comparing the performance of two indices for spatial model selection application to two mortality data the statistical analysis of spatially correlated data has become an important scientific research topic lately the analysis of the mortality or morbidity rates observed at different areas may help to decide if people living in certain locations are considered at higher risk than others once the statistical model for the data of interest has been chosen further effort can be devoted to identifying the areas under higher risks many scientists including statisticians have tried the conditional autoregressive car model to describe the spatial autocorrelation among the observed data this model has greater smoothing effect than the exchangeable models such as the poisson gamma model for spatial data this paper focuses on comparing the two types of models using the index lg the ratio of local to global variability two applications taiwan asthma mortality and scotland lip cancer are considered and the use of lg is illustrated the estimated values for both data sets are small implying a poisson gamma model may be favoured over the car model we discuss the implications for the two applications respectively to evaluate the performance of the index lg we also compute the bayes factor a bayesian model selection criterion to see which model is preferred for the two applications and simulation data to derive the value of lg we estimate its posterior mode based on samples derived from the bugs program while for bayes factor we use the double laplace metropolis method schwarz criterion and a modified harmonic mean for approximations the results of lg and bayes factor are consistent we conclude that lg is fairly accurate as an index for selection between poisson gamma and car model when easy and fast computation is of concern we recommend using lg as the first and less costly index

robust linear static panel data models using e contamination abstract the paper develops a general bayesian framework for robust linear static panel data models using e contamination a two step approach is employed to derive the conditional type ii maximum likelihood ml ii posterior distribution of the coefficients and individual effects the ml ii posterior means are weighted averages of the bayes estimator under a base prior and the data dependent empirical bayes estimator two stage and three stage hierarchy estimators are developed and their finite sample performance is investigated through a series of monte carlo experiments these include standard random effects as well as mundlak type chamberlain type and hausman taylor type models the simulation results underscore the relatively good performance of the three stage hierarchy estimator within a single theoretical framework our bayesian approach encompasses a variety of specifications while conventional methods require separate estimators for each case

variable selection in bayesian smoothing spline anova models a bayesian nonparametric regression model for curve fitting and variable selection is proposed for analyzing complex computer model output the smoothing splines anova framework is used to separate the regression function into main effects and interacti

decision analysis by augmented probability simulation we provide a generic monte carlo method to find the alternative of maximum expected utility in a decision analysis we define an artificial distribution on the product space of alternatives and states and show that the optimal alternative is the mode of the implied marginal distribution on the alternatives after drawing a sample from the artificial distribution we may use exploratory data analysis tools to approximately identify the optimal alternative we illustrate our method for some important types of influence diagrams

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

bayes factor covariance testing in item response models two marginal one parameter item response theory models are introduced by integrating out the latent variable or random item parameter it is shown that both marginal response models are multivariate probit models with a compound symmetry covariance structure several common hypotheses concerning the underlying covariance structure are evaluated using fractional bayes factor tests the support for a unidimensional factor i e assumption of local independence and differential item functioning are evaluated by testing the covariance components the posterior distribution of common covariance components is obtained in closed form by transforming latent responses with an orthogonal helmert matrix this posterior distribution is defined as a shifted inverse gamma thereby introducing a default prior and a balanced prior distribution based on that an mcmc algorithm is described to estimate all model parameters and to compute fractional bayes factor tests simulation studies are used to show that the fractional bayes factor tests have good properties for testing the underlying covariance structure of binary response data the method is illustrated with two real data studies

decision network polynomials and the sensitivity of decision support models abstract decision makers benefit from the utilization of decision support models in several applications obtaining managerial insights is essential to better inform the decision process this work offers an in depth investigation into the structural properties of decision support models we show that the input output mapping in influence diagrams decision trees and decision networks is piecewise multilinear the conditions under which sensitivity information cannot be extracted through differentiation are examined in detail by complementing high order derivatives with finite change sensitivity indices we obtain a systematic approach that allows analysts to gain a wide range of managerial insights a well known case study in the medical sector illustrates the findings

