the gaussian measure of shifted balls let be a centered gaussian measure on a hilbert spaceh and let b r subseteq h be the centered ball of radiusr 0 fora h and mathop lim limits t mathbf to mathbf infty mathbf r t t mathbf a we give the exact asymptotics of b r t t a ast also upper and lower bounds are given when is defined on an arbitrary separable banach space our results range from small deviation estimates to large deviation estimates

cluster analysis for cognitive diagnosis theory and applications latent class models for cognitive diagnosis often begin with specification of a matrix that indicates which attributes or skills are needed for each item then by imposing restrictions that take this into account along with a theory governing how subjects interact with items parametric formulations of item response functions are derived and fitted cluster analysis provides an alternative approach that does not require specifying an item response model but does require an item by attribute matrix after summarizing the data with a particular vector of sum scores k means cluster analysis or hierarchical agglomerative cluster analysis can be applied with the purpose of clustering subjects who possess the same skills asymptotic classification accuracy results are given along with simulations comparing effects of test length and method of clustering an application to a language examination is provided to illustrate how the methods can be implemented in practice

optimal bayesian estimation in random covariate design with a rescaled gaussian process prior in bayesian nonparametric models gaussian processes provide a popular prior choice for regression function estimation existing literature on the theoretical investigation of the resulting posterior distribution almost exclusively assume a fixed design for covariates the only random design result we are aware of van der vaart and van zanten 2011 assumes the assigned gaussian process to be supported on the smoothness class specified by the true function with probability one this is a fairly restrictive assumption as it essentially rules out the gaussian process prior with a squared exponential kernel when modeling rougher functions in this article we show that an appropriate rescaling of the above gaussian process leads to a rate optimal posterior distribution even when the covariates are independently realized from a known density on a compact set the proofs are based on deriving sharp concentration inequalities for frequentist kernel estimators the results might be of independent interest

on estimating a density using hellinger distance and some other strange facts on s interesse ici aux possibles vitesses d estimation d une densite a support compact dans m sous des hypotheses de regularite lorsque la perte est mesuree par le carre de la distance de hellinger on regardera aussi le cas connu des normes mathbb l q pour 1 q 2 et le risque est le risque minimax sur la famille on donne une methode generale permettant de traiter les problemes dans le cadre de la theorie de l approximation sous des conditions concernant l entropie metrique et l e capacite des familles a estimer les rapports entre regularite et entropie metrique etant bien connus nous pourrons aussi traiter les cas classiques et d autres qui le sont moins sous des conditions de bornes inferieures les vitesses sont celles observees pour la norme mathbb l q mais elles different dans le cas general on montre aussi que les restrictions sur la compacite du support ou la regularite sont indispensables et que leur absence mene a l impossibilite d obtenir une estimation raisonnable en ce sens que n importe quelle suite d estimateurs sera arbitrairement mauvaise en un point au moins un resultat analogue est vrai sous des conditions de regularite

random rates in anisotropic regression in the context of minimax theory we propose a new kind of risk normalized by a random variable measurable with respect to the data we present a notion of optimality and a method to construct optimal procedures accordingly we apply this general setup to the problem of selecting significant variables in gaussian white noise in particular we show that our method essentially improves the accuracy of estimation in the sense of giving explicit improved confidence sets in l2 norm links to adaptive estimation are discussed 1 introduction searching for significant variables is certainly one of the oldest and most popular problems in statistics one of the simplest models where the issue of selecting significant variables was first stated mathematically is linear regression a vast literature has been devoted to this topic since and different approaches have been proposed over the last forty years both for estimation and for hypothesis testing among many authors we refer to akaike 1 breiman and freedman 3 chernoff 5 csiszar and korner 6 dychakov 10 patel 42 renyi 46 freidlina 13 meshalkin 35 malyutov and tsitovich 34 schwarz 47 and stone 48 in classical parametric regression if we consider a linear model we first have to measure the possible gain of searching for a limited number of significant variables if the model comes from a specific field of application then only an adequate description together with its solution is relevant however from a mathematical point of view a theory of selecting significant variables does not lead at least asymptotically to a substantial improvement of the accuracy of estimation in a regular parametric model the classical n rate of convergence is not affected by the number of significant variables however even in this setup let us emphasize that asymptotically has to be understood as up to a constant and that the correct choice of significant variables may possibly improve this constant if instead of a linear model we consider a nonparametric regression model the search for significant variables becomes crucial for estimating the regression function the rate of convergence explicitly depends on the set of significant variables let us develop this statement with the following example of multivariate regression suppose we observe z n xi y i i 1 n in the model

