the bayesian formulation using posterior predictive replications of the data allows a test statistic to depend on both data and unknown nuisance parameters and thus permits a direct measure of the discrepancy between sample and population quantities the tail area probability for a test statistic is then found under the joint posterior distribution of replicate data and the nuisance parameters both conditional on the null hypothesis this posterior predictive p value can also be viewed as the posterior mean of a classical p value averaging over the posterior distribution of nuisance parameters under the null hypothesis and thus it provides one general method for dealing with nuisance parameters two classical examples including the behrens fisher problem are used to illustrate the posterior predictive p value and some of its interesting properties which also reveal a new bayesian interpretation for some classical p values an application to multiple imputation inference is also presented a frequency evaluation shows that in general if the replication is defined by new nuisance parameters and new data then the type i frequentist error of an level posterior predictive test is often close to but less than and will never exceed 2

on priors and bayes factors abstract by analogy to the authors measure of the influence of observations on bayes factors a new measure is proposed to determine the influence of a prior distribution on a bayes factor this is developed for linear models the method is illustrated for a consumption model using houthakker s international income and consumption data

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

the intrinsic bayes factor for model selection and prediction abstract in the bayesian approach to model selection or hypothesis testing with models or hypotheses of differing dimensions it is typically not possible to utilize standard noninformative or default prior distributions this has led bayesians to use conventional proper prior distributions or crude approximations to bayes factors in this article we introduce a new criterion called the intrinsic bayes factor which is fully automatic in the sense of requiring only standard noninformative priors for its computation and yet seems to correspond to very reasonable actual bayes factors the criterion can be used for nested or nonnested models and for multiple model comparison and prediction from another perspective the development suggests a general definition of a reference prior for model comparison

bayes and likelihood calculations from confidence intervals recently there has been considerable progress on setting good approximate confidence intervals for a single parameter in a multi parameter family here we use these frequentist results as a convenient device for making bayes empirical bayes and likelihood inferences about a simple formula is given that produces an approximate likelihood function l x for with all nuisance parameters eliminated based on any system of approximate confidence intervals the statistician can then modify l x with bayes or empirical bayes information for without worrying about nuisance parameters

simultaneous bayesian frequentist sequential testing of nested hypotheses summary conditional frequentist tests of a precise hypothesis versus a composite alternative have recently been developed and have been shown to be equivalent to conventional bayes tests in the very strong sense that the reported frequentist error probabilities equal the posterior probabilities of the hypotheses these results are herein extended to sequential testing and yield fully frequentist sequential tests that are considerably easier to use than are conventional sequential tests among the interesting properties of these new tests is the lack of dependence of the reported error probabilities on the stopping rule seeming to lend frequentist support to the stopping rule principle

automatic bayes factors for testing equality and inequality constrained hypotheses on variances in comparing characteristics of independent populations researchers frequently expect a certain structure of the population variances these expectations can be formulated as hypotheses with equality and or inequality constraints on the variances in this article we consider the bayes factor for testing such in equality constrained hypotheses on variances application of bayes factors requires specification of a prior under every hypothesis to be tested however specifying subjective priors for variances based on prior information is a difficult task we therefore consider so called automatic or default bayes factors these methods avoid the need for the user to specify priors by using information from the sample data we present three automatic bayes factors for testing variances the first is a bayes factor with equal priors on all variances where the priors are specified automatically using a small share of the information in the sample data the second is the fractional bayes factor where a fraction of the likelihood is used for automatic prior specification the third is an adjustment of the fractional bayes factor such that the parsimony of inequality constrained hypotheses is properly taken into account the bayes factors are evaluated by investigating different properties such as information consistency and large sample consistency based on this evaluation it is concluded that the adjusted fractional bayes factor is generally recommendable for testing equality and inequality constrained hypotheses on variances

discreteness of spectrum and positivity criteria for schrodinger operators we provide a class of necessary and sufficient conditions for the discreteness of spectrum of schrodinger operators with scalar potentials which are semibounded below the classical discreteness of spectrum criterion by a m molchanov 1953 uses a notion of negligible set in a cube as a set whose wiener capacity is less than a small constant times the capacity of the cube we prove that this constant can be taken arbitrarily between 0 and 1 this solves a problem formulated by i m gelfand in 1953 moreover we extend the notion of negligibility by allowing the constant to depend on the size of the cube we give a complete description of all negligibility conditions of this kind the a priori equivalence of our conditions involving different negligibility classes is a nontrivial property of the capacity we also establish similar strict positivity criteria for the schrodinger operators with nonnegative potentials

