bayesian analysis of dichotomous quantal response models abstract qualitative response models qrm s are analyzed from the bayesian point of view using diffuse and informative prior distributions exact finite sample bayesian and large sample bayesian and non bayesian estimation results are compared in addition the paper provides 1 plots and discussion of the properties of likelihood functions for qrm s 2 posterior distributions for logit models derivatives and elasticities 3 bayesian prediction procedures for qrm s 4 new estimates for the median and other fractiles of the logistic distribution 5 posterior odds ratios for model selection problems and 6 comparisons of two alternative monte carlo numerical integration procedures it is concluded that asymptotic approximations are not accurate for small to moderate sized samples even when only a single input variable is used and that operational bayesian methods are available for providing both exact small sample and large sample approximate inferences for drm s

some aspects of polya tree distributions for statistical modelling the definition and elementary properties of polya tree distributions are reviewed two theorems are presented showing that polya trees can be constructed to concentrate arbitrarily closely about any desired pdf and that polya tree priors can put positive mass in every relative entropy neighborhood of every positive density with finite entropy thereby satisfying a consistency condition such theorems are false for dirichlet processes models are constructed combining partially specified polya trees with other information such as monotonicity or unimodality it is shown how to compute bounds on posterior expectations over the class of all priors with the given specifications a numerical example is given a theorem of diaconis and freedman about dirichlet processes is generalized to polya trees allowing polya trees to be the models for errors in regression problems finally empirical bayes models using dirichlet processes are generalized to polya trees an example from berry and christensen is reanalyzed with a polya tree model

a comparison of two model averaging techniques with an application to growth empirics parameter estimation under model uncertainty is a difficult and fundamental issue in econometrics this paper compares the performance of various model averaging techniques in particular it contrasts bayesian model averaging bma currently one of the standard methods used in growth empirics with a new method called weighted average least squares wals the new method has two major advantages over bma its computational burden is trivial and it is based on a transparent definition of prior ignorance the theory is applied to and sheds new light on growth empirics where a high degree of model uncertainty is typically present

comments with reply on robust stability of an interval plant with respect to a convex region in the complex plane by j shaw and j jayasuriya the commenter points out that the results of j shaw and j jayasuriya ibid vol 38 no 2 pp 284 287 feb 1993 were published by f j kraus and w truol int j contr vol 52 no 4 pp 967 983 1991 for robust stability of polynomial families with affline linear parameter dependence the authors agree with the commenter and point out that their own results derive from a paper they submitted for publication in 1990 but which has not yet been printed which contained arguments similar to those of the commenters

model averaged double robust estimation researchers estimating causal effects are increasingly challenged with decisions on how to best control for a potentially high dimensional set of confounders typically a single propensity score model is chosen and used to adjust for confounding while the uncertainty surrounding which covariates to include into the propensity score model is often ignored and failure to include even one important confounder will results in bias we propose a practical and generalizable approach that overcomes the limitations described above through the use of model averaging we develop and evaluate this approach in the context of double robust estimation more specifically we introduce the model averaged double robust ma dr estimators which account for model uncertainty in both the propensity score and outcome model through the use of model averaging the ma dr estimators are defined as weighted averages of double robust estimators where each double robust estimator corresponds to a specific choice of the outcome model and the propensity score model the ma dr estimators extend the desirable double robustness property by achieving consistency under the much weaker assumption that either the true propensity score model or the true outcome model be within a specified possibly large class of models using simulation studies we also assessed small sample properties and found that ma dr estimators can reduce mean squared error substantially particularly when the set of potential confounders is large relative to the sample size we apply the methodology to estimate the average causal effect of temozolomide plus radiotherapy versus radiotherapy alone on one year survival in a cohort of 1887 medicare enrollees who were diagnosed with glioblastoma between june 2005 and december 2009

a bayesian approach for predicting with polynomial regression of unknown degree this article compares three methods for computing the posterior probabilities of the possible orders in polynomial regression models these posterior probabilities are used for forecasting using bayesian model averaging it is shown that bayesian model averaging provides a closer relationship between the theoretical coverage of the high density predictive interval hdpi and the observed coverage than those corresponding to selecting the best model the performance of the different procedures is illustrated with simulations and some known engineering data

the variable selection problem abstract the problem of variable selection is one of the most pervasive model selection problems in statistical applications often referred to as the problem of subset selection it arises when one wants to model the relationship between a variable of interest and a subset of potential explanatory variables or predictors but there is uncertainty about which subset to use this vignette reviews some of the key developments that have led to the wide variety of approaches for this problem

bayesian computation via the gibbs sampler and related markov chain monte carlo methods the use of the gibbs sampler for bayesian computation is reviewed and illustrated in the context of some canonical examples other markov chain monte carlo simulation methods are also briefly described and comments are made on the advantages of sample based approaches for bayesian inference summaries

