on the uniform consistency of bayes estimates for multinomial probabilities a k sided die is thrown n times to estimate the probabilities 1 k of landing on the various sides the mle of is the vector of empirical proportions p p 1 p k consider a set of bayesians that put uniformly positive prior mass on all reasonable subsets of the parameter space their posterior distributions will be uniformly concentrated near p sharp bounds are given using entropy these bounds apply to all sample sequences there are no exceptional null sets

objective bayes model selection in probit models we describe a new variable selection procedure for categorical responses where the candidate models are all probit regression models the procedure uses objective intrinsic priors for the model parameters which do not depend on tuning parameters and ranks the models for the different subsets of covariates according to their model posterior probabilities when the number of covariates is moderate or large the number of potential models can be very large and for those cases we derive a new stochastic search algorithm that explores the potential sets of models driven by their model posterior probabilities the algorithm allows the user to control the dimension of the candidate models and thus can handle situations when the number of covariates exceed the number of observations we assess through simulations the performance of the procedure and apply the vari able selector to a gene expression data set where the response is whether a patient exhibits pneumonia software needed to run the procedures is available in the r package varselectip copyright 2011 john wiley sons ltd categorical responses often appear in the analysis of the effect of a medical treatment which is applied to a collection of patients these patients are typically characterized by a set of p potential covariates and a first important statistical problem in the presence of covariates is to reduce the dimension of the model by retaining from the original p covariates those that have some influence in the observed responses thus we have a model selection problem in the class of 2 p models in this paper we describe a new variable selection procedure applicable to dichotomous responses that are modeled with a probit regression although under the 0 1 loss function an optimal solution is the model having the highest posterior probability the 0 1 loss function might not be realistic for many applications one potential shortcoming is that submodels different from the true one are assigned the same loss regardless of how far or close to the true one they are therefore it seems appropriate to not choose a single optimal model but a subset of them having posterior probabilities over a given threshold thus the output of our model selection procedure will be a ranking of the models according to their posterior probabilities the literature on variable selection in logistic or probit models is not large researchers have car ried out variable selection in logistic regression with lasso type procedures for instance meier et al 1 used a group lasso for logistic regression whereas kyung et al 2 used the latent variable probit model for bayesian lasso variable selection the bayesian approaches tend to dominate and researchers typically solve the main difficulties in its formulation the choice of the prior for the models involved and for the model parameters by using subjective priors swartz and shete 3 did a simulation study of the five types of variable selection in case control logistic regression and found that a bayesian stochastic search based on a spike and slab prior was the best performer chen and dey 4 also used a bayesian approach and were particularly concerned with correlated binary responses in multivariate logistic regression in their study the researchers developed subjective priors and selected models on the

tweedie s formula and selection bias we suppose that the statistician observes some large number of estimates zi each with its own unobserved expectation parameter i the largest few of the zi s are likely to substantially overestimate their corresponding i s this being an example of selection bias or regression to the mean tweedie s formula first reported by robbins in 1956 offers a simple empirical bayes approach for correcting selection bias this article investigates its merits and limitations in addition to the methodology tweedie s formula raises more general questions concerning empirical bayes theory discussed here as relevance and empirical bayes information there is a close connection between applications of the formula and james stein estimation

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

the logistic normal distribution for bayesian nonparametric predictive densities abstract this article models the common density of an exchangeable sequence of observations by a generalization of the process derived from a logistic transform of a gaussian process the support of the logistic normal includes all distributions that are absolutely continuous with respect to the dominating measure of the observations the logistic normal family is closed in the prior to posterior bayes analysis with the observations entering the posterior distribution through the covariance function of the gaussian process the covariance of the gaussian process plays the role of a smoothing kernel three features of the model provide a flexible structure for computing the predictive density a the mean of the gaussian process corresponds to the prior mean of the random density b the prior variance of the gaussian process controls the influence of the data in the posterior process as the variance increases the predictive density has greater fidelity to the data c the prior covariance of the gau