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

comparing and weighting imperfect models using d probabilities abstractwe propose a new approach for assigning weights to models using a divergence based method d probabilities relying on evaluating parametric models relative to a nonparametric bayesian ref

valuation based systems for bayesian decision analysis this paper proposes a new method for representing and solving bayesian decision problems the representation is called a valuation based system and has some similarities to influence diagrams however unlike influence diagrams which emphasize conditional independence among random variables valuation based systems emphasize factorizations of joint probability distributions also whereas influence diagram representation allows only conditional probabilities valuation based system representation allows all probabilities the solution method is a hybrid of local computational methods for the computation of marginals of joint probability distributions and the local computational methods for discrete optimization problems we briefly compare our representation and solution methods to those of influence diagrams

gaussian influence diagrams an influence diagram is a network representation of probabilistic inference and decision analysis models the nodes correspond to variables that can be either constants uncertain quantities decisions or objectives the arcs reveal probabilistic dependence of the uncertain quantities and information available at the time of the decisions the influence diagram focuses attention on relationships among the variables as a result it is increasingly popular for eliciting and communicating the structure of a decision or probabilistic model this paper develops the framework for assessment and analysis of linear quadratic gaussian models within the influence diagram representation the gaussian influence diagram exploits conditional independence in a model to simplify elicitation of parameters for the multivariate normal distribution it is straightforward to assess and maintain a positive semi definite covariance matrix problems of inference and decision making can be analyzed using simple transformations to the assessed model and these procedures have attractive numerical properties algorithms are also provided to translate between the gaussian influence diagram and covariance matrix representations for the normal distribution

laplace approximations for posterior expectations when the mode occurs at the boundary of the parameter space abstract this article gives asymptotic expansions for posterior expectations when the mode is on the boundary of the parameter space the idea based on the divergence theorem is to reduce the high dimensional integrals over the parameters space to surface integrals over the boundary of the parameter space and then apply the usual interior mode laplace method to the latter integrals it is shown that these approximations have second order accuracy the method is illustrated with applications to a two sample binomial problem and a random eflects model

statistical mechanical approaches to models with many poorly known parameters models of biochemical regulation in prokaryotes and eukaryotes typically consisting of a set of first order nonlinear ordinary differential equations have become increasingly popular of late these systems have large numbers of poorly known parameters simplified dynamics and uncertain connectivity three key features of a class of problems we call sloppy models which are shared by many other high dimensional multiparameter nonlinear models we use a statistical ensemble method to study the behavior of these models in order to extract as much useful predictive information as possible from a sloppy model given the available data used to constrain it we discuss numerical challenges that emerge in using the ensemble method for a large system we characterize features of sloppy model parameter fluctuations by various spectral decompositions and find indeed that five parameters can be used to fit an elephant we also find that model entropy is as important to the problem of model choice as model energy is to parameter choice

soft tissue artifact distribution on lower limbs during treadmill gait influence of skin markers location on cluster design segment poses and joint kinematics estimated from skin markers are highly affected by soft tissue artifact sta and its rigid motion component starm while four marker clusters could decrease the sta non rigid motion during gait activity other data such as marker location or starm patterns would be crucial to compensate for sta in clinical gait analysis the present study proposed 1 to devise a comprehensive average map illustrating the spatial distribution of sta for the lower limb during treadmill gait and 2 to analyze starm from four marker clusters assigned to areas extracted from spatial distribution all experiments were realized using a stereophotogrammetric system to track the skin markers and a bi plane fluoroscopic system to track the knee prosthesis computation of the spatial distribution of sta was realized on 19 subjects using 80 markers apposed on the lower limb three different areas were extracted from the distribution map of the thigh the marker displacement reached a maximum of 24 9 mm and 15 3 mm in the proximal areas of thigh and shank respectively starm was larger on thigh than the shank with rms error in cluster orientations between 1 2 and 8 1 the translation rms errors were also large 3 0 mm to 16 2 mm no marker cluster correctly compensated for starm however the coefficient of multiple correlations exhibited excellent scores between skin and bone kinematics as well as for starm between subjects these correlations highlight dependencies between starm and the kinematic components this study provides new insights for modeling starm for gait activity

a bayesian information criterion for singular models summary we consider approximate bayesian model choice for model selection problems that involve models whose fisher information matrices may fail to be invertible along other competing submodels such singular models do not obey the regularity conditions underlying the derivation of schwarz s bayesian information criterion bic and the penalty structure in bic generally does not reflect the frequentist large sample behaviour of the marginal likelihood although large sample theory for the marginal likelihood of singular models has been developed recently the resulting approximations depend on the true parameter value and lead to a paradox of circular reasoning guided by examples such as determining the number of components in mixture models the number of factors in latent factor models or the rank in reduced rank regression we propose a resolution to this paradox and give a practical extension of bic for singular model selection problems