on the consistency of bayes estimates on etudie les proprietes de frequence des regles de bayes avec une attention particuliere a la consistence

comparing bayes model averaging and stacking when model approximation error cannot be ignored we compare bayes model averaging bma to a non bayes form of model averaging called stacking in stacking the weights are no longer posterior probabilities of models they are obtained by a technique based on cross validation when the correct data generating model dgm is on the list of models under consideration bma is never worse than stacking and often is demonstrably better provided that the noise level is of order commensurate with the coefficients and explanatory variables here however we focus on the case that the correct dgm is not on the model list and may not be well approximated by the elements on the model list we give a sequence of computed examples by choosing model lists and dgm s to contrast the risk performance of stacking and bma in the first examples the model lists are chosen to reflect geometric principles that should give good performance in these cases stacking typically outperforms bma sometimes by a wide margin in the second set of examples we examine how stacking and bma perform when the model list includes all subsets of a set of potential predictors when we standardize the size of terms and coefficients in this setting we find that bma outperforms stacking when the deviant terms in the dgm point in directions accommodated by the model list but that when the deviant term points outside the model list stacking seems to do better overall our results suggest the stacking has better robustness properties than bma in the most important settings

evaluating predictors of dispersion a comparison of dominance analysis and bayesian model averaging conventional measures of predictor importance in linear models are applicable only when the assumption of homoscedasticity is satisfied moreover they cannot be adapted to evaluating predictor importance in models of heteroscedasticity i e dispersion an issue that seems not to have been systematically addressed in the literature we compare two suitable approaches dominance analysis da and bayesian model averaging bma for simultaneously evaluating predictor importance in models of location and dispersion we apply them to the beta general linear model as a test case illustrating this with an example using real data simulations using several different model structures sample sizes and degrees of multicollinearity suggest that both da and bma largely agree on the relative importance of predictors of the mean but differ when ranking predictors of dispersion the main implication of these findings for researchers is that the choice between da and bma is most important when they wish to evaluate the importance of predictors of dispersion

multivariate adaptive regression splines a new method is presented for flexible regression modeling of high dimensional data the model takes the form of an expansion in product spline basis functions where the number of basis functions as well as the parameters associated with each one product degree and knot locations are automatically determined by the data this procedure is motivated by the recursive partitioning approach to regression and shares its attractive properties unlike recursive partitioning however this method produces continuous models with continuous derivatives it has more power and flexibility to model relationships that are nearly additive or involve interactions in at most a few variables in addition the model can be represented in a form that separately identifies the additive contributions and those associated with the different multivariable interactions

least squares model averaging this paper considers the problem of selection of weights for averaging across least squares estimates obtained from a set of models existing model average methods are based on exponential akaike information criterion aic and bayesian information criterion bic weights in distinction this paper proposes selecting the weights by minimizing a mallows criterion the latter an estimate of the average squared error from the model average fit we show that our new mallows model average mma estimator is asymptotically optimal in the sense of achieving the lowest possible squared error in a class of discrete model average estimators in a simulation experiment we show that the mma estimator compares favorably with those based on aic and bic weights the proof of the main result is an application of the work of li 1987

analyzing nonstationary spatial data using piecewise gaussian processes in many problems in geostatistics the response variable of interest is strongly related to the underlying geology of the spatial location in these situations there is often little correlation in the responses found in different rock strata so the underlying covariance structure shows sharp changes at the boundaries of the rock types conventional stationary and nonstationary spatial methods are inappropriate because they typically assume that the covariance between points is a smooth function of distance in this article we propose a generic method for the analysis of spatial data with sharp changes in the underlying covariance structure our method works by automatically decomposing the spatial domain into disjoint regions within which the process is assumed to be stationary but the data are assumed independent across regions uncertainty in the number of disjoint regions their shapes and the model within regions is dealt with in a fully bayesian fashion we illustrate our approach on a previously

prediction via orthogonalized model mixing abstract we introduce an approach and algorithms for model mixing in large prediction problems with correlated predictors we focus on the choice of predictors in linear models and mix over possible subsets of candidate predictors our approach is based on expressing the space of models in terms of an orthogonalization of the design matrix advantages are both statistical and computational statistically orthogonalization often leads to a reduction in the number of competing models by eliminating correlations computationally large model spaces cannot be enumerated recent approaches are based on sampling models with high posterior probability via markov chains based on orthogonalization of the space of candidate predictors we can approximate the posterior probabilities of models by products of predictor specific terms this leads to an importance sampling function for sampling directly from the joint distribution over the model space without resorting to markov chains compared to the latter ortho