model selection principles in misspecified models summary model selection is of fundamental importance to high dimensional modelling featured in many contemporary applications classical principles of model selection include the bayesian principle and the kullback leibler divergence principle which lead to the bayesian information criterion and akaike information criterion respectively when models are correctly specified yet model misspecification is unavoidable in practice we derive novel asymptotic expansions of the two well known principles in misspecified generalized linear models which give the generalized bayesian information criterion and generalized akaike information criterion a specific form of prior probabilities motivated by the kullback leibler divergence principle leads to the generalized bayesian information criterion with prior probability gbicp which can be naturally decomposed as the sum of the negative maximum quasi log likelihood a penalty on model dimensionality and a penalty on model misspecification directly numerical studies demonstrate the advantage of the new methods for model selection in both correctly specified and misspecified models

model order selection based on information theoretic criteria design of the penalty information theoretic criteria itc have been widely adopted in engineering and statistics for selecting among an ordered set of candidate models the one that better fits the observed sample data the selected model minimizes a penalized likelihood metric where the penalty is determined by the criterion adopted while rules for choosing a penalty that guarantees a consistent estimate of the model order are known theoretical tools for its design with finite samples have never been provided in a general setting in this paper we study model order selection for finite samples under a design perspective focusing on the generalized information criterion gic which embraces the most common itc the theory is general and as case studies we consider a the problem of estimating the number of signals embedded in additive white gaussian noise awgn by using multiple sensors b model selection for the general linear model glm which includes e g the problem of estimating the number of sinusoids in awgn the analysis reveals a trade off between the probabilities of overestimating and underestimating the order of the model we then propose to design the gic penalty to minimize underestimation while keeping the overestimation probability below a specified level for the considered problems this method leads to analytical derivation of the optimal penalty for a given sample size a performance comparison between the penalty optimized gic and common aic and bic is provided demonstrating the effectiveness of the proposed design strategy

stochastic model specification search for gaussian and partial non gaussian state space models model specification for state space models is a difficult task as one has to decide which components to include in the model and to specify whether these components are fixed or time varying to this aim a new model space mcmc method is developed in this paper it is based on extending the bayesian variable selection approach which is usually applied to variable selection in regression models to state space models for non gaussian state space models stochastic model search mcmc makes use of auxiliary mixture sampling we focus on structural time series models including seasonal components trend or intervention the method is applied to various well known time series

the stability of special symmetric solutions of x t αf x t 1 with small amplitudes we consider the differential delay equation f x t f x t 1 with odd f c 3 r f 0 0 f 0 0 f 0 0 f 0 0 a theorem of kaplan and yorke 4 implies that 2f 0 is a bifurcation point where a certain class of periodic solutions bifurcates from the trivial zero solutions

the markov moment problem and de finetti s theorem part ii the markov moment problem is to characterize sequences admitting the representation s n 0 1 x nf x dx where f x is a probability density on 0 1 and 0 f x c for almost all x there are well known characterizations through complex systems of non linear inequalities on s n necessary and sufficient linear conditions are the following s 0 1 and for all and here is the forward difference operator this result is due to hausdorff we give a new proof with some ancillary results for example characterizing monotone densities then we make the connection to de finetti s theorem with characterizations of the mixing measure

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

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

nonparametric bayes estimators based on beta processes in models for life history data the article studies the problem of finding bayes estimators for cumulative hazard rates and related quantities w r t prior distributions that correspond to cumulative hazard rate processes with nonnegative independent increments a particular class of prior processes termed beta processes is introduced and is shown to constitute a conjugate class an important bonus of the approach based on cumulative hazards is that more complicated models for life history data than the simple life table situation can be treated for example time inhomogeneous markov chains we find posterior distributions and derive bayes estimators in such models and present a semiparametric bayesian analysis of the cox regression model

needles and straw in haystacks empirical bayes estimates of possibly sparse sequences an empirical bayes approach to the estimation of possibly sparse sequences observed in gaussian white noise is set out and investigated the prior considered is a mixture of an atom of probability at zero and a heavy tailed density y with the mixing weight chosen by marginal maximum likelihood in the hope of adapting between sparse and dense sequences if estimation is then carried out using the posterior median this is a random thresholding procedure other thresholding rules employing the same threshold can also be used probability bounds on the threshold chosen by the marginal maximum likelihood approach lead to overall risk bounds over classes of signal sequences of length n allowing for sparsity of various kinds and degrees the signal classes considered are nearly black sequences where only a proportion is allowed to be nonzero and sequences with normalized p norm bounded by for 0 and 0 1 simulations show excellent performance for appropriately chosen functions y the method is computationally tractable and software is available the extension to a modified thresholding method relevant to the estimation of very sparse sequences is also considered