criteria for bayesian model choice with application to variable selection in objective bayesian model selection no single criterion has emerged as dominant in defining objective prior distributions indeed many criteria have been separately proposed and utilized to propose differing prior choices we first formalize the most general and compelling of the various criteria that have been suggested together with a new criterion we then illustrate the potential of these criteria in determining objective model selection priors by considering their application to the problem of variable selection in normal linear models this results in a new model selection objective prior with a number of compelling properties

computing bayes factors by combining simulation and asymptotic approximations abstract the bayes factor is a ratio of two posterior normalizing constants which may be difficult to compute we compare several methods of estimating bayes factors when it is possible to simulate observations from the posterior distributions via markov chain monte carlo or other techniques the methods that we study are all easily applied without consideration of special features of the problem provided that each posterior distribution is well behaved in the sense of having a single dominant mode we consider a simulated version of laplace s method a simulated version of bartlett correction importance sampling and a reciprocal importance sampling technique we also introduce local volume corrections for each of these in addition we apply the bridge sampling method of meng and wong we find that a simulated version of laplace s method with local volume correction furnishes an accurate approximation that is especially useful when likelihood function evaluations are costly a simple bridge sampli

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

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

assessing a mixture model for clustering with the integrated completed likelihood we propose an assessing method of mixture model in a cluster analysis setting with integrated completed likelihood for this purpose the observed data are assigned to unknown clusters using a maximum a posteriori operator then the integrated completed likelihood icl is approximated using the bayesian information criterion bic numerical experiments on simulated and real data of the resulting icl criterion show that it performs well both for choosing a mixture model and a relevant number of clusters in particular icl appears to be more robust than bic to violation of some of the mixture model assumptions and it can select a number of dusters leading to a sensible partitioning of the data

a principal component analysis approach to correcting the knee flexion axis during gait abstract accurate and precise knee flexion axis identification is critical for prescribing and assessing tibial and femoral derotation osteotomies but is highly prone to marker misplacement induced error the purpose of this study was to develop an efficient algorithm for post hoc correction of the knee flexion axis and test its efficacy relative to other established algorithms gait data were collected on twelve healthy subjects using standard marker placement as well as intentionally misplaced lateral knee markers the efficacy of the algorithm was assessed by quantifying the reduction in knee angle errors crosstalk error was quantified from the coefficient of determination r 2 between knee flexion and adduction angles mean rotation offset error o was quantified from the knee and hip rotation kinematics across the gait cycle the principal component analysis pca based algorithm significantly reduced r 2 p o knee to converge toward 11 9 8 0 of external rotation demonstrating improved certainty of the knee kinematics the within subject standard deviation of o hip between marker placements was reduced from 13 5 1 5 to 0 7 0 2 p baker et al 1999 and better than a null space algorithm schwartz and rozumalski 2005 for this healthy subject population

bayes factors and approximations for variance component models abstract in this article we consider tests of variance components using bayes factors such tests arise in many fields of application including medicine agriculture and engineering when using bayes factors the choice of prior distribution on the parameter of interest is of great importance we propose a unit information reference method for variance component models the calculation of bayes factors in this context is not straightforward there are well documented difficulties with markov chain monte carlo approaches such as gibbs sampling and the usual laplace approximation is not appropriate due to the boundary null hypothesis we describe both an importance sampling approach and an analytical approximation for calculating the numerator and denominator of the bayes factor the importance sampling approach is straightforward to implement and also forms the basis for a rejection algorithm that allows generation of samples from the posterior distributions under the null and alternative hypotheses

the deviance information criterion 12 years on summary the essentials of our paper of 2002 are briefly summarized and compared with other criteria for model comparison after some comments on the paper s reception and influence we consider criticisms and proposals forimprovement made by us and others

bayesian model selection for partially observed diffusion models summary we present an approach to bayesian model selection for finitely observed diffusion processes we use data augmentation by treating the paths between observed points as missing data for a fixed model formulation the strong dependence between the missing paths and the volatility of the diffusion can be broken down by adopting the method of roberts stramer 2001 we describe how this method may be extended to the case of model selection via reversible jump markov chain monte carlo in addition we extend the formulation of a diffusion model to capture a potential non markov state dependence in the drift issues of appropriate choices of priors and efficient transdimensional proposal distributions for the reversible jump algorithm are also addressed the approach is illustrated using simulated data and an example from finance