identification of the 1pl model with guessing parameter parametric and semi parametric results in this paper we study the identification of a particular case of the 3pl model namely when the discrimination parameters are all constant and equal to 1 we term this model 1pl g model the identification analysis is performed under three different specifications the first specification considers the abilities as unknown parameters it is proved that the item parameters and the abilities are identified if a difficulty parameter and a guessing parameter are fixed at zero the second specification assumes that the abilities are mutually independent and identically distributed according to a distribution known up to the scale parameter it is shown that the item parameters and the scale parameter are identified if a guessing parameter is fixed at zero the third specification corresponds to a semi parametric 1pl g model where the distribution g generating the abilities is a parameter of interest it is not only shown that after fixing a difficulty parameter and a guessing parameter at zero the item parameters are identified but also that under those restrictions the distribution g is not identified it is finally shown that after introducing two identification restrictions either on the distribution g or on the item parameters the distribution g and the item parameters are identified provided an infinite quantity of items is available

preserving poisson structure and orthogonality in numerical integration of differential equations we consider the numerical integration of two types of systems of differential equations we first consider hamiltonian systems of differential equations with a poisson structure we show that symplectic runge kutta methods preserve this structure when the poisson tensor is constant using nonlinear changes of coordinates this structure can also be preserved for nonconstant poisson tensors as exemplified on the euler equations for the free rigid body we also consider orthogonal flows and the closely related class of isospectral flows to numerically preserve the orthogonality property we take the approach of formulating an equivalent system of differential algebraic equations daes and of integrating the system with a special combination of a particular class of runge kutta methods this approach requires only matrix matrix products and can preserve geometric properties of the flow such as reversibility

information consistency of nonparametric gaussian process methods bayesian nonparametric models are widely and successfully used for statistical prediction while posterior consistency properties are well studied in quite general settings results have been proved using abstract concepts such as metric entropy and they come with subtle conditions which are hard to validate and not intuitive when applied to concrete models furthermore convergence rates are difficult to obtain by focussing on the concept of information consistency for bayesian gaussian process gp models consistency results and convergence rates are obtained via a regret bound on cumulative log loss these results depend strongly on the covariance function of the prior process thereby giving a novel interpretation to penalization with reproducing kernel hilbert space norms and to commonly used covariance function classes and their parameters the proof of the main result employs elementary convexity arguments only a theorem of widom is used in order to obtain precise convergence rates for several covariance functions widely used in practice

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

anisotropic function estimation using multi bandwidth gaussian processes in nonparametric regression problems involving multiple predictors there is typically interest in estimating an anisotropic multivariate regression surface in the important predictors while discarding the unimportant ones our focus is on defining a bayesian procedure that leads to the minimax optimal rate of posterior contraction up to a log factor adapting to the unknown dimension and anisotropic smoothness of the true surface we propose such an approach based on a gaussian process prior with dimension specific scalings which are assigned carefully chosen hyperpriors we additionally show that using a homogenous gaussian process with a single bandwidth leads to a sub optimal rate in anisotropic cases

on bayesian supremum norm contraction rates building on ideas from castillo and nickl 6 a method is pro vided to study nonparametric bayesian posterior convergence rates when strong measures of distances such as the sup norm are considered in particular we show that likelihood methods can achieve optimal minimax sup norm rates in density estimation on the unit interval the introduced methodology is used to prove that commonly used families of prior distri butions on densities namely log density priors and dyadic random density histograms can indeed achieve optimal sup norm rates of convergence new results are also derived in the gaussian white noise model as a further il lustration of the presented techniques ams 2000 subject classications primary 62g20 secondary 62g05 62g07