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

a new bayesian test to test for the intractability countering hypothesis abstractwe present a new test of hypothesis in which we seek the probability of the null conditioned on the data where the null is a simplification undertaken to counter the intractability of the more complex model that the simpler null model is nested within with the more complex model rendered intractable the null model uses a simplifying assumption that capacitates the learning of an unknown parameter vector given the data bayes factors are shown to be known only up to a ratio of unknown data dependent constants a problem that cannot be cured using prescriptions similar to those suggested to solve the problem caused to bayes factor computation by noninformative priors thus a new test is needed in which we can circumvent bayes factor computation in this test we undertake generation of data from the model in which the null hypothesis is true and can achieve support in the measured data for the null by comparing the marginalized posterior of the model parameter given the measured data to that gi

the bayesian lasso the lasso estimate for linear regression parameters can be interpreted as a bayesian posterior mode estimate when the regression parameters have independent laplace i e double exponential priors gibbs sampling from this posterior is possible using an expanded hierarchy with conjugate normal priors for the regression parameters and independent exponential priors on their variances a connection with the inverse gaussian distribution provides tractable full conditional distributions the bayesian lasso provides interval estimates bayesian credible intervals that can guide variable selection moreover the structure of the hierarchical model provides both bayesian and likelihood methods for selecting the lasso parameter slight modifications lead to bayesian versions of other lasso related estimation methods including bridge regression and a robust variant

better bootstrap confidence intervals abstract we consider the problem of setting approximate confidence intervals for a single parameter in a multiparameter family the standard approximate intervals based on maximum likelihood theory can be quite misleading in practice tricks based on transformations bias corrections and so forth are often used to improve their accuracy the bootstrap confidence intervals discussed in this article automatically incorporate such tricks without requiring the statistician to think them through for each new application at the price of a considerable increase in computational effort the new intervals incorporate an improvement over previously suggested methods which results in second order correctness in a wide variety of problems in addition to parametric families bootstrap intervals are also developed for nonparametric situations

conditioning likelihood and coherence a review of some foundational concepts 2000 conditioning likelihood and coherence a review of some foundational concepts journal of the american statistical association vol 95 no 452 pp 1340 1346

misspecification in infinite dimensional bayesian statistics we consider the asymptotic behavior of posterior distributions if the model is misspecified given a prior distribution and a random sample from a distribution p 0 which may not be in the support of the prior we show that the posterior concentrates its mass near the points in the support of the prior that minimize the kullback leibler divergence with respect to p 0 an entropy condition and a prior mass condition determine the rate of convergence the method is applied to several examples with special interest for infinite dimensional models these include gaussian mixtures nonparametric regression and parametric models

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

bayes variable selection in semiparametric linear models there is a rich literature on bayesian variable selection for parametric models our focus is on generalizing methods and asymptotic theory established for mixtures of g priors to semiparametric linear regression models having unknown residual densities using a dirichlet process location mixture for the residual density we propose a semiparametric g prior which incorporates an unknown matrix of cluster allocation indicators for this class of priors posterior computation can proceed via a straightforward stochastic search variable selection algorithm in addition bayes factor and variable selection consistency is shown to result under a class of proper priors on g even when the number of candidate predictors p is allowed to increase much faster than sample size n while making sparsity assumptions on the true model size

inference from intrinsic bayes procedures under model selection and uncertainty in this article we present a fully coherent and consistent objective bayesian analysis of the linear regression model using intrinsic priors the intrinsic prior is a scaled mixture of g priors and promotes shrinkage toward the subspace defined by a base or null model while it has been established that the intrinsic prior provides consistent model selectors across a range of models the posterior distribution of the model parameters has not previously been investigated we prove that the posterior distribution of the model parameters is consistent under both model selection and model averaging when the number of regressors is fixed further we derive tractable expressions for the intrinsic posterior distribution as well as sampling algorithms for both a selected model and model averaging we compare the intrinsic prior to other mixtures of g priors and provide details on the consistency properties of modified versions of the zellner siow prior and hyper g priors supplementary materials for this arti