bayes model averaging with selection of regressors when a number of distinct models contend for use in prediction the choice of a single model can offer rather unstable predictions in regression stochastic search variable selection with bayesian model averaging offers a cure for this robustness issue but at the expense of requiring very many predictors here we look at bayes model averaging incorporating variable selection for prediction this offers similar mean square errors of prediction but with a vastly reduced predictor space this can greatly aid the interpretation of the model it also reduces the cost if measured variables have costs the development here uses decision theory in the context of the multivariate general linear model in passing this reduced predictor space bayes model averaging is contrasted with single model approximations a fast algorithm for updating regressions in the markov chain monte carlo searches for posterior inference is developed allowing many more variables than observations to be contemplated we discuss the merits of absolute rather than proportionate shrinkage in regression especially when there are more variables than observations the methodology is illustrated on a set of spectroscopic data used for measuring the amounts of different sugars in an aqueous solution

inference for deterministic simulation models the bayesian melding approach abstract deterministic simulation models are used in many areas of science engineering and policy making typically these are complex models that attempt to capture underlying mechanisms in considerable detail and they have many user specified inputs the inputs are often specified by some form of trial and error approach in which plausible values are postulated the corresponding outputs inspected and the inputs modified until plausible outputs are obtained here we address the issue of more formal inference for such models a probabilistic approach called bayesian synthesis was shown to suffer from the borel paradox according to which the results can depend on the parameterization of the model we propose a modified approach called bayesian melding which takes into full account information and uncertainty about both inputs and outputs to the model while avoiding the borel paradox this is done by recognizing the existence of two priors one implicit and one explicit on each input and output

a statistical concept to assess the uncertainty in bayesian model weights and its impact on model ranking bayesian model averaging bma ranks the plausibility of alternative conceptual models according to bayes theorem a prior belief about each model s adequacy is updated to a posterior model probability based on the skill to reproduce observed data and on the principle of parsimony the posterior model probabilities are then used as model weights for model ranking selection or averaging despite the statistically rigorous bma procedure model weights can become uncertain quantities due to measurement noise in the calibration data set or due to uncertainty in model input uncertain weights may in turn compromise the reliability of bma results we present a new statistical concept to investigate this weighting uncertainty and thus to assess the significance of model weights and the confidence in model ranking our concept is to resample the uncertain input or output data and then to analyze the induced variability in model weights in the special case of weighting uncertainty due to measurement noise in the calibration data set we interpret statistics of bayesian model evidence to assess the distance of a model s performance from the theoretical upper limit to illustrate our suggested approach we investigate the reliability of soil plant model selection following up on a study by wohling et al 2015 results show that the bma routine should be equipped with our suggested upgrade to 1 reveal the significant but otherwise undetected impact of measurement noise on model ranking results and 2 to decide whether the considered set of models should be extended with better performing alternatives

towards a coherent philosophy for modelling the environment the predominant philosophy underlying most environmental modelling is a form of pragmatic realism the limitations of this approach in practical applications are discussed in particular in relati

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

frequentist model average estimators the traditional use of model selection methods in practice is to proceed as if the final selected model had been chosen in advance without acknowledging the additional uncertainty introduced by model selection this often means underreporting of variability and too optimistic confidence intervals we build a general large sample likelihood apparatus in which limiting distributions and risk properties of estimators post selection as well as of model average estimators are precisely described also explicitly taking modeling bias into account this allows a drastic reduction in complexity as competing model averaging schemes may be developed discussed and compared inside a statistical prototype experiment where only a few crucial quantities matter in particular we offer a frequentist view on bayesian model averaging methods and give a link to generalized ridge estimators our work also leads to new model selection criteria the methods are illustrated with real data applications

stochastic complexity and modeling on demontre un theoreme fondamental qui donne une borne inferieure pour la longueur de code et donc pour les erreurs de prediction on definit les notions d information a priori et d information utile dans les donnees

bayesian inference in a simultaneous equation model with limited dependent variables abstract this paper develops a bayesian approach for inference in a simultaneous equation model with limited dependent variables sldv by employing a combination of gibbs sampling and data augmentation we can draw from the exact posterior of this sldv model and avoid direct evaluation of the non trivial likelihood function a by product from our posterior simulation is the savage dickey density ratio which is used for computing the bayes factor the practicality and efficiency of the proposed method are illustrated through an example in corporate finance

model validation and predictive capability for the thermal challenge problem we address the thermal problem posed at the sandia validation challenge workshop unlike traditional approaches that confound calibration with validation and prediction our approach strictly distinguishes these activities and produces a quantitative measure of model form uncertainty in the face of available data we introduce a general validation metric that can be used to characterize the disagreement between the quantitative predictions from a model and relevant empirical data when either or both predictions and data are expressed as probability distributions by considering entire distributions this approach generalizes traditional approaches to validation that focus only on the mean behaviors of predictions and observations the proposed metric has several desirable properties that should make it practically useful in engineering including objectiveness and robustness retaining the units of the data themselves and generalizing the deterministic difference the metric can be used to assess the overall performance of a model against all the experimental observations in the validation domain and it can be extrapolated to express predictive capability of the model under conditions for which direct experimental observations are not available we apply the metric and the scheme for characterizing predictive capability to the thermal problem