inequalities of the brunn minkowski type for gaussian measures let m 2 be an integer let be the standard gaussian measure on rn and let t t exp s2 2 ds sqrt 2 small t le given 1 l m 0 this paper gives a necessary and sufficient condition such that the inequality 1 1 a1 m a m 1 1 a 1 m 1 a m is true for all borel sets a 1 a m in hbfrn of strictly positive measure or all convex borel sets a 1 a m in bfrn of strictly positive measure respectively in particular the paper exhibits inequalities of the brunn minkowski type for which are true for all convex sets but not for all measurable sets

adaptive bayesian estimation using a gaussian random field with inverse gamma bandwidth we consider nonparametric bayesian estimation inference using a rescaled smooth gaussian field as a prior for a multidimensional function the rescaling is achieved using a gamma variable and the procedure can be viewed as choosing an inverse gamma bandwidth the procedure is studied from a frequentist perspective in three statistical settings involving replicated observations density estimation regression and classification we prove that the resulting posterior distribution shrinks to the distribution that generates the data at a speed which is minimax optimal up to a logarithmic factor whatever the regularity level of the data generating distribution thus the hierachical bayesian procedure with a fixed prior is shown to be fully adaptive

building accurate emulators for stochastic simulations via quantile kriging computer simulation has increasingly become popular for analysis of systems that cannot be feasibly changed because of costs or scale this work proposes a method to construct an emulator for stochastic simulations by performing a designed experiment on the simulator and developing an emulative distribution existing emulators have focused on estimation of the mean of the simulation output but this work presents an emulator for the distribution of the output this construction provides both an explicit distribution and a fast sampling scheme beyond the emulator description this work demonstrates the emulator s efficiency that is its convergence rate is the asymptotically optimal among all possible emulators using the same sample size under certain conditions an example of its practical use is demonstrated using a stochastic simulation of fracture mechanics supplementary materials for this article are available online

rates of contraction of posterior distributions based on gaussian process priors we derive rates of contraction of posterior distributions on nonparametric or semiparametric models based on gaussian processes the rate of contraction is shown to depend on the position of the true parameter relative to the reproducing kernel hilbert space of the gaussian process and the small ball probabilities of the gaussian process we determine these quantities for a range of examples of gaussian priors and in several statistical settings for instance we consider the rate of contraction of the posterior distribution based on sampling from a smooth density model when the prior models the log density as a fractionally integrated brownian motion we also consider regression with gaussian errors and smooth classification under a logistic or probit link function combined with various priors

supremum norm posterior contraction and credible sets for nonparametric multivariate regression in the setting of nonparametric multivariate regression with unknown error variance we study asymptotic properties of a bayesian method for estimating a regression function f and its mixed partial derivatives we use a random series of tensor product of b splines with normal basis coefficients as a prior for f and the error variance is either estimated using the empirical bayes approach or is endowed with a suitable prior in a hierarchical bayes approach we establish pointwise l2 and supremum norm posterior contraction rates for f and its mixed partial derivatives and show that they coincide with the minimax rates our results cover even the anisotropic situation where the true regression function may have different smoothness in different directions using the convergence bounds we show that pointwise l2 and supremum norm credible sets for f and its mixed partial derivatives have guaranteed frequentist coverage with optimal size new results on tensor products of b splines are also obtained in the course

equivalent linear logistic test models this paper is about the linear logistic test model lltm we demonstrate that there are infinitely many equivalent ways to specify a model an implication is that there may well be many ways to change the specification of a given lltm and achieve the same improvement in model fit to illustrate this phenomenon we analyze a real data set using a lagrange multiplier test for the specification of the model this lagrange multiplier test is similar to the modification index used in structural equation modeling