parametric empirical bayes inference theory and applications abstract this article reviews the state of multiparameter shrinkage estimators with emphasis on the empirical bayes viewpoint particularly in the case of parametric prior distributions some successful applications of major importance are considered recent results concerning estimates of error and confidence intervals are described and illustrated with data

local sensitivity of posterior expectations we investigate the degree to which posterior expectations are sensitive to prior distributions using a local method based on functional differentiation invariance considerations suggest a family of norms which can be used to measure perturbations to the prior the sensitivity measure is seen to depend heavily on the choice of norm asymptotic results suggest which norm will yield the most useful results in practice we find that to maintain asymptotically sensible behaviour it is necessary to reduce the richness of the class of prior perturbations as the dimension of the parameter space increases jeffreys prior is characterized as the prior to which inference is least sensitive

the effect of oscillations in the dynamics of differential equations with delay abstract the purpose of this paper is to study the dynamic behavior of delay differential equations of the form x t f x t 1 a e sin t e cos t e r provided that a and f meet some hypotheses by augmenting the above equation the explicit time dependent terms are replaced by state dependent terms the augmented system is autonomous and has a pair of purely imaginary and simple zero eigenvalues applying the center manifold reduction the existence of an attractive integral manifold with periodic structure for the original equation is shown furthermore we give a description of the flow on the obtained manifold this allows us to determine the sufficient conditions for existence of saddle node bifurcation to illustrate our results we consider an autonomous equation perturbed by a periodic function

filtered variate prior distributions for histogram smoothing abstract we develop prior distributions for histogram inference favoring smooth population frequencies that is probability vectors with small differences for neighboring categories we give a theory of prior random probability vectors representable as a linear transform or filter of a standard random probability vector or equivalently a random weighted average of nonrandom smooth probability vectors promising methods of prior assessment are given based on elicitation of a list of typically smooth probability vectors the empirical moments of which can then be matched by the mean vector and variance matrix of a constructed continuous type filtered variate prior distribution

a new class of b w halftoning algorithms a new class of dithering algorithms for black and white b w images is presented the basic idea behind the technique is to divide the image into small blocks and minimize the distortion between the original continuous tone image and its low pass filtered halftone this corresponds to a quadratic programming problem with linear constraints which is solved via standard optimization techniques examples of b w halftone images obtained by this technique are compared to halftones obtained via existing dithering algorithms

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

optimal treatments in cost effectiveness analysis in the presence of covariates improving patient subgroup definition abstract in the presence of covariates the cost effectiveness analysis of medical treatments shows that the optimal treatment varies across the patient population subgroups and hence to accurately define the subgroups is a crucial step in the analysis a patient subgroup definition using only influential covariates within the potential set of patients covariates established by the expert has recently been proposed and the influential covariates were chosen from the univariate distributions of the effectiveness and the cost conditional on the effectiveness in this paper we argue that the bayesian variable selection procedure should be developed using the bivariate distribution of the cost and the effectiveness which is not the usual practice this new approach provides results with wider applicability and more understandable without a significative increase in the complexity of the procedure for real and simulated data sets optimal treatments for subgroups are found and compared with that from previous methods

estimation and accuracy after model selection classical statistical theory ignores model selection in assessing estimation accuracy here we consider bootstrap methods for computing standard errors and confidence intervals that take model selection into account the methodology involves bagging also known as bootstrap smoothing to tame the erratic discontinuities of selection based estimators a useful new formula for the accuracy of bagging then provides standard errors for the smoothed estimators two examples nonparametric and parametric are carried through in detail a regression model where the choice of degree linear quadratic cubic is determined by the cp criterion and a lasso based estimation problem