the hierarchical logistic regression model for multilevel analysis abstract a hierarchical logistic regression model is proposed for studying data with group structure and a binary response variable the group structure is defined by the presence of micro observations embedded within contexts macro observations and the specification is at both of these levels at the first micro level the usual logistic regression model is defined for each context the same regressors are used in each context but the micro regression coefficients are free to vary over contexts at the second level the micro coefficients are treated as functions of macro regressors an empirical bayes estimation procedure is proposed for estimating the micro and macro coefficients explicit formulas are provided that are computationally feasible for large scale data analyses these include an algorithm for finding the maximum likelihood estimates of the covariance components representing within and between macro equation error variability the methodology is applied to world fertility survey data

maximum likelihood bayesian averaging of spatial variability models in unsaturated fractured tuff 1 hydrologic analyses typically rely on a single conceptual mathematical model yet hydrologic environments are open and complex rendering them prone to multiple interpretations and mathematical descriptions adopting only one of these may lead to statistical bias and underestimation of uncertainty bayesian model averaging bma hoeting et al 1999 provides an optimal way to combine the predictions of several competing models and to assess their joint predictive uncertainty however it tends to be computationally demanding and relies heavily on prior information about model parameters neuman 2002 2003 proposed a maximum likelihood version mlbma of bma to render it computationally feasible and to allow dealing with cases where reliable prior information is lacking we apply mlbma to seven alternative variogram models of log air permeability data from single hole pneumatic injection tests in six boreholes at the apache leap research site alrs in central arizona unbiased ml estimates of variogram and drift parameters are obtained using adjoint state maximum likelihood cross validation samper and neuman 1989a in conjunction with universal kriging and generalized least squares standard information criteria provide an ambiguous ranking of the models which does not justify selecting one of them and discarding all others as is commonly done in practice instead we eliminate some of the models based on their negligibly small posterior probabilities and use the rest to project the measured log permeabilities by kriging onto a rock volume containing the six boreholes we then average these four projections and associated kriging variances using the posterior probability of each model as weight finally we cross validate the results by eliminating from consideration all data from one borehole at a time repeating the above process and comparing the predictive capability of mlbma with that of each individual model we find that mlbma is superior to any individual geostatistical model of log permeability among those we consider at the alrs

multivariate logit copula model with an application to dental data applications of copulas for multivariate continuous data abound but there are only a few that treat multivariate binary data in the present paper we model multivariate binary data based on copulas using mixtures of max infinitely divisible copulas introduced by joe and hu j multivar anal 1996 57 2 240 265 when applying copulas to binary data the marginal distributions also contribute to the dependence measures we propose the use of covariate information in the copula parameters to obtain a direct effect of a covariate on dependence to deal with model uncertainty due to selecting among several candidate models we use a model averaging technique we apply the model to data from the signal tandmobiel dental study and in particular to four binary responses that refer to caries experience in the mandibular and maxillary left and right molars we aim to model kendall s tau associations between them and examine how covariate information affects these associations we found that there are systematically larger associations between the two mandibular and the two maxillary molars using covariates to model these associations more closely we found that the systematic fluoride and age of the children affect the associations note that such relationships could not have been revealed by methods that focus on the marginal models copyright 2008 john wiley sons ltd

stability of multivariable fractional polynomial models with selection of variables and transformations a bootstrap investigation sauerbrei and royston have recently described an algorithm based on fractional polynomials for the simultaneous selection of variables and of suitable transformations for continuous predictors in a multivariable regression setting they illustrated the approach by analyses of two breast cancer data sets here we extend their work by considering how to assess possible instability in such multivariable fractional polynomial models we first apply the algorithm repeatedly in many bootstrap replicates we then use log linear models to investigate dependencies among the inclusion fractions for each predictor and among the simplified classes of fractional polynomial function chosen in the bootstrap samples to further evaluate the results we define measures of instability based on a decomposition of the variability of the bootstrap selected functions in relation to a reference function from the original model for each data set we are able to identify large reasonably stable subsets of the bootstrap replications in which the functional forms of the predictors appear fairly stable despite the considerable flexibility of the family of fractional polynomials and the consequent risk of overfitting when several variables are considered we conclude that the multivariable selection algorithm can find stable models