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

risk bounds for model selection via penalization performance bounds for criteria for model selection are devel oped using recent theory for sieves the model selection criteria are based on an empirical loss or contrast function with an added penalty term moti vated by empirical process theory and roughly proportional to the number of parameters needed to describe the model divided by the number of ob servations most of our examples involve density or regression estimation settings and we focus on the problem of estimating the unknown density or regression function we show that the quadratic risk of the minimum penal ized empirical contrast estimatoris bounded by an index of the accuracy of the sieve this accuracy index quanties the trade off among the candidate models between the approximation error and parameter dimension relative to sample size if we choose a list of models which exhibit good approximation prop erties with respect to different classes of smoothness the estimator can be simultaneously minimax rate optimal in each of those classes this is what is usually called adaptation the type of classes of smoothness in which one gets adaptation depends heavily on the list of models if too many models are involved in order to get accurate approximation of many wide classes of functions simultaneously it may happen that the estimator is only approx

the generalized dina model framework the g dina generalized deterministic inputs noisy and gate model is a generalization of the dina model with more relaxed assumptions in its saturated form the g dina model is equivalent to other general models for cognitive diagnosis based on alternative link functions when appropriate constraints are applied several commonly used cognitive diagnosis models cdms can be shown to be special cases of the general models in addition to model formulation the g dina model as a general cdm framework includes a component for item by item model estimation based on design and weight matrices and a component for item by item model comparison based on the wald test the paper illustrates the estimation and application of the g dina model as a framework using real and simulated data it concludes by discussing several potential implications of and relevant issues concerning the proposed framework

identifying latent structures in restricted latent class models this article focuses on a family of restricted latent structure models with wide applications in psychological and educational assessment where the model parameters are restricted via a latent str

statistical analysis of q matrix based diagnostic classification models diagnostic classification models dmcs have recently gained prominence in educational assessment psychiatric evaluation and many other disciplines central to the model specification is the so called q matrix that provides a qualitative specification of the item attribute relationship in this article we develop theories on the identifiability for the q matrix under the dina and the dino models we further propose an estimation procedure for the q matrix through the regularized maximum likelihood the applicability of this procedure is not limited to the dina or the dino model and it can be applied to essentially all q matrix based dmcs simulation studies show that the proposed method admits high probability recovering the true q matrix furthermore two case studies are presented the first case is a dataset on fraction subtraction educational application and the second case is a subsample of the national epidemiological survey on alcohol and related conditions concerning the social anxiety disord

a conditional density estimation partition model using logistic gaussian processes conditional density estimation density regression estimates the distribution of a response variable y conditional on covariates x utilizing a partition model framework a conditional density estimation method is proposed using logistic gaussian processes the partition is created using a voronoi tessellation and is learned from the data using a reversible jump markov chain monte carlo algorithm the markov chain monte carlo algorithm is made possible through a laplace approximation on the latent variables of the logistic gaussian process model this approximation marginalizes the parameters in each partition element allowing an efficient search of the posterior distribution of the tessellation the method has desirable consistency properties in simulation and applications the model successfully estimates the partition structure and conditional distribution of y

objective bayesian variable selection a novel fully automatic bayesian procedure for variable selection in normal regression models is proposed the procedure uses the posterior probabilities of the models to drive a stochastic search the posterior probabilities are computed using intrinsic priors which can be considered default priors for model selection problems that is they are derived from the model structure and are free from tuning parameters thus they can be seen as objective priors for variable selection the stochastic search is based on a metropolis hastings algorithm with a stationary distribution proportional to the model posterior probabilities the procedure is illustrated on both simulated and real examples

thomas bayes walk on manifolds convergence of the bayes posterior measure is considered in canonical statistical settings where observations sit on a geometrical object such as a compact manifold or more generally on a compact metric space verifying some conditions a natural geometric prior based on randomly rescaled solutions of the heat equation is considered upper and lower bound posterior contraction rates are derived