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

testing hardy weinberg equilibrium an objective bayesian analysis we analyze the general multiallelic hardy weinberg equilibrium problem from an objective bayesian testing standpoint we argue that for small or moderate sample sizes the answer is rather sensitive to the prior chosen and this suggests to carry out a sensitivity analysis with respect to the prior this goal is achieved through the identification of a class of priors specifically designed for this testing problem in this paper we consider the class of intrinsic priors under the full model indexed by a tuning quantity the training sample size these priors are objective satisfy savage s continuity condition and have proved to behave extremely well for many statistical testing problems we compute the posterior probability of the hardy weinberg equilibrium model for the class of intrinsic priors assess robustness over the range of plausible answers as well as stability of the decision in favor of either hypothesis copyright 2010 john wiley sons ltd

smooth bifurcation of symmetric periodic solutions of functional differential equations with odd f c r in contrast to the one dimensional ordinary differential equation this equation can have periodic solutions as the example i t 7r 2 x t 1 with solutions x t z sin rc 2 t z 0 shows numerical bifurcation diagrams e g 6 143 indicate a complicated structure of the period solutions depending on the delay parameter u one finds various intersecting smooth curves one task of mathematics is to prove the existence of periodic solutions in this paper we are mainly interested in symmetric periodic solutions i e solutions x of c which fultil the symmetry property

the selection of prior distributions by formal rules abstract subjectivism has become the dominant philosophical foundation for bayesian inference yet in practice most bayesian analyses are performed with so called noninformative priors that is priors constructed by some formal rule we review the plethora of techniques for constructing such priors and discuss some of the practical and philosophical issues that arise when they are used we give special emphasis to jeffreys s rules and discuss the evolution of his viewpoint about the interpretation of priors away from unique representation of ignorance toward the notion that they should be chosen by convention we conclude that the problems raised by the research on priors chosen by formal rules are serious and may not be dismissed lightly when sample sizes are small relative to the number of parameters being estimated it is dangerous to put faith in any default solution but when asymptotics take over jeffreys s rules and their variants remain reasonable choices we also provide an annotated b

frequentist accuracy of bayesian estimates in the absence of relevant prior experience popular bayesian estimation techniques usually begin with some form of uninformative prior distribution intended to have minimal inferential influence bayes rule will still produce nice looking estimates and credible intervals but these lack the logical force attached to experience based priors and require further justification this paper concerns the frequentist assessment of bayes estimates a simple formula is shown to give the frequentist standard deviation of a bayesian point estimate the same simulations required for the point estimate also produce the standard deviation exponential family models make the calculations particularly simple and bring in a connection to the parametric bootstrap

more accurate confidence intervals in exponential families summary fisher s theory of maximum likelihood estimation routinely provides approximate confidence intervals for a parameter of interest 0 the standard intervals 0 za5 s where 0 is the maximum likelihood estimator 5 is an estimate of standard error based on differentiation of the log likelihood function and za is a normal percentile point recent work has produced systems of better approximate confidence intervals which look more like exact intervals when exact intervals exist and in general have coverage probabilities an order of magnitude more accurate than the standard intervals this paper develops an efficient and dependable algorithm for calculating highly accurate approximate intervals on a routine basis for parameters 0 defined in the framework of a multiparameter exponential family the better intervals require only a few times as much computational effort as the standard intervals a variety of numerical and theoretical arguments are used to show that the algorithm works well and that the improvement over the standard intervals can be striking in realistic situations

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

consistent model selection criteria on high dimensions asymptotic properties of model selection criteria for high dimensional regression models are studied where the dimension of covariates is much larger than the sample size several sufficient conditions for model selection consistency are provided non gaussian error distributions are considered and it is shown that the maximal number of covariates for model selection consistency depends on the tail behavior of the error distribution also sufficient conditions for model selection consistency are given when the variance of the noise is neither known nor estimated consistently results of simulation studies as well as real data analysis are given to illustrate that finite sample performances of consistent model selection criteria can be quite different

bayesian posteriors for arbitrarily rare events we study how much data a bayesian observer needs to correctly infer the relative likelihoods of two events when both events are arbitrarily rare each period either a blue die or a red die is tossed the two dice land on side 1 with unknown probabilities p 1 and q 1 which can be arbitrarily low given a data generating process where p 1 c q 1 we are interested in how much data are required to guarantee that with high probability the observer s bayesian posterior mean for p 1 exceeds 1 c times that for q 1 if the prior densities for the two dice are positive on the interior of the parameter space and behave like power functions at the boundary then for every 0 there exists a finite n so that the observer obtains such an inference after n periods with probability at least 1 whenever n p 1 n the condition on n and p 1 is the best possible the result can fail if one of the prior densities converges to zero exponentially fast at the boundary