evaluation of knee functional calibration with and without the effect of soft tissue artefact functional calibration methods were devised to improve repeatability and accuracy of the knee flexion extension axis which is used to define the medio lateral axis of the femur coordinate system in gait analysis repeatability of functional calibration methods has been studied extensively in healthy individuals but not accuracy in the absence of a benchmark knee axis we captured bi plane fluoroscopy data of the knee joint in 17 subjects with unilateral total knee arthroplasty during treadmill walking the prosthesis provided a benchmark knee axis to evaluate the functional calibration methods stereo photogrammetry data of thigh and shank marker clusters were captured simultaneously to investigate the effect of soft tissue artefact sta three methods were tested the axis transformation technique att finds the best single fixed axis of rotation 2dofknee finds the axis that minimises knee varus valgus and trajajc finds the axis perpendicular to the trajectory in the transverse plane of the femur of a point located on the longitudinal axis of the tibia using fluoroscopy data functional axes formed an angle of less than 2 in the transverse plane with the benchmark axis true internal external range of movement was correlated with decreased accuracy for att while varus valgus range of movement was correlated with decreased accuracy for 2dofknee and trajajc sta had negative impact on accuracy and variability using stereo photogrammetry data the accuracy of 2dofknee was 1 7 sd 5 1 smaller than att 2 9 sd 5 1 but not to trajajc 1 7 sd 5 2 our results confirm that of previous studies which utilised the femur condylar axis as reference

trend and fluctuations analysis and design of population dynamics measurements in replicate ecosystems the dynamical evolution of complex systems is often intrinsically stochastic and subject to external random forces in order to study the resulting variability in dynamics it is essential to make measurements on replicate systems and to separate arbitrary variation of the average dynamics of these replicates from fluctuations around the average dynamics here we do so for population time series data from replicate ecosystems sharing a common average dynamics or common trend we explain how model parameters including the effective interactions between species and dynamical noise can be estimated from the data and how replication reduces errors in these estimates for this it is essential that the model can fit a variety of average dynamics we then show how one can judge the quality of a model compare alternate models and determine which combinations of parameters are poorly determined by the data in addition we show how replicate population dynamics experiments could be designed to optimize the acquired information of interest about the systems our approach is illustrated on a set of time series gathered from replicate microbial closed ecosystems

multistage monte carlo method for solving influence diagrams using local computation the main goal of this paper is to describe a new multistage monte carlo mmc simulation method for solving influence diagrams using local computation global methods have been proposed by others that sample from the joint probability distribution of all the variables in the influence diagram however for influence diagrams having many variables the state space of all variables grows exponentially and the sample sizes required for good estimates may be too large to be practical in this paper we develop a mmc method which samples only a small set of chance variables for each decision node in the influence diagram mmc is akin to methods developed for exact solution of influence diagrams in that we limit the number of chance variables sampled at any time because influence diagrams model each chance variable with a conditional probability distribution the mmc method lends itself well to influence diagram representations

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

bayesian analysis of binary and polychotomous response data abstract a vast literature in statistics biometrics and econometrics is concerned with the analysis of binary and polychotomous response data the classical approach fits a categorical response regression model using maximum likelihood and inferences about the model are based on the associated asymptotic theory the accuracy of classical confidence statements is questionable for small sample sizes in this article exact bayesian methods for modeling categorical response data are developed using the idea of data augmentation the general approach can be summarized as follows the probit regression model for binary outcomes is seen to have an underlying normal regression structure on latent continuous data values of the latent data can be simulated from suitable truncated normal distributions if the latent data are known then the posterior distribution of the parameters can be computed using standard results for normal linear models draws from this posterior are used to sample new latent data and t

mixtures of g priors for bayesian model averaging with economic application this paper examines the issue of variable selection in linear regression modeling where there is a potentially large amount of possible covariates and economic theory offers insufficient guidance on how to select the appropriate subset in this context bayesian model averaging presents a formal bayesian solution to dealing with model uncertainty the main interest here is the effect of the prior on the results such as posterior inclusion probabilities of regressors and predictive performance the authors combine a binomial beta prior on model size with a g prior on the coefficients of each model in addition they assign a hyperprior to g as the choice of g has been found to have a large impact on the results for the prior on g they examine the zellner siow prior and a class of beta shrinkage priors which covers most choices in the recent literature the authors propose a benchmark beta prior inspired by earlier findings with fixed g and show it leads to consistent model selection inference is conducted through a markov chain monte carlo sampler over model space and g the authors examine the performance of the various priors in the context of simulated and real data for the latter they consider two important applications in economics namely cross country growth regression and returns to schooling recommendations for applied users are provided