sensitivity analysis and assessment of prior model probabilities in mlbma with application to unsaturated fractured tuff 1 previous application of maximum likelihood bayesian model averaging mlbma neuman 2002 2003 to alternative variogram models of log air permeability data in fractured tuff has demonstrated its effectiveness in quantifying conceptual model uncertainty and enhancing predictive capability ye et al 2004 a question remained how best to ascribe prior probabilities to competing models in this paper we examine the extent to which lead statistics of posterior log permeability predictions are sensitive to prior probabilities of seven corresponding variogram models we then explore the feasibility of quantifying prior model probabilities by 1 maximizing shannon s entropy h shannon 1948 subject to constraints reflecting a single analyst s or a group of analysts prior perception about how plausible each alternative model or a group of models is relative to others and 2 selecting a posteriori the most likely among such maxima corresponding to alternative prior perceptions of various analysts or groups of analysts another way to select among alternative prior model probability sets which however is not guaranteed to yield optimum predictive performance though it did so in our example and would therefore not be our preferred option is a minimum maximum approach according to which one selects a priori the set corresponding to the smallest value of maximum entropy whereas maximizing h subject to the prior perception of a single analyst or group maximizes the potential for further information gain through conditioning selecting the smallest among such maxima gives preference to the most informed prior perception among those of several analysts or groups we use the same variogram models and log permeability data as ye et al 2004 to demonstrate that our proposed approach yields the least amount of posterior entropy residual uncertainty after conditioning and enhances predictive model performance as compared to 1 the noninformative neutral case in which all prior model probabilities are set equal to each other and 2 an informed case that nevertheless violates the principle of parsimony

a copula based model for multivariate non normal longitudinal data analysis of a dose titration safety study on a new antidepressant a new model for multivariate non normal longitudinal data is proposed in a first step each longitudinal series of data corresponding to a given response is modelled separately using a copula to relate the marginal distributions of the response at each time of observation in a second step at each observation time the conditional on the past distributions of each response are related using another copula describing the relationship between the corresponding variables note that there is no need to consider the same family of distributions for these response variables the technique is illustrated in a dose titration safety study on a new antidepressant the haemodynamic effect on diastolic blood pressure systolic blood pressure and heart rate is studied these three responses are measured repeatedly over time on ten healthy volunteers during the dose escalation the available covariates are sex and the concentration of drug in the plasma at time of measurement

markov chain monte carlo methods for computing bayes factors a comparative review the problem of calculating posterior probabilities for a collection of competing models and associated bayes factors continues to be a formidable challenge for applied bayesian statisticians current approaches that take advantage of modern markov chain monte carlo computing methods include those that attempt to sample over some form of the joint space created by the model indicators and the parameters for each model others that sample over the model space alone and still others that attempt to estimate the marginal likelihood of each model directly because the collection of these is equivalent to the collection of model probabilities themselves we review several methods and compare them in the context of three examples a simple regression example a more challenging hierarchical longitudinal model and a binary data latent variable model we find that the joint model parameter space search methods perform adequately but can be difficult to program and tune whereas the marginal likelihood methods o

noncircular principal component analysis and its application to model selection one of the most commonly used data analysis tools principal component analysis pca since is based on variance maximization assumes a circular model and hence cannot account for the potential noncircularity of complex data in this paper we introduce noncircular pca ncpca which extends the traditional pca to the case where there can be both circular and noncircular gaussian signals in the subspace we study the properties of ncpca introduce an efficient algorithm for its computation and demonstrate its application to model selection i e the detection of both the signal subspace order and the number of circular and noncircular signals we present numerical results to demonstrate the advantages of ncpca over regular pca when there are noncircular signals in the subspace at the same time we note that since a noncircular model has more degrees of freedom than a circular one there are cases where a circular model might be preferred even though the underlying problem is noncircular in particular we show that a circular model is preferred when the signal to noise ratio snr is low number of samples is small or the degree of noncircularity of the signals is low hence ncpca inherently provides guidance as to when to take noncircularity into account

calibration based empirical probability probability forecasts for a sequence of uncertain events may be compared with the outcomes of those events by means of a natural criterion of empirical validity calibration it is shown that any two sequences of forecasts which both meet this criterion must be in asymptotic agreement these agreed values can then be considered as correct objective probability forecasts for the particular sequence of outcome results obtained however the objective forecasts vary with the extent of the information taken into account when they are formulated we thus obtain a general theory of empirical probability relative to an information base this theory does not require that such probabilities be interpreted in terms of repeated trials of the same event some implications of this theory are discussed

