incorporating spatial dependence into bayesian multiple testing of statistical parametric maps in functional neuroimaging abstract the analysis of functional neuroimaging data often involves the simultaneous testing for activation at thousands of voxels leading to a massive multiple testing problem this is true whether the data analyzed are time courses observed at each voxel or a collection of summary statistics such as statistical parametric maps spms it is known that classical multiplicity corrections become strongly conservative in the presence of a massive number of tests some more popular approaches for thresholding imaging data such as the benjamini hochberg step up procedure for false discovery rate control tend to lose precision or power when the assumption of independence of the data does not hold bayesian approaches to large scale simultaneous inference also often rely on the assumption of independence we introduce a spatial dependence structure into a bayesian testing model for the analysis of spms by using spms rather than the voxel time courses much of the computational burden of bayesian analysis is mitigated increased power is demonstrated by using the dependence model to draw inference on a real dataset collected in a fmri study of cognitive control the model also is shown to lead to improved identification of neural activation patterns known to be associated with eye movement tasks

equivalence relations on separated nets arising from linear toral flows in 1998 burago kleiner and mcmullen independently proved the existence of separated nets in r which are not bi lipschitz equivalent bl to a lattice a finer equivalence relation than bl is bounded displacement bd separated nets arise naturally as return times to a section for minimal r actions we analyze the separated nets which arise via these constructions focusing particularly on nets arising from linear r actions on tori we show that generically these nets are bl to a lattice and for some choices of dimensions and sections they are generically bd to a lattice we also show the existence of such nets which are not bd to a lattice

an observer design for linear systems with unknown inputs a method is presented for designing an observer for the state of a linear time invariant system with unknown inputs the structure algorithm developed by silverman is applied to obtain the observer which estimates the maximum estimable subspace of the state it is shown that the observer equation can be derived from the maximum uncontrollable subspace of the original system with the aid of a left inverse for a transposed linear system this leads us to the necessary and sufficient condition for the existence of a state observer an application to the insensitivity observer synthesis is also included

entire majorants via euler maclaurin summation it is the aim of this article to give extremal majorants of type 2 for the class of functions f n x sgn x x n where n n as applications we obtain positive definite extensions to r of it m defined on r 1 1 where m n optimal bounds in hilbert type inequalities for the class of functions it m and majorants of type 2 for functions whose graphs are trapezoids

extremal functions in de branges and euclidean spaces ii this paper presents the gaussian subordination framework to generate optimal one sided approximations to multidimensional real valued functions by functions of prescribed exponential type such extremal problems date back to the works of beurling and selberg and provide a variety of applications in analysis and analytic number theory here we majorize and minorize on bbb r n the gaussian bf x mapsto e pi lambda bf x 2 where lambda0 is a free parameter by functions with distributional fourier transforms supported on euclidean balls optimizing weighted l 1 errors by integrating the parameter lambda against suitable measures we solve the analogous problem for a wide class of radial functions applications to inequalities and periodic analogues are discussed the constructions presented here rely on the theory of de branges spaces of entire functions and on new interpolation tools derived from the theory of laplace transforms of laguerre p o lya functions

decoupling shrinkage and selection in bayesian linear models a posterior summary perspective selecting a subset of variables for linear models remains an active area of research this article reviews many of the recent contributions to the bayesian model selection and shrinkage prior literature a posterior variable selection summary is proposed which distills a full posterior distribution over regression coefficients into a sequence of sparse linear predictors

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

interpolation formulas with derivatives in de branges spaces ii abstract we investigate necessary and sufficient conditions under which entire functions in de branges spaces can be recovered from function values and values of derivatives our main focus is on spaces with a structure function whose logarithmic derivative is bounded in the upper half plane

a predictive approach to the analysis of designed experiments abstract viewing the analysis of designed experiments as a model selection problem we introduce the use of a predictive bayesian criterion in this context based on the predictive density of a replicate experiment pdre a calibration of the criterion is provided to assist in the model choice the relationships of the proposed criterion to other prevalent criteria such as aic bic and mallows s c p are given an information theoretic criterion based on the pdre s of two competing models is also introduced and compared with the usual f statistic for two nested models examples are given to illustrate the proposed methodology