a new method for estimating joint parameters from motion data joint centers and axes of rotation joint parameters are central to all branches of movement analysis in gait analysis the standard protocol used to determine hip and knee joint parameters is prone to errors arising from palpation anthropometric regression equations and misplaced alignment devices several alternative methods have been proposed but to date none have been shown to be accurate and reliable enough for use in the clinical setting this article describes a new method for joint parameter estimation the new method can be summarized as follows i the motions of two adjacent segments spanning a single joint are tracked ii the axis of rotation between every pair of observed segment configurations is computed iii the most likely intersection of all axes effective joint center and most likely orientation of the axes effective joint axis is found initial validation of the method was conducted on a hinged mechanical analog and a single healthy adult subject for the analog the center was found to be within 3 8 mm of the geometric center and 2 0 degrees of the geometric axis standard deviation for the adult subject hip centers varied on the order of 1 3 mm knee centers by 3 9 mm and knee axes by 2 0 degrees the results suggest that the new method is an objective precise and practical alternative to the standard clinical approach

binary regression using an extended beta binomial distribution with discussion of correlation induced by covariate measurement errors abstract the beta binomial distribution is extended to allow negative correlations among binary variates within an experimental unit regression models are proposed for both the binary variate response rate and for the pairwise correlation between binary variates and corresponding likelihood estimation procedures are described binary regression problems are also considered in which measurement errors in the regression variables represent the sole source of overdispersion some corresponding response rate estimation procedures are described and illustrated

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

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

decision making with hybrid influence diagrams using mixtures of truncated exponentials mixtures of truncated exponentials mte potentials are an alternative to discretization for representing continuous chance variables in influence diagrams also mte potentials can be used to approximate utility functions this paper introduces mte influence diagrams which can represent decision problems without restrictions on the relationships between continuous and discrete chance variables without limitations on the distributions of continuous chance variables and without limitations on the nature of the utility functions in mte influence diagrams all probability distributions and the joint utility function or its multiplicative factors are represented by mte potentials and decision nodes are assumed to have discrete state spaces mte influence diagrams are solved by variable elimination using a fusion algorithm

on the choice of a model to fit data from an exponential family let x 1 cdots x n be iid observations coming from an exponential family the problem of interest is this given a finite number of models m j smoothly curved manifolds in mathbb r k choose the best model to fit the observations with some penalty for choosing models with dimensions which are too large a result of schwarz is made more specific and is extended to the case where the models are curved manifolds if s y n j is up to a constant c n independent of the model the log of the posterior probability of the j th model where the sample mean y n 1 n sum n i 1 x i has been replaced by y schwarz suggested an asymptotic expansion of s y n j whose leading terms are gamma y n j n sup psi in m j cap theta y psi b psi frac 1 2 k j log n in the case where the models are affine subspaces of mathbb r k we establish a similar asymptotic expansion including the next term with uniform bounds for y in a compact neighborhood of nabla b theta where theta is the true value of the parameter we suggest a criterion for the choice of the best model that consists of maximizing the three leading terms in the expansion s y n j we show that the criterion gives the correct model with probabilities p n theta rightarrow 1 as n rightarrow infty

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

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

approximate bayes factors when a mode occurs on the boundary abstract bayes factors measuring the strength of evidence in favor of the null are often used in the bayesian approach to testing hypotheses laplace approximations to bayes factors are convenient and quite accurate in many contexts however one usual assumption the existence of an interior mode does not always hold the posterior mode can occur at the boundary of the parameter space this article discusses the boundary mode laplace s method for boundary modes uses it to approximate the bayes factor and presents a modification to the schwarz criterion the sensitivity of bayes factor to the choice of prior on the nuisance parameter in testing nested models is also investigated results are illustrated for the case of testing extrabinomial variability