tail probabilities from observed likelihoods summary an exponential model not in standard form is fully characterized by an observed likelihood function and its first sample space derivative up to one one transformations of the observable variable this property is used to modify the lugannani rice 1980 tail probability approximation to make it parameterization invariant then for general continuous models a version of tangent exponential model is defined and used to derive a general tail probability approximation that uses only the observed likelihood and its first sample space derivative the analysis extends from density functions to distribution functions the tangent exponential model methods of fraser 1988 a related tail probability approximation has been reported barndorff nielsen 1988b in the discussion to reid 1988

an intrinsic limiting procedure for model selection and hypotheses testing abstract improper priors typically arise in default bayesian estimation problems in the bayesian approach to model selection or hypothesis testing the main tool is the bayes factor when improper priors for the parameters appearing in the models are used the bayes factor is not well defined the intrinsic bayes factor introduced by berger and pericchi is an interesting method for overcoming that difficulty that method is of particular interest as a means for generating proper prior distributions intrinsic priors for model comparison from the improper priors typically used in estimation the goal of this article is to develop a limiting procedure that provides a solid justification for the use of bayes factor with intrinsic priors the procedure is formalized and discussed for nested and nonnested models illustrations and comparisons with other approximations to bayes factors such as the bayesian information criterion of schwarz and the fractional bayes factor of o hagan are provided

constructing summary statistics for approximate bayesian computation semi automatic approximate bayesian computation summary many modern statistical applications involve inference for complex stochastic models where it is easy to simulate from the models but impossible to calculate likelihoods approximate bayesian computation abc is a method of inference for such models it replaces calculation of the likelihood by a step which involves simulating artificial data for different parameter values and comparing summary statistics of the simulated data with summary statistics of the observed data here we show how to construct appropriate summary statistics for abc in a semi automatic manner we aim for summary statistics which will enable inference about certain parameters of interest to be as accurate as possible theoretical results show that optimal summary statistics are the posterior means of the parameters although these cannot be calculated analytically we use an extra stage of simulation to estimate how the posterior means vary as a function of the data and we then use these estimates of our summary statistics within abc empirical results show that our approach is a robust method for choosing summary statistics that can result in substantially more accurate abc analyses than the ad hoc choices of summary statistics that have been proposed in the literature we also demonstrate advantages over two alternative methods of simulation based inference

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

consistent model selection criteria for quadratically supported risks in this paper we study asymptotic properties of model selection criteria for high dimensional regression models where the number of covariates is much larger than the sample size in particular we consider a class of loss functions called the class of quadratically supported risks which is large enough to include the quadratic loss huber loss quantile loss and logistic loss we provide sufficient conditions for the model selection criteria which are applicable to the class of quadratically supported risks our results extend most previous sufficient conditions for model selection consistency in addition sufficient conditions for pathconsistency of the lasso and nonconvex penalized estimators are presented here pathconsistency means that the probability of the solution path that includes the true model converges to 1 pathconsistency makes it practically feasible to apply consistent model selection criteria to high dimensional data the data adaptive model selection procedure is proposed which is selection consistent and performs well for finite samples results of simulation studies as well as real data analysis are presented to compare the finite sample performances of the proposed data adaptive model selection criterion with other competitors