bayesian variable selection in linear regression abstract this article is concerned with the selection of subsets of predictor variables in a linear regression model for the prediction of a dependent variable it is based on a bayesian approach intended to be as objective as possible a probability distribution is first assigned to the dependent variable through the specification of a family of prior distributions for the unknown parameters in the regression model the method is not fully bayesian however because the ultimate choice of prior distribution from this family is affected by the data it is assumed that the predictors represent distinct observables the corresponding regression coefficients are assigned independent prior distributions for each regression coefficient subject to deletion from the model the prior distribution is a mixture of a point mass at 0 and a diffuse uniform distribution elsewhere that is a spike and slab distribution the random error component is assigned a normal distribution with mean 0 and standard deviation

characterization of strong observability and construction of an observer abstract for given matrices a b c there is considered the time invariant linear system x ax bu y cx with state x input u and output y it is called strongly observable if x ax bu cx t 0 with a piecewise continuous control u t always implies x t 0 this means that for any piecewise continuous input u t the output y t can vanish identically only if the state x t vanishes already so that the state x t can be expressed observed by the output y t alone without knowing u t the derivation of such a formula observer which expresses x t in terms of y t alone for time invariant systems i e constant matrices a b c is one part of the contents of this note the other part consists of characterizations of strong observability by rank conditions concerning the matrices a b and c similarly to the well known rank condition for controllability or observability

nonparametric regression using bayesian variable selection abstract this paper estimates an additive model semiparametrically while automatically selecting the significant independent variables and the appropriate power transformation of the dependent variable the nonlinear variables are modeled as regression splines with significant knots selected from a large number of candidate knots the estimation is made robust by modeling the errors as a mixture of normals a bayesian approach is used to select the significant knots the power transformation and to identify outliers using the gibbs sampler to carry out the computation empirical evidence is given that the sampler works well on both simulated and real examples and that in the univariate case it compares favorably with a kernel weighted local linear smoother the variable selection algorithm in the paper is substantially faster than previous bayesian variable selection algorithms

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

the influence of variable selection a bayesian diagnostic perspective abstract variable selection is ubiquitous in statistical practice forward selection backwards elimination or a combination thereof are among the most popular techniques statistical texts teach direct interpretation of coefficients after variable selection for scientific purposes the desired interpretation is unconditional on the variable selection in this article an influence analysis of variable selection is performed from a bayesian viewpoint variable selection is shown to be surprisingly influential a new statistic is recommended for routine examination during variable selection

lindley s paradox 1982 lindley s paradox journal of the american statistical association vol 77 no 378 pp 325 334

bayesian variable selection in structured high dimensional covariate spaces with applications in genomics we consider the problem of variable selection in regression modeling in high dimensional spaces where there is known structure among the covariates this is an unconventional variable selection problem for two reasons 1 the dimension of the covariate space is comparable and often much larger than the number of subjects in the study and 2 the covariate space is highly structured and in some cases it is desirable to incorporate this structural information in to the model building process we approach this problem through the bayesian variable selection framework where we assume that the covariates lie on an undirected graph and formulate an ising prior on the model space for incorporating structural information certain computational and statistical problems arise that are unique to such high dimensional structured settings the most interesting being the phenomenon of phase transitions we propose theoretical and computational schemes to mitigate these problems we illustrate our methods on two

predictive model selection we consider the problem of selecting one model from a large class of plausible models a predictive bayesian viewpoint is advocated to avoid the specification of prior probabilities for the candidate models and the detailed interpretation of the parameters in each model using criteria derived from a certain predictive density and a prior specification that emphasizes the observables we implement the proposed methodology for three common problems arising in normal linear models variable subset selection selection of a transformation of predictor variables and estimation of a parametric variance function interpretation of the relative magnitudes of the criterion values for various models is facilitated by a calibration of the criteria relationships between the proposed criteria and other well known criteria are examined

detecting differentially expressed genes in microarrays using bayesian model selection dna microarrays open up a broad new horizon for investigators interested in studying the genetic determinants of disease the high throughput nature of these arrays where differential expression for thousands of genes can be measured simultaneously creates an enormous wealth of information but also poses a challenge for data analysis because of the large multiple testing problem involved the solution has generally been to focus on optimizing false discovery rates while sacrificing power the drawback of this approach is that more subtle expression differences will be missed that might give investigators more insight into the genetic environment necessary for a disease process to take hold we introduce a new method for detecting differentially expressed genes based on a high dimensional model selection technique bayesian anova for microarrays bam which strikes a balance between false rejections and false nonrejections the basis of the new approach involves a weighted average of generalized ridge