bayesian model comparison with the hyvarinen score computation and consistency the bayes factor is a widely used criterion in model comparison and its logarithm is a difference of out of sample predictive scores under the logarithmic scoring rule however when some of the ca

multiple bayes factors for testing hypotheses abstract partial and multiple bayes factors are introduced to obtain pairwise comparisons of hypotheses in a statistical experiment with a partition on the parameter space robust bayesian analyses are performed by introducing suitable classes of priors and by calculating lower and upper bounds of bayes factors and posterior probabilities classes of intuitively meaningful priors are introduced including unimodal densities without the constraint of symmetry for the case of precise hypotheses procedures for the corresponding optimizations are specified and examples are given

discreteness of spectrum criteria for schrodinger operators with non negative polynomial or homogeneous potentials abstract we consider schrodinger operators v defined in l 2 r n with non negative potentials v which are either polynomials or homogeneous functions we provide new yet simple necessary and sufficient conditions for the discreteness of spectrum of such schrodinger operators

bayesian inference for the common location parameter of several shifted exponential populations abstract consider k random samples which are independently drawn from k shifted exponential distributions with respective scale parameters 1 2 k and common location parameter on the basis of the given samples and in a bayesian framework we address the problem of point and interval estimation of the location parameter under the conjugate priors which are usually proper priors moreover we also address the problem of testing the equality of the location parameters we propose bayesian hypothesis testing procedures for the equality of the location parameters under the noninformative prior the noninformative prior is usually improper which yields a calibration problem that makes the bayes factor to be defined up to a multiplicative constant so we propose the default bayesian hypothesis testing procedures based on the fractional bayes factor and the intrinsic bayes factors under the reference priors our proposed bayesian procedures are compared and contrasted via a comparison study a simulation study and a real world data analysis to the existing classical exact procedures proposed by tippett tippett s method tippett 1931 fisher fisher s method fisher 1932 stouffer inverse normal method stouffer et al 1949 and george logit method george 1977 and to the generalized variable procedures proposed by tsui and weerahandi generalized p value method tsui and weerahandi 1989

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

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

bayes factors for outlier models using the device of imaginary observations abstract suppose we think that most observations in a sample have been generated from a distribution with density f x but we fear that a few outliers from a distribution with density g x may have contaminated our sample in many situations we might assume that f x is a density depending on a parameter and that g x is of the same form as f but with parameter or a number of bayesian models for this problem when f is normal have been discussed by freeman he points out that with a vague improper prior for contaminating parameters most posterior weight is put on the model allowing for the largest number of outliers he therefore confines attention to proper priors when trying to answer the question of how many outliers however in many situations we do not have very certain information on the contaminating parameters and would like to make inferences about outliers when using improper priors for the parameters of the model in this article we apply the ideas of spiegelhalter and smith to

variable selection and function estimation in additive nonparametric regression using a data based prior abstract a hierarchical bayesian approach is proposed for variable selection and function estimation in additive nonparametric gaussian regression models and additive nonparametric binary regression models the prior for each component function is an integrated wiener process resulting in a posterior mean estimate that is a cubic smoothing spline each of the explanatory variables is allowed to be in or out of the model and the regression functions are estimated by model averaging to allow variable selection and model averaging data based priors are used for the smoothing parameter and the slope at 0 of each component function a two step markov chain monte carlo method is used to efficiently obtain the data based prior and to carry out variable selection and function estimation it is shown by simulation that significant improvements in the function estimators can be obtained over an approach that estimates all the unknown functions simultaneously the methodology is illustrated for a binary regressio

bayesian estimation of a multilevel irt model using gibbs sampling in this article a two level regression model is imposed on the ability parameters in an item response theory irt model the advantage of using latent rather than observed scores as dependent variables of a multilevel model is that it offers the possibility of separating the influence of item difficulty and ability level and modeling response variation and measurement error another advantage is that contrary to observed scores latent scores are test independent which offers the possibility of using results from different tests in one analysis where the parameters of the irt model and the multilevel model can be concurrently estimated the two parameter normal ogive model is used for the irt measurement model it will be shown that the parameters of the two parameter normal ogive model and the multilevel model can be estimated in a bayesian framework using gibbs sampling examples using simulated and real data are given