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

on inclusion driven learning of bayesian networks two or more bayesian network structures are markov equivalent when the corresponding acyclic digraphs encode the same set of conditional independencies therefore the search space of bayesian network structures may be organized in equivalence classes where each of them represents a different set of conditional independencies the collection of sets of conditional independencies obeys a partial order the so called inclusion order this paper discusses in depth the role that the inclusion order plays in learning the structure of bayesian networks in particular this role involves the way a learning algorithm traverses the search space we introduce a condition for traversal operators the inclusion boundary condition which when it is satisfied guarantees that the search strategy can avoid local maxima this is proved under the assumptions that the data is sampled from a probability distribution which is faithful to an acyclic digraph and the length of the sample is unbounded the previous discussion leads to the design of a new traversal operator and two new learning algorithms in the context of heuristic search and the markov chain monte carlo method we carry out a set of experiments with synthetic and real world data that show empirically the benefit of striving for the inclusion order when learning bayesian networks from data

variable selection and bayesian model averaging in case control studies covariate and confounder selection in case control studies is often carried out using a statistical variable selection method such as a two step method or a stepwise method in logistic regression inference is then carried out conditionally on the selected model but this ignores the model uncertainty implicit in the variable selection process and so may underestimate uncertainty about relative risks we report on a simulation study designed to be similar to actual case control studies this shows that p values computed after variable selection can greatly overstate the strength of conclusions for example for our simulated case control studies with 1000 subjects of variables declared to be significant with p values between 0 01 and 0 05 only 49 per cent actually were risk factors when stepwise variable selection was used we propose bayesian model averaging as a formal way of taking account of model uncertainty in case control studies this yields an easily interpreted summary the posterior probability that a variable is a risk factor and our simulation study indicates this to be reasonably well calibrated in the situations simulated the methods are applied and compared in the context of a case control study of cervical cancer

focused information criterion and model averaging for generalized additive partial linear models we study model selection and model averaging in generalized additive partial linear models gaplms polynomial spline is used to approximate nonparametric functions the corresponding estimators of the linear parameters are shown to be asymptotically normal we then develop a focused information criterion fic and a frequentist model average fma estimator on the basis of the quasi likelihood principle and examine theoretical properties of the fic and fma the major advantages of the proposed procedures over the existing ones are their computational expediency and theoretical reliability simulation experiments have provided evidence of the superiority of the proposed procedures the approach is further applied to a real world data example

bayesian predictions of reynolds averaged navier stokes uncertainties using maximum a posteriori estimates computational fluid dynamics analyses of high reynolds number flows mostly rely on the reynolds averaged navier stokes equations the associated closure models are based on multiple simplifying ass

multimodel bayesian analysis of groundwater data worth we explore the way in which uncertain descriptions of aquifer heterogeneity and groundwater flow impact one s ability to assess the worth of collecting additional data we do so on the basis of maximum likelihood bayesian model averaging mlbma by accounting jointly for uncertainties in geostatistical and flow model structures and parameter hydraulic conductivity as well as system state hydraulic head estimates given uncertain measurements of one or both variables previous description of our approach was limited to geostatistical models based solely on hydraulic conductivity data here we implement the approach on a synthetic example of steady state flow in a two dimensional random log hydraulic conductivity field with and without recharge by embedding an inverse stochastic moment solution of groundwater flow in mlbma a moment equations based geostatistical inversion method is utilized to circumvent the need for computationally expensive numerical monte carlo simulations the approach is compatible with either deterministic or stochastic flow models and consistent with modern statistical methods of parameter estimation admitting but not requiring prior information about the parameters it allows but does not require approximating lead predictive statistical moments of system states by linearization while updating model posterior probabilities and parameter estimates on the basis of potential new data both before and after such data are actually collected

adjustment uncertainty in effect estimation summary often there is substantial uncertainty in the selection of confounders when estimating the association between an exposure and health we define this type of uncertainty as adjustment uncertainty we propose a general statistical framework for handling adjustment uncertainty in exposure effect estimation for a large number of confounders we describe a specific implemen tation and we develop associated visualization tools theoretical results and simulation studies show that the proposed method provides consistent estimators of the exposure effect and its variance we also show that when the goal is to estimate an exposure effect accounting for ad justment uncertainty bayesian model averaging with posterior model probabilities approximated using information criteria can fail to estimate the exposure effect and can over or underestimate its variance we compare our approach to bayesian model averaging using time series data on levels of fine particulate matter and mortality

bayesian multivariate hierarchical transformation models for roc analysis a bayesian multivariate hierarchical transformation model bmhtm is developed for receiver operating characteristic roc curve analysis based on clustered continuous diagnostic outcome data with covariates two special features of this model are that it incorporates non linear monotone transformations of the outcomes and that multiple correlated outcomes may be analysed the mean variance and transformation components are all modelled parametrically enabling a wide range of inferences the general framework is illustrated by focusing on two problems 1 analysis of the diagnostic accuracy of a covariate dependent univariate test outcome requiring a box cox transformation within each cluster to map the test outcomes to a common family of distributions 2 development of an optimal composite diagnostic test using multivariate clustered outcome data in the second problem the composite test is estimated using discriminant function analysis and compared to the test derived from logistic regression analysis where the gold standard is a binary outcome the proposed methodology is illustrated on prostate cancer biopsy data from a multi centre clinical trial