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

a statistical model for positron emission tomography abstract positron emission tomography pet still in its research stages is a technique that promises to open new medical frontiers by enabling physicians to study the metabolic activity of the body in a pictorial manner much as in x ray transmission tomography and other modes of computerized tomography the quality of the reconstructed image in pet is very sensitive to the mathematical algorithm to be used for reconstruction in this article we tailor a mathematical model to the physics of positron emissions and we use the model to describe the basic image reconstruction problem of pet as a standard problem in statistical estimation from incomplete data we describe various estimation procedures such as the maximum likelihood ml method using the em algorithm the method of moments and the least squares method a computer simulation of a pet experiment is then used to demonstrate the ml and the least squares reconstructions the main purposes of this article are to report on what we believe is an

steady state learning and nash equilibrium we study the steady states of a system in which players learn about the strategies their opponents are playing by updating their bayesian priors in light of their observations players are matched at random to play a fixed extensive form game and each player observes the realized actions in his own matches but not the intended off path play of his opponents or the realized actions in other matches because players are assumed to live finite lives there are steady states in which learning continually takes place if lifetimes are long and players are very patient the steady state distribution of actions approximates that of a nash equilibrium

consistency of objective bayes factors as the model dimension grows in the class of normal regression models with a finite number of regressors and for a wide class of prior distributions a bayesian model selection procedure based on the bayes factor is consistent casella and moreno j amer statist assoc 104 2009 1261 1271 however in models where the number of parameters increases as the sample size increases properties of the bayes factor are not totally understood here we study consistency of the bayes factors for nested normal linear models when the number of regressors increases with the sample size we pay attention to two successful tools for model selection schwarz ann statist 6 1978 461 464 approximation to the bayes factor and the bayes factor for intrinsic priors berger and pericchi j amer statist assoc 91 1996 109 122 moreno bertolino and racugno j amer statist assoc 93 1998 1451 1460 we find that the the schwarz approximation and the bayes factor for intrinsic priors are consistent when the rate of growth of the dimension of the bigger model is o n b for b 1 when b 1 the schwarz approximation is always inconsistent under the alternative while the bayes factor for intrinsic priors is consistent except for a small set of alternative models which is characterized

inverse halftoning using binary permutation filters the problem of reconstructing a continuous tone image given its ordered dithered halftone or its error diffused halftone image is considered we develop a modular class of nonlinear filters that can reconstruct the continuous tone information preserving image details and edges that provide important visual cues the proposed nonlinear reconstruction algorithms denoted as binary permutation filters are based on the space and rank orderings of the halftone samples provided by the multiset permutation of the on pixels in a halftone observation window for a given window size we obtain a wide range of filters by varying the amount of space rank ordering information utilized in the estimate for image reconstructions from ordered dithered halftones we develop periodically space varying filters that can account for the periodical nature of the underlying screening process a class of suboptimal but simpler space invariant reconstruction filters are also proposed and tested constrained lms type algorithms are employed for the design of reconstruction filters that minimize the reconstruction mean squared error we present simulations showing that binary permutation filters are modular robust to image source characteristics and that they produce high visual quality image reconstruction

consistency of bernstein polynomial posteriors a bernstein prior is a probability measure on the space of all the distribution functions on 0 1 under very general assumptions it selects absolutely continuous distribution functions whose densities are mixtures of known beta densities the bemstein prior is of interest in bayesian nonparametric inference with continuous data we study the consistency of the posterior from a bernstein prior we first show that under mild assumptions the posterior is weakly consistent for any distribution function p o on 0 1 with continuous and bounded lebesgue density with slightly stronger assumptions on the prior the posterior is also hellinger consistent this implies that the predictive density from a bernstein prior which is a bayesian density estimate converges in the hellinger sense to the true density assuming that it is continuous and bounded we also study a sieve maximum likelihood version of the density estimator and show that it is also hellinger consistent under weak assumptions when the order of the bernstein polynomial i e the number of components in the beta distribution mixture is truncated we show that under mild restrictions the posterior concentrates on the set of pseudotrue densities finally we study the behaviour of the predictive density numerically and we also study a hybrid bayes maximum likelihood density estimator

analysis of ordered dither for arbitrary sampling lattices and screen periodicities ordered dither is considered a simple and effective method of image binarization for rectangular lattices it has been shown that this method of spatially periodic quantization causes aliasing in the frequency domain resulting in a degradation of halftone image quality this result is extended by deriving an expression for the spectrum of the halftoned image generalized for any sampling lattice and any screen periodicity this expression will show the explicit relationship between screen function and halftone image quality using this expression the nonlinearities associated with four optimized screens are analyzed to determine which screens are best suited to halftone rectangularly and hexagonally sampled images