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

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

on the use of non local prior densities in bayesian hypothesis tests we examine philosophical problems and sampling deficiencies that are associated with current bayesian hypothesis testing methodology paying particular attention to objective bayes methodology because the prior densities that are used to define alternative hypotheses in many bayesian tests assign non negligible probability to regions of the parameter space that are consistent with null hypotheses resulting tests provide exponential accumulation of evidence in favour of true alternative hypotheses but only sublinear accumulation of evidence in favour of true null hypotheses thus it is often impossible for such tests to provide strong evidence in favour of a true null hypothesis even when moderately large sample sizes have been obtained we review asymptotic convergence rates of bayes factors in testing precise null hypotheses and propose two new classes of prior densities that ameliorate the imbalance in convergence rates that is inherited by most bayesian tests using members of these classes we obtain analytic expressions for bayes factors in linear models and derive approximations to bayes factors in large sample settings

bayesian model selection analysis of a survival model with a surviving fraction we describe a methodology for model comparison in a bayesian framework as applied to survival with a surviving fraction this is illustrated using a case study of a randomized and controlled clinical trial investigating time until recurrence of depression posterior distributions are simulated using metropolis within gibbs markov chain methods models reflecting the effects of covariates on the log odds of being in the surviving fraction the log of the hazard rate as well as both and neither are compared bayes factors for comparing the models are obtained by using the bridge sampling method of calculating normalizing constants

robust bayesian inference via coarsening the standard approach to bayesian inference is based on the assumption that the distribution of the data belongs to the chosen model class however even a small violation of this assumption can ha

a point interpolation meshless method based on radial basis functions a point interpolation meshless method is proposed based on combining radial and polynomial basis functions involvement of radial basis functions overcomes possible singularity associated with the meshless methods based on only the polynomial basis this non singularity is useful in constructing well performed shape functions furthermore the interpolation function obtained passes through all scattered points in an influence domain and thus shape functions are of delta function property this makes the implementation of essential boundary conditions much easier than the meshless methods based on the moving least squares approximation in addition the partial derivatives of shape functions are easily obtained thus improving computational efficiency examples on curve surface fittings and solid mechanics problems show that the accuracy and convergence rate of the present method is high copyright 2002 john wiley sons ltd

training samples in objective bayesian model selection central to several objective approaches to bayesian model selection is the use of training samples subsets of the data so as to allow utilization of improper objective priors the most common prescription for choosing training samples is to choose them to be as small as possible subject to yielding proper posteriors these are called minimal training samples when data can vary widely in terms of either information content or impact on the improper priors use of minimal training samples can be inadequate important examples include certain cases of discrete data the presence of censored observations and certain situations involving linear models and explanatory variables such situations require more sophisticated methods of choosing training samples a variety of such methods are developed in this paper and successfully applied in challenging situations

bayesian analysis of a poisson process with a change point summary a bayesian approach to estimation and hypothesis testing for a poisson process with a change point is developed and an example given

a simultaneous estimation and variable selection rule a new data based method of estimation and variable selection in linear statistical models is proposed this method is based on a generalized maximum entropy formalism and makes use of both sample and non sample information in determining a basis for coefficient shrinkage and extraneous variable identification in contrast to tradition shrinkage and variable selection are achieved on a coordinate by coordinate basis and the procedure works well for both ill and well posed statistical models analytical asymptotic results are presented and sampling experiments are used as a basis for determining finite sample behavior and comparing the sampling performance of the new estimation rule with traditional competitors solution algorithms for the non linear inversion problem that results are simple to implement

smoothed particle electromagnetics a mesh free solver for transients in this paper an advanced mesh free particle method for electromagnetic transient analysis is presented the aim is to obtain efficient simulations by avoiding the use of a mesh such as in the most popular grid based numerical methods the basic idea is to obtain numerical solutions for partial differential equations describing the electromagnetic problem by using a set of particles arbitrarily placed in the problem domain the mesh free smoothed particle hydrodynamics method has been adopted co obtain numerical solution of time domain maxwell s curl equations an explicit finite difference scheme has been employed for time integration details about the numerical treatment of electromagnetic vector fields components are discussed two case studies in one and in two dimensions are reported in order to validate the new proposed methodology named as smoothed particle electromagnetics a comparison with the standard finite difference time domain method results is performed the intrinsic adaptive capability of the proposed method has been exploited by introducing irregular particles distribution