a bayesian method for combining results from several binomial experiments abstract the problem of combining information related to i binomial experiments each having a distinct probability of success i is considered instead of using a standard exchangeable prior for 1 i we propose a more flexible distribution that takes into account various degrees of similarity among the i s using ideas developed by malec and sedransk we consider a partition g of the experiments and take the i s belonging to the same partition subset to be exchangeable and the i s belonging to distinct subsets to be independent next we perform bayesian inference on conditional on g of course one is typically uncertain about which partition to use and so a prior distribution is assigned on a set of plausible partitions g the final inference on is obtained by combining the conditional inferences according to the posterior distribution of g the methodology adopted in this article offers a wide flexibility in structuring the dependence among the i s this allows the

assessment and propagation of model uncertainty in most examples of inference and prediction the expression of uncertainty about unknown quantities y on the basis of known quantities x is based on a model m that formalizes assumptions about how x and y are related m will typically have two parts structural assumptions s such as the form of the link function and the choice of error distribution in a generalized linear model and parameters whose meaning is specific to a given choice of s it is common in statistical theory and practice to acknowledge parametric uncertainty about given a particular assumed structure s it is less common to acknowledge structural uncertainty about s itself a widely used approach involves enlisting the aid of x to specify a plausible single best choice s for s and then proceeding as if s were known to be correct in general this approach fails to assess and propagate structural uncertainty fully and may lead to miscalibrated uncertainty assessments about y given x when miscalibration occurs it will often result in understatement of inferential or predictive uncertainty about y leading to inaccurate scientific summaries and overconfident decisions that do not incorporate sufficient hedging against uncertainty in this paper i discuss a bayesian approach to solving this problem that has long been available in principle but is only now becoming routinely feasible by virtue of recent computational advances and examine its implementation in examples that involve forecasting the price of oil and estimating the chance of catastrophic failure of the us space shuttle

a nondegenerate penalized likelihood estimator for variance parameters in multilevel models group level variance estimates of zero often arise when fitting multilevel or hierarchical linear models especially when the number of groups is small for situations where zero variances are implausible a priori we propose a maximum penalized likelihood approach to avoid such boundary estimates this approach is equivalent to estimating variance parameters by their posterior mode given a weakly informative prior distribution by choosing the penalty from the log gamma family with shape parameter greater than 1 we ensure that the estimated variance will be positive we suggest a default log gamma 2 penalty with 0 which ensures that the maximum penalized likelihood estimate is approximately one standard error from zero when the maximum likelihood estimate is zero thus remaining consistent with the data while being nondegenerate we also show that the maximum penalized likelihood estimator with this default penalty is a good approximation to the posterior median obtained under a noninformative prior

the cost of generalizing logistic regression abstract binary response regression models in which the link function is a family defined by one or more unknown shape parameters are considered detailed attention is given to the two single parameter families proposed by aranda ordaz 1981 that incorporate the logistic linear and complementary log log link functions as special cases one model is a symmetric family of alternatives to the logistic shape and the second model is an asymmetric family the increase in variance of quantities of interest due to the addition of an extra parameter is calculated using asymptotic methods this inflation in variance is interpreted as the cost of adding an additional parameter to the model in considering the seriousness of this cost the reduction in bias due to the estimation of the additional parameter should also be taken into account three quantities of interest predicted probabilities for a fixed value of the explanatory variable predicted value of the explanatory variable to obtain a specific probability

case studies on the application of the stable manifold approach for nonlinear optimal control design this paper presents application results of a recently developed method for approximately solving the hamilton jacobi equation in nonlinear control theory the method is based on stable manifold theory and consists of a successive approximation algorithm which is suitable for computer calculations numerical approach for this algorithm is advantageous in that the computational complexity does not increase with respect to the accuracy of approximation and non analytic nonlinearities such as saturation can be handled first the stable manifold approach for approximately solving the hamilton jacobi equation is reviewed from the computational viewpoint and next the detailed applications are reported for the problems such as swing up and stabilization of a 2 dimensional inverted pendulum simulation stabilization of systems with input saturation simulation and a sub optimal servo system design for magnetic levitation system experiment

bayesian inference in econometric models using monte carlo integration methods for the systematic application of monte carlo integration with importance sampling to bayesian inference in econometric models are developed conditions under which the numerical approximation of a posterior moment converges almost surely to the true value as the number of monte carlo replications increases and the numerical accuracy of this approximation may be assessed reliably are set forth methods for the analytical verification of these conditions are discussed

robust bayes and empirical bayes analysis with _ epsilon contaminated priors on considere deux directions dans l analyse bayesienne robuste la premiere consiste a determiner l etendue des probabilites a posteriori d un ensemble la seconde consiste a selectionner une bonne distribution a priori