testing random effects in the linear mixed model using approximate bayes factors deciding which predictor effects may vary across subjects is a difficult issue standard model selection criteria and test procedures are often inappropriate for comparing models with different numbers of random effects due to constraints on the parameter space of the variance components testing on the boundary of the parameter space changes the asymptotic distribution of some classical test statistics and causes problems in approximating bayes factors we propose a simple approach for testing random effects in the linear mixed model using bayes factors we scale each random effect to the residual variance and introduce a parameter that controls the relative contribution of each random effect free of the scale of the data we integrate out the random effects and the variance components using closed form solutions the resulting integrals needed to calculate the bayes factor are low dimensional integrals lacking variance components and can be efficiently approximated with laplace s method we propose a default prior distribution on the parameter controlling the contribution of each random effect and conduct simulations to show that our method has good properties for model selection problems finally we illustrate our methods on data from a clinical trial of patients with bipolar disorder and on data from an environmental study of water disinfection by products and male reproductive outcomes

a correction for axis misalignment in the joint angle curves representing knee movement in gait analysis this paper derives a simple mathematical model relating changes in the orientations of the two cartesian coordinate systems involved in recording knee movement and the varus valgus and the internal external rotation angles for describing the knee s motion rotation matrices are given for changing the orientations of the two cartesian coordinate systems in such a way that the quadratic variations in the varus valgus and in the external internal angles are minimal these estimated rotation matrices are used to correct for axis misalignment the correction is calibrated by considering the impact of the new orientation of the thigh cartesian coordinate system on the hip joint angles the procedure is applied to kinematic data collected on normal subjects the uncertainty about the specification of the thigh cartesian coordinate system is shown to explain some of the between subject variability in the varus valgus and in the internal external rotation angles curves

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

consistency of bayesian procedures for variable selection it has long been known that for the comparison of pairwise nested models a decision based on the bayes factor produces a consistent model selector in the frequentist sense here we go beyond the usual consistency for nested pairwise models and show that for a wide class of prior distributions including intrinsic priors the corresponding bayesian procedure for variable selection in normal regression is consistent in the entire class of normal linear models we find that the asymptotics of the bayes factors for intrinsic priors are equivalent to those of the schwarz bic criterion also recall that the jeffreys lindley paradox refers to the well known fact that a point null hypothesis on the normal mean parameter is always accepted when the variance of the conjugate prior goes to infinity this implies that some limiting forms of proper prior distributions are not necessarily suitable for testing problems intrinsic priors are limits of proper prior distributions and for finite sample sizes they have been proved to behave extremely well for variable selection in regression a consequence of our results is that for intrinsic priors lindley s paradox does not arise

estimation of the false negative fraction of a diagnostic kit through bayesian regression model averaging in modelling we usually endeavour to find a single best model that explains the relationship between independent and dependent variables selection of a single model fails to take into account the prior uncertainty in the model space the bayesian model averaging bma approach tackles this problem by considering the set of all possible models we apply bma approach to the estimation of the false negative fraction fnf in a particular case of a two stage multiple screening test for bowel cancer we find that after taking model uncertainty into consideration the estimate of the fnf obtained is largely dependent on the covariance structure of the priors results obtained when the zellner g prior for the prior variance is used is largely influenced by the magnitude of g

mixtures of g priors for bayesian variable selection zellner s g prior remains a popular conventional prior for use in bayesian variable selection despite several undesirable consistency issues in this article we study mixtures of g priors as an alternative to default g priors that resolve many of the problems with the original formulation while maintaining the computational tractability that has made the g prior so popular we present theoretical properties of the mixture g priors and provide real and simulated examples to compare the mixture formulation with fixed g priors empirical bayes approaches and other default procedures please see arnold zellner s letter and the author s response

hierarchical models for assessing variability among functions summary in many applications of functional data analysis summarising functional variation based on fits without taking account of the estimation process runs the risk of attributing the estimation variation to the functional variation thereby overstating the latter for example the first eigenvalue of a sample covariance matrix computed from estimated functions may be biased upwards we display a set of estimated neuronal poisson process intensity functions where this bias is substantial and we discuss two methods for account ing for estimation variation one method uses a random coefficient model which requires all functions to be fitted with the same basis functions an alternative method removes the same basis restriction by means of a hierarchical gaussian process model in a small simulation study the hierarchical gaussian process model outperformed the random coefficient model and greatly reduced the bias in the estimated first eigenvalue that would result from ignoring estimation variability for the neuronal data the hierarchical gaussian process estimate of the first eigenvalue was much smaller than the naive estimate that ignored variability due to function estimation the neuronal setting also illustrates the benefit of incorporating alignment parameters into the hierarchical scheme