practical bayesian density estimation using mixtures of normals abstract mixtures of normals provide a flexible model for estimating densities in a bayesian framework there are some difficulties with this model however first standard reference priors yield improper posteriors second the posterior for the number of components in the mixture is not well defined if the reference prior is used third posterior simulation does not provide a direct estimate of the posterior for the number of components we present some practical methods for coping with these problems finally we give some results on the consistency of the method when the maximum number of components is allowed to grow with the sample size

a new approach to bayesian hypothesis testing abstract in this paper a new bayesian approach is proposed to test a point null hypothesis based on the deviance in a decision theoretical framework the proposed test statistic may be regarded as the bayesian version of the likelihood ratio test and appeals in practical applications with three desirable properties first it is immune to jeffreys concern about the use of improper priors second it avoids jeffreys lindley s paradox third it is easy to compute and its threshold value is easily derived facilitating the implementation in practice the method is illustrated using some real examples in economics and finance it is found that the leverage effect is insignificant in an exchange time series and that the fama french three factor model is rejected

nonparametric seemingly unrelated regression abstract a method is presented for simultaneously estimating a system of nonparametric regressions which may seem unrelated but where the errors are potentially correlated between equations we show that the advantage of estimating such a seemingly unrelated system of nonparametric regressions is that less observations can be required to obtain reliable function estimates than if each of the regression equations is estimated separately and the correlation ignored this increase in efficiency is investigated empirically using both simulated and real data the method uses a bayesian hierarchical framework where each regression function is represented as a linear combination of a large number of basis terms all the regression coefficients and the variance matrix of the errors are estimated simultaneously by their posterior means the computation is carried out using a markov chain monte carlo sampling scheme that employs a focused sampling step to combat the high dimensional representation of the unknown regression functions the methodology extends easily to other nonparametric multivariate regression models

bayesianly justifiable and relevant frequency calculations for the applied statistician a common reaction among applied statisticians is that the bayesian statistician s energies in an applied problem must be directed at the a priori elicitation of one model specification from which an optimal design and all inferences follow automatically by applying bayes s theorem to calculate conditional distributions of unknowns given knowns i feel however that the applied bayesian statistician s tool kit should be more extensive and include tools that may be usefully labeled frequency calculations three types of bayesianly justifiable and relevant frequency calculations are presented using examples to convey their use for the applied statistician

schrodinger operators in the twentieth century this paper reviews the past fifty years of work on spectral theory and related issues in nonrelativistic quantum mechanics

a general approach for the stability analysis of the time domain finite element method for electromagnetic simulations this paper presents a general approach for the stability analysis of the time domain finite element method tdfem for electromagnetic simulations derived from the discrete system analysis the approach determines the stability by analyzing the root locus map of a characteristic equation and evaluating the spectral radius of the finite element system matrix the approach is applicable to the tdfem simulation involving dispersive media and to various temporal discretization schemes such as the central difference forward difference backward difference and newmark methods it is shown that the stability of the tdfem is determined by the material property and by the temporal and spatial discretization schemes the proposed approach is applied to a variety of tdfem schemes which include 1 time domain finite element modeling of dispersive media 2 time domain finite element boundary integral method 3 higher order tdfem and 4 orthogonal tdfem numerical results demonstrate the validity of the proposed approach for stability analysis

objective bayesian model selection in gaussian graphical models summary this paper presents a default model selection procedure for gaussian graphical models that involves two new developments first we develop a default version of the hyper inverse wishart prior for restricted covariance matrices called the hyper inverse wishart g prior and show how it corresponds to the implied fractional prior for selecting a graph using fractional bayes factors second we apply a class of priors that automatically handles the problem of multiple hypothesis testing we demonstrate our methods on a variety of simulated examples concluding with a real example analyzing covariation in mutual fund returns these studies reveal that the combined use of a multiplicity correction prior on graphs and fractional bayes factors for computing marginal likelihoods yields better performance than existing bayesian methods