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

on priors with a kullback leibler property in this paper we highlight properties of bayesian models in which the prior puts positive mass on all kullback leibler neighborhoods of all densities these properties are concerned with model choice via the bayes factor density estimation and the maximization of expected utility for decision problems in four illustrations we focus on the bayes factor and show that whatever models are being compared the log bayes factor sample size converges to a non random number which has a nice interpretation a parametric versus semiparametric model comparison provides a fifth illustration

variable selection via gibbs sampling abstract a crucial problem in building a multiple regression model is the selection of predictors to include the main thrust of this article is to propose and develop a procedure that uses probabilistic considerations for selecting promising subsets this procedure entails embedding the regression setup in a hierarchical normal mixture model where latent variables are used to identify subset choices in this framework the promising subsets of predictors can be identified as those with higher posterior probability the computational burden is then alleviated by using the gibbs sampler to indirectly sample from this multinomial posterior distribution on the set of possible subset choices those subsets with higher probability the promising ones can then be identified by their more frequent appearance in the gibbs sample

avoiding zero between study variance estimates in random effects meta analysis fixed effects meta analysis has been criticized because the assumption of homogeneity is often unrealistic and can result in underestimation of parameter uncertainty random effects meta analysis and meta regression are therefore typically used to accommodate explained and unexplained between study variability however it is not unusual to obtain a boundary estimate of zero for the residual between study standard deviation resulting in fixed effects estimates of the other parameters and their standard errors to avoid such boundary estimates we suggest using bayes modal bm estimation with a gamma prior on the between study standard deviation when no prior information is available regarding the magnitude of the between study standard deviation a weakly informative default prior can be used with shape parameter 2 and rate parameter close to 0 that produces positive estimates but does not overrule the data leading to only a small decrease in the log likelihood from its maximum we review the most commonly used estimation methods for meta analysis and meta regression including classical and bayesian methods and apply these methods as well as our bm estimator to real datasets we then perform simulations to compare bm estimation with the other methods and find that bm estimation performs well by i avoiding boundary estimates ii having smaller root mean squared error for the between study standard deviation and iii better coverage for the overall effects than the other methods when the true model has at least a small or moderate amount of unexplained heterogeneity

clinical heterogeneity in random effect meta analysis between study boundary estimate problem random effect meta analysis is commonly applied to estimate overall effects with unexplained heterogeneity across studies however standard methods including restricted maximum likelihood ml or reml frequently produce near zero estimates for between study variance parameters consequently these methods are reduced to simple and unrealistic fixed effect models resulting in an ignorance of the substantial clinical heterogeneity and sometimes leading to incorrect conclusions to solve the boundary estimate problem we propose 1 an adjusted maximum likelihood method for the between study variance that maximizes a likelihood defined as a product of a standard likelihood and a gaussian class of adjustment factor and 2 a framework using sensitivity analysis by developing a new criterion to check for the occurrence of the boundary estimate although the adjustment introduces bias to the overall effects to ensure strictly positive estimates of the between study variance when the number of studies k is small the bias asymptotically approaches zero resulting in the same estimates derived from the reml method moreover the adjusted maximum likelihood estimator of the between study variance is consistent for large k and interestingly the reml method and our method are equivalent in terms of mean squared error criterion up to o k 1 we illustrate our approach with a motivating example to examine the controversial result of a meta analysis for 24 randomized controlled trials of human albumin numerical evaluations show that our approach produces no boundary estimates but similar synthesized results with the standard maximum likelihood methods as those produced by conventional methods especially with a small number of studies

predictive rans simulations via bayesian model scenario averaging the turbulence closure model is the dominant source of error in most reynolds averaged navier stokes simulations yet no reliable estimators for this error component currently exist here we develop a stochastic a posteriori error estimate calibrated to specific classes of flow it is based on variability in model closure coefficients across multiple flow scenarios for multiple closure models the variability is estimated using bayesian calibration against experimental data for each scenario and bayesian model scenario averaging bmsa is used to collate the resulting posteriors to obtain a stochastic estimate of a quantity of interest qoi in an unmeasured prediction scenario the scenario probabilities in bmsa are chosen using a sensor which automatically weights those scenarios in the calibration set which are similar to the prediction scenario the methodology is applied to the class of turbulent boundary layers subject to various pressure gradients for all considered prediction scenarios the standard deviation of the stochastic estimate is consistent with the measurement ground truth furthermore the mean of the estimate is more consistently accurate than the individual model predictions

parameterization and bayesian modeling progress in statistical computation often leads to advances in statistical modeling for example it is surprisingly common that an existing model is reparameterized solely for computational purposes but then this new configuration motivates a new family of models that is useful in applied statistics one reason why this phenomenon may not have been noticed in statistics is that reparameterizations do not change the likelihood in a bayesian framework however a transformation of parameters typically suggests a new family of prior distributions we discuss examples in censored and truncated data mixture modeling multivariate imputation stochastic processes and multilevel models