equivalent mirid models it is shown that in the context of the model with internal restrictions on the item difficulties mirid different componential theories about an item set may lead to equivalent models furthermore we provide conditions for the identifiability of the mirid model parameters and it will be shown how the mirid model relates to the linear logistic test model lltm while it is known that the lltm is a special case of the mirid we show that it is possible to construct an lltm that encompasses the mirid the mirid model places a bilinear restriction on the item parameters of the rasch model it is explained how this fact is used to simplify the results of bechger verhelst and verstralen 2001 and bechger verstralen and verhelst 2002 and extend their scope to a wider class of models

metric entropy and the small ball problem for gaussian measures abstract we establish a precise link between the small ball problem for a gaussian measure on a separable banach space and the metric entropy of the unit ball of the hubert space h generating this link allows us to compute small ball probabilities from metric entropy results and vice versa

bayesian manifold regression there is increasing interest in the problem of nonparametric regression with high dimensional predictors when the number of predictors d is large one encounters a daunting problem in attempting to estimate a d dimensional surface based on limited data fortunately in many applications the support of the data is concentrated on a d dimensional subspace with d ll d manifold learning attempts to estimate this subspace our focus is on developing computationally tractable and theoretically supported bayesian nonparametric regression methods in this context when the subspace corresponds to a locally euclidean compact riemannian manifold we show that a gaussian process regression approach can be applied that leads to the minimax optimal adaptive rate in estimating the regression function under some conditions the proposed model bypasses the need to estimate the manifold and can be implemented using standard algorithms for posterior computation in gaussian processes finite sample performance is illustrated in a data analysis example

information rates of nonparametric gaussian process methods we consider the quality of learning a response function by a nonparametric bayesian approach using a gaussian process gp prior on the response function we upper bound the quadratic risk of the learning procedure which in turn is an upper bound on the kullback leibler information between the predictive and true data distribution the upper bound is expressed in small ball probabilities and concentration measures of the gp prior we illustrate the computation of the upper bound for the matern and squared exponential kernels for these priors the risk and hence the information criterion tends to zero for all continuous response functions however the rate at which this happens depends on the combination of true response function and gaussian prior and is expressible in a certain concentration function in particular the results show that for good performance the regularity of the gp prior should match the regularity of the unknown response function

unitary integrators and applications to continuous orthonormalization techniques in this paper the issue of integrating matrix differential systems whose solutions are unitary matrices is addressed such systems have skew hermitian coefficient matrices in the linear case and a

a class of renyi information estimators for multidimensional densities a class of estimators of the renyi and tsallis entropies of an unknown distribution f in r m is presented these estimators are based on the k th nearest neighbor distances computed from a sample of n i i d vectors with distribution f we show that entropies of any order q including shannon s entropy can be estimated consistently with minimal assumptions on f moreover we show that it is straightforward to extend the nearest neighbor method to estimate the statistical distance between two distributions using one i i d sample from each

maximum likelihood estimation for stochastic differential equations using sequential gaussian process based optimization abstractstochastic differential equations sdes are used as statistical models in many disciplines however intractable likelihood functions for sdes make inference challenging and we need to resort to simulation based techniques to estimate and maximize the likelihood function while importance sampling methods have allowed for the accurate evaluation of likelihoods at fixed parameter values there is still a question of how to find the maximum likelihood estimate in this article we propose an efficient gaussian process based method for exploring the parameter space using estimates of the likelihood from an importance sampler our technique accounts for the inherent monte carlo variability of the estimated likelihood and does not require knowledge of gradients the procedure adds potential parameter values by maximizing the so called expected improvement leveraging the fact that the likelihood function is assumed to be smooth our simulations demonstrate that our method has significant computation

identifiability of diagnostic classification models diagnostic classification models dcms are important statistical tools in cognitive diagnosis in this paper we consider the issue of their identifiability in particular we focus on one basic and popular model the dina model we propose sufficient and necessary conditions under which the model parameters are identifiable from the data the consequences in terms of the consistency of parameter estimates of fulfilling or failing to fulfill these conditions are illustrated via simulation the results can be easily extended to the dino model through the duality of the dina and dino models moreover the proposed theoretical framework could be applied to study the identifiability issue of other dcms