bayesian identification selection and estimation of semiparametric functions in high dimensional additive models in this paper we propose an approach to both estimate and select unknown smooth functions in an additive model with potentially many functions each function is written as a linear combination of basis terms with coefficients regularized by a proper linearly constrained gaussian prior given any potentially rank deficient prior precision matrix we show how to derive linear constraints so that the corresponding effect is identified in the additive model this allows for the use of a wide range of bases and precision matrices in priors for regularization by introducing indicator variables each constrained gaussian prior is augmented with a point mass at zero thus allowing for function selection posterior inference is calculated using markov chain monte carlo and the smoothness in the functions is both the result of shrinkage through the constrained gaussian prior and model averaging we show how using non degenerate priors on the shrinkage parameters enables the application of substantially more computationally efficient sampling schemes than would otherwise be the case we show the favourable performance of our approach when compared to two contemporary alternative bayesian methods to highlight the potential of our approach in high dimensional settings we apply it to estimate two large seemingly unrelated regression models for intra day electricity load both models feature a variety of different univariate and bivariate functions which require different levels of smoothing and where component selection is meaningful priors for the error disturbance covariances are selected carefully and the empirical results provide a substantive contribution to the electricity load modelling literature in their own right

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

bounding s t and s_1 t on the riemann hypothesis let pi s t denote the argument of the riemann zeta function zeta s at the point s frac 1 2 it assuming the riemann hypothesis we present two proofs of the bound begin aligned s t le left frac 1 4 o 1 right frac log t log log t end aligned for large t this improves a result of goldston and gonek by a factor of 2 the first method consists of bounding the auxiliary function s 1 t int 0 t s u text d u using extremal functions constructed by carneiro littmann and vaaler we then relate the size of s t to the size of the functions s 1 t pm h s 1 t when h asymp 1 log log t the alternative approach bounds s t directly relying on the solution of the beurling selberg extremal problem for the odd function f x arctan left frac 1 x right frac x 1 x 2 this draws upon recent work by carneiro and littmann

hilbert spaces and the pair correlation of zeros of the riemann zeta function montgomery s pair correlation conjecture predicts the asymptotic behavior of the function n t defined to be the number of pairs and of ordinates of nontrivial zeros of the riemann zetafunction satisfying 0 0 using montgomery s formula and some extremal functions of exponential type these functions are optimal in the sense that they majorize and minorize the characteristic function of the interval in a way to minimize the l1 r 1 sin x x 2 dx error we give a complete solution for this extremal problem using the framework of reproducing kernel hilbert spaces of entire functions this extends previous work of p x gallagher 18 in 1985 where the case 1 2 n was considered using non extremal majorants and minorants

some comments on cp i study the typical configuration of a cp plot when the number of variables in the regression problem is large and there are many weak effects i show that a particular configuration that is very commonly seen can arise in a simple way i give a formula by means of which the risk incurred by the minimum cp rule can be estimated

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

model averaging and dimension selection for the singular value decomposition many multivariate data analysis techniques for an m n matrix y are related to the model y m e where y is an m n matrix of full rank and m is an unobserved mean matrix of rank k m n typically the rank of m is estimated in a heuristic way and then the least squares estimate of m is obtained via the singular value decomposition of y yielding an estimate that can have a very high variance in this article we suggest a model based alternative to the preceding approach by providing prior distributions and posterior estimation for the rank of m and the components of its singular value decomposition in addition to providing more accurate inference such an approach has the advantage of being extendable to more general data analysis situations such as inference in the presence of missing data and estimation in a generalized linear modeling framework

extremal functions in de branges and euclidean spaces abstract in this work we obtain optimal majorants and minorants of exponential type for a wide class of radial functions on r n these extremal functions minimize the l 1 r n x 2 2 n d x distance to the original function where 1 is a free parameter to achieve this result we develop new interpolation tools to solve an associated extremal problem for the exponential function f x e x where 0 in the general framework of de branges spaces of entire functions we then specialize the construction to a particular family of homogeneous de branges spaces to approach the multidimensional euclidean case finally we extend the result from the exponential function to a class of subordinated radial functions via integration on the parameter 0 against suitable measures applications of the results presented here include multidimensional versions of hilbert type inequalities extremal one sided approximations by trigonometric polynomials for a class of even periodic functions and extremal one sided approximations by polynomials for a class of functions on the sphere s n 1 with an axis of symmetry