bayesian model choice asymptotics and exact calculations abstract model determination is a fundamental data analytic task here we consider the problem of choosing amongst a finite with loss of generality we assume two set of models after briefly reviewing classical and bayesian model choice strategies we present a general predictive density which includes all proposed bayesian approaches we are aware of using laplace approximations we can conveniently assess and compare asymptotic behavior of these approaches concern regarding the accuracy of these approximation for small to moderate sample sizes encourages the use of monte carlo techniques to carry out exact calculations a data set fit with nested non linear models enables comparison between proposals and between exact and asymptotic values

empirical bayes methods for combining likelihoods abstract suppose that several independent experiments are observed each one yielding a likelihood l k k for a real valued parameter of interest k for example k might be the log odds ratio for a 2 2 table relating to the kth population in a series of medical experiments this article concerns the following empirical bayes question how can we combine all of the likelihoods l k to get an interval estimate for any one of the k s say 1 the results are presented in the form of a realistic computational scheme that allows model building and model checking in the spirit of a regression analysis no special mathematical forms are required for the priors or the likelihoods this scheme is designed to take advantage of recent methods that produce approximate numerical likelihoods l k k even in very complicated situations with all nuisance parameters eliminated the empirical bayes likelihood theory is extended to situations where the k s have a regression structure as well as an empiri

variable selection and functional form uncertainty in cross country growth regressions regression analyses of cross country economic growth data are complicated by two main forms of model uncertainty the uncertainty in selecting explanatory variables and the uncertainty in specifying the functional form of the regression function most discussions in the literature address these problems independently yet a joint treatment is essential we present a new framework that makes such a joint treatment possible using flexible nonlinear models specified by gaussian process priors and addressing the variable selection problem by means of bayesian model averaging using this framework we extend the linear model to allow for parameter heterogeneity of the type suggested by new growth theory while taking into account the uncertainty in selecting explanatory variables controlling for variable selection uncertainty we confirm the evidence in favor of parameter heterogeneity presented in several earlier studies however controlling for functional form uncertainty we find that the effects of many of the explanatory variables identified in the literature are not robust across countries and variable selections

optimal predictive model selection often the goal of model selection is to choose a model for future prediction and it is natural to measure the accuracy of a future prediction by squared error loss under the bayesian approach it is commonly perceived that the optimal predictive model is the model with highest posterior probability but this is not necessarily the case in this paper we show that for selection among normal linear models the optimal predictive model is often the median probability model which is defined as the model consisting of those variables which have overall posterior probability greater than or equal to 1 2 of being in a model the median probability model often differs from the highest probability model

default bayes factors for nonnested hypothesis testing abstract bayesian hypothesis testing for nonnested hypotheses is studied using various default bayes factors such as the fractional bayes factor the median intrinsic bayes factor and the encompassing and expected intrinsic bayes factors the different default methods are first compared with each other and with the p value in normal one sided testing to illustrate the basic issues general results for one sided testing in location and scale models are then presented the default bayes factors are also studied for specific models involving multiple hypotheses in particular a multiple hypothesis testing example involving a sequential clinical trial is discussed in most of the examples presented we also derive the intrinsic prior this is the prior distribution which if used directly would yield answers asymptotically equivalent to those for the given default bayes factor

a time domain collocation meshless method with local radial basis functions for electromagnetic transient analysis a meshless method with local radial basis functions is proposed for solving the time domain electromagnetic wave equations in comparison with the conventional radial point interpolation meshless rpim method that employs and positions dual sets of nodes of both electric and magnetic field nodes the proposed method uses only one set of the nodes electric field nodes where electric fields are also collocated in space with this feature implementation complexity of the rpim method is significantly reduced and conformal modeling and muti scale capabilities of the rpim method can now be further explored with higher efficiency the time marching formulations of the proposed method are derived and stability analysis of the method is presented comparisons of the proposed method with the conventional meshless method are also presented the accuracy and efficiency of the proposed method are demonstrated through simulation of an h shaped cavity and a quarter ring resonator

implementation of material interface conditions in the radial point interpolation meshless method we propose a systematic approach to accurate imposition of material interface conditions for the meshless radial point interpolation method rpim a new set of equations for updating fields at interfaces is derived a piecewise polynomial and a modified radial basis function are proposed and applied to account for field discontinuities near the interfaces in contrast to the previous work the proposed approach incorporates material properties and reduces geometrical dependence of the interpolation function numerical results show that the proposed approach significantly improves the simulation accuracy of the rpim at a cost of a small overhead in computational time