entropy search for information efficient global optimization contemporary global optimization algorithms are based on local measures of utility rather than a probability measure over location and value of the optimum they thus attempt to collect low function values not to learn about the optimum the reason for the absence of probabilistic global optimizers is that the corresponding inference problem is intractable in several ways this paper develops desiderata for probabilistic optimization algorithms then presents a concrete algorithm which addresses each of the computational intractabilities with a sequence of approximations and explicitly addresses the decision problem of maximizing information gain from each evaluation

on the numerical integration of orthogonal flows with runge kutta methods abstract this paper deals with the numerical integration of matrix differential equations of type y t f t y t y t where f maps for all t orthogonal to skew symmetric matrices it has been shown dieci et al siam j numer anal 31 1994 261 281 iserles and zanna technical report na5 univ of cambridge 1995 that gauss legendre runge kutta glrk methods preserve the orthogonality of the flow generated by y f t y y whenever f t y is a skew symmetric matrix but the implicit nature of the methods is a serious drawback in practical applications recently higham appl numer math 22 1996 217 223 has shown that there exist linearly implicit methods based on the glrk methods with orders 2 which preserve the orthogonality of the flow the aim of this paper is to study the order and stability properties of a class of linearly implicit orthogonal methods of glrk type obtained by extending higham s approach also two particular linearly implicit schemes with orders 3 and 4 based on the two stage glrk method that minimize the local truncation error are proposed in addition the results of several numerical experiments are presented to test the behaviour of the new methods

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

explicit methods based on a class of four stage fourth order runge kutta methods for preserving quadratic laws this paper concerns the study of explicit 4 stage fourth order runge kutta methods which preserve quadratic conservation laws when their weights are made solution dependent application to orthogonal and hamiltonian linear systems and numerical tests are also reported

control functionals for monte carlo integration summary a non parametric extension of control variates is presented these leverage gradient information on the sampling density to achieve substantial variance reduction it is not required that the sampling density be normalized the novel contribution of this work is based on two important insights a trade off between random sampling and deterministic approximation and a new gradient based function space derived from stein s identity unlike classical control variates our estimators improve rates of convergence often requiring orders of magnitude fewer simulations to achieve a fixed level of precision theoretical and empirical results are presented the latter focusing on integration problems arising in hierarchical models and models based on non linear ordinary differential equations

asymptotic behavior of bayes estimates under possibly incorrect models we prove that the posterior distribution in a possibly incorrect parametric model a s concentrates in a strong sense on the set of pseudotrue parameters determined by the true distribution as a consequence we obtain in the case of a unique pseudotrue parameter the strong consistency of pseudo bayes estimators w r t general loss functions further we present a simple example based on normal distributions and having two different pseudotrue parameters where pseudo bayes estimators have an essentially different asymptotic behavior than the pseudomaximum likelihood estimator while the mle is strongly consistent the sequence of posterior means is strongly inconsistent and a s almost all its accumulation points are not pseudotrue finally we give conditions under which a pseudo bayes estimator for a unique pseudotrue parameter has an asymptotic normal distribution

convergence rates of posterior distributions we consider the asymptotic behavior of posterior distributions and bayes estimators for infinite dimensional statistical models we give general results on the rate of convergence of the posterior measure these are applied to several examples including priors on finite sieves log spline models dirichlet processes and interval censoring

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

optimality of poisson processes intensity learning with gaussian processes in this paper we provide theoretical support for the so called sigmoidal gaussian cox process approach to learning the intensity of an inhomogeneous poisson process on a d dimensional domain this method was proposed by adams murray and mackay icml 2009 who developed a tractable computational approach and showed in simulation and real data experiments that it can work quite satisfactorily the results presented in the present paper provide theoretical underpinning of the method in particular we show how to tune the priors on the hyper parameters of the model in order for the procedure to automatically adapt to the degree of smoothness of the unknown intensity and to achieve optimal convergence rates

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