partial factor modeling predictor dependent shrinkage for linear regression we develop a modified gaussian factor model for the purpose of inducing predictor dependent shrinkage for linear regression the new model predicts well across a wide range of covariance structures on real and simulated data furthermore the new model facilitates variable selection in the case of correlated predictor variables which often stymies other methods

observers for linear systems with unknown inputs a direct design procedure of a full order observer for a linear system with unknown inputs is presented using straightforward matrix calculations some examples are given in these examples a reduced order observer is also derived it is shown that this may restrict the rate of convergence of some state estimates

a bayesian approach to nonparametric bivariate regression abstract this article outlines a general bayesian approach to estimating a bivariate regression function in a nonparametric manner it models the function using a bivariate regression spline basis with many terms binary indicator variables corresponding to these terms are introduced to explicitly model the uncertainty of whether or not the terms provide a significant contribution to the regression the regression function is estimated using an estimate of its posterior mean smoothing over the distribution of these binary indicator variables to make the computations tractable all estimates are obtained using markov chain monte carlo sampling extensive simulated comparisons are provided that demonstrate the competitive performance of this approach against other data driven bivariate surface estimators prominent in the literature it is then shown how the procedure can be extended to provide a general approach to nonparametric bivariate surface estimation in two difficult regression settings the first

gaussian subordination for the beurling selberg extremal problem we determine extremal entire functions for the problem of ma jorizing minorizing and approximating the gaussian function e x 2 by en tire functions of exponential type the combination of the gaussian and a general distribution approach provides the solution of the extremal problem for a wide class of even functions that includes most of the previously known examples for instance 3 4 10 and 17 plus a variety of new interesting functions such asjxj for 1 log x 2 2 x 2 2 for 0 log x2 2 andx2n logx2 forn2 n further applications to number theory include optimal approximations of theta functions by trigonometric polynomi als and optimal bounds for certain hilbert type inequalities related to the discrete hardy littlewood sobolev inequality in dimension one

a class of extremal functions for the fourier transform we determine a class of real valued integrable functions f x and corresponding functions m x such that f x 1 and the value of mfo is miimi several applications of these functions to number theory and analysis are given

extremal problems in de branges spaces the case of truncated and odd functions in this paper we find extremal one sided approximations of exponential type for a class of truncated and odd functions with a certain exponential subordination these approximations optimize the l 1 mathbb r e x 2 mathrm d x error where e is an arbitrary hermite biehler entire function of bounded type in the upper half plane this extends the work of holt and vaaler duke math j 83 203 247 1996 for the signum function we also provide periodic analogues of these results finding optimal one sided approximations by trigonometric polynomials of a given degree to a class of periodic functions with exponential subordination these extremal trigonometric polynomials optimize the l 1 mathbb r mathbb z mathrm d vartheta error where vartheta is an arbitrary nontrivial measure on mathbb r mathbb z the periodic results extend the work of li and vaaler indiana univ math j 48 1 183 236 1999 who considered this problem for the sawtooth function with respect to jacobi measures our techniques are based on the theory of reproducing kernel hilbert spaces of entire functions and of polynomials and on the construction of suitable interpolations with nodes at the zeros of laguerre polya functions

a two stage bayesian model selection strategy for supersaturated designs in early stages of experimentation one often has many candidate factors of which only few have significant influence on the response supersaturated designs can offer important advantages however standard regression techniques of fitting a prediction line using all candidate variables fail to analyze data from such designs stepwise regression may be used but has drawbacks as reported in the literature a two stage bayesian model selection strategy able to keep all possible models under consideration while providing a level of robustness akin to bayesian analyses incorporating noninformative priors is proposed the strategy is demonstrated on a well known dataset and compared to competing methods via simulation

on the pitman yor process with spike and slab base measure summaryfor the most popular discrete nonparametric models beyond the dirichlet process the prior guess at the shape of the data generating distribution also known as the base measure is assumed to be diffuse such a specification greatly simplifies the derivation of analytical results allowing for a straightforward implementation of bayesian nonparametric inferential procedures however in several applied problems the available prior information leads naturally to the incorporation of an atom into the base measure and then the dirichlet process is essentially the only tractable choice for the prior in this paper we fill this gap by considering the pitman yor process with an atom in its base measure we derive computable expressions for the distribution of the induced random partitions and for the predictive distributions these findings allow us to devise an effective generalized polya urn gibbs sampler applications to density estimation clustering and curve estimation with both simulated and real data serve as an illustration of our results and allow comparisons with existing methodology in particular we tackle a functional data analysis problem concerning basal body temperature curves

a local characterization of observability abstract we consider time dependent linear systems of the form x ax bu y cx with state x rn control input u rm and output y rp the main results are local characterizations of observability and strong observability or observability with unknown inputs of a c and a b c these criteria are pointwise rank conditions on a certain matrix which is explicitly built up from the first n 2 derivatives of a and b and the first n 1 derivatives of c the results generalize well known theorems for time invariant systems the proofs lead also to observers with and without the input and the main tool is a generalized product rule for the differentiation of a product of matrices where only one factor and the product itself are known to be differentiable

on the strong observability in linear time varying singular systems abstract in this article the strong observability conditions necessary and sufficient conditions originally stated for the state reconstruction of linear systems with unknown inputs are extended to the class of linear time varying singular systems this novel local in a time interval of interest conditions are given under a constant rank assumption a method to reconstruct the state in finite time in linear time varying singular systems despite the presence of unknown inputs is presented the proposed observer consists of two parts one part that ensures the asymptotic convergence of the estimation errors to a bounded region around the origin and another one that ensures the finite time convergence of the state estimates to their actual values

regressions by leaps and bounds this paper describes several algorithms for computing the residual sums of squares for all possible regressions with what appears to be a minimum of arithmetic less than six floating point operations per regression and shows how two of these algorithms can be combined to form a simple leap and bound technique for finding the best subsets without examining all possible subsets the result is a reduction of several orders of magnitude in the number of operations required to find the best subsets

a truncated em approach for spike and slab sparse coding we study inference and learning based on a sparse coding model with spike and slab prior as in standard sparse coding the model used assumes independent latent sources that linearly combine to generate data points however instead of using a standard sparse prior such as a laplace distribution we study the application of a more flexible spike and slab distribution which models the absence or presence of a source s contribution independently of its strength if it contributes we investigate two approaches to optimize the parameters of spike and slab sparse coding a novel truncated em approach and for comparison an approach based on standard factored variational distributions the truncated approach can be regarded as a variational approach with truncated posteriors as variational distributions in applications to source separation we find that both approaches improve the state of the art in a number of standard benchmarks which argues for the use of spike and slab priors for the corresponding data domains furthermore we find that the truncated em approach improves on the standard factored approach in source separation tasks which hints to biases introduced by assuming posterior independence in the factored variational approach likewise on a standard benchmark for image denoising we find that the truncated em approach improves on the factored variational approach while the performance of the factored approach saturates with increasing numbers of hidden dimensions the performance of the truncated approach improves the state of the art for higher noise levels

some extremal functions in fourier analysis ii we obtain extremal majorants and minorants of exponential type for a class of even functions on which includes log x and x where 1 1 we also give periodic versions of these results in which the majorants and minorants are trigonometric polynomials of bounded degree as applications we obtain optimal estimates for certain hermitian forms which include discrete analogues of the one dimensional hardy littlewood sobolev inequalities a further application provides an erdos turan type inequality that estimates the sup norm of algebraic polynomials on the unit disc in terms of power sums in the roots of the polynomials

a note on the zeros of zeta and l functions let pi s t denote the argument of the riemann zeta function at the point s tfrac 1 2 it assuming the riemann hypothesis we give a new and simple proof of the sharpest known bound for s t we discuss a generalization of this bound for a large class of l functions including those which arise from cuspidal automorphic representations of gl m over a number field we also prove a number of related results including bounding the order of vanishing of an l function at the central point and bounding the height of the lowest zero of an l function

an alternative to the standard bayesian procedure for discrimination between normal linear models summary we consider the standard bayesian procedure for discrimination focusing on its tendency to give low posterior probabilities to some correct models even in conditions of moderate to large sampling sizes furthermore in some situations that could well occur in practice the standard procedure is even asymptotically inconsistent we claim that the explanation is that the standard procedure inflates the posterior probability of the model that has the smallest expected increase in information about its parameters we then propose to modify it by an amount related to the expected gain in information and show that the lindley paradox does not occur finally we obtain the limiting alternative criterion that results by letting the prior densities become noninformative which enables us to measure how sensitive the discrimation is to the informative prior densities used

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

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