adaptive bayesian wavelet shrinkage abstract when fitting wavelet based models shrinkage of the empirical wavelet coefficients is an effective tool for denoising the data this article outlines a bayesian approach to shrinkage obtained by placing priors on the wavelet coefficients the prior for each coefficient consists of a mixture of two normal distributions with different standard deviations the simple and intuitive form of prior allows us to propose automatic choices of prior parameters these parameters are chosen adaptively according to the resolution level of the coefficients typically shrinking high resolution frequency coefficients more heavily assuming a good estimate of the background noise level we obtain closed form expressions for the posterior means and variances of the unknown wavelet coefficients the latter may be used to assess uncertainty in the reconstruction several examples are used to illustrate the method and comparisons are made with other shrinkage methods

graphical models for marked point processes based on local independence summary a new class of graphical models capturing the dependence structure of events that occur in time is proposed the graphs represent so called local independences meaning that the intensities of certain types of events are independent of some but not necessarilly all events in the past this dynamic concept of independence is asymmetric similar to granger non causality so the corresponding local independence graphs differ considerably from classical graphical models hence a new notion of graph separation which is called separation is introduced and implications for the underlying model as well as for likelihood inference are explored benefits regarding facilitation of reasoning about and understanding of dynamic dependences as well as computational simplifications are discussed

eigenvalues invariant factors highest weights and schubert calculus we describe recent work of klyachko totaro knutson and tao that characterizes eigenvalues of sums of hermitian matrices and decomposition of tensor products of representations of gln c we explain related applications to invariant factors of products of matrices intersections in grassmann varieties and singular values of sums and products of arbitrary matrices recent breakthroughs primarily by a klyachko b totaro a knutson and t tao with contributions by p belkale and c woodward have led to complete solutions of several old problems involving the various notions in the title our aim here is to describe this work and especially to show how these solutions are derived from it along the way we will see that these problems are also related to other areas of mathematics including geometric invariant theory symplectic geometry and combinatorics in addition we present some related applications to singular values of arbitrary matrices although many of the theorems we state here have not appeared elsewhere their proofs are mostly soft algebra based on the hard geometric or combinatorial work of others indeed this paper emphasizes concrete elementary arguments we do give some new examples and counterexamples and raise some new open questions we have attempted to point to the sources and to some of the key partial results that had been conjectured or proved before however there is a very large literature particularly for linear algebra problems about eigenvalues singular values and invariant factors we have listed only a few of these articles from whose bibliographies we hope an interested reader can trace the history we apologize to the many whose work is not cited directly we begin in the first five sections by describing each of the problems together with some of their early histories and we state as theorems the new solutions to these problems in section 6 we describe the steps toward these solutions that were carried out before the recent breakthroughs then we discuss the recent solutions and explain how these theorems follow from the work of the above mathematicians sections 7 8 9 and 10 also contain variations and generalizations of some of the theorems stated in the first five sections as well as attributions of the theorems to their authors one of our fascinations with this subject even now that we have proofs of the theorems is the challenge to understand in a deeper way why all these subjects are received by the editors in july 1999 and in revised form january 3 2000 2000 mathematics subject classification primary 15a42 22e46 14m15 secondary 05e15 13f10 14c17 15a18 47b07 the author was partly supported by nsf grant dms9970435 c 2000 american mathematical society

debate concerning the mean velocity profile of a turbulent boundary layer there has been considerable controversy during the past few years concerning the validity of the classical log law that describes the overlap region of the mean velocity profile in the canonical turbulent boundary layer alternative power laws have been proposed by barenblatt chorin george and castillo to name just a few advocates of either law typically have used selected data sets to foster their claims the experimental and direct numerical simulation data sets from six independent groups are analyzed for the range of momentum thickness reynolds numbers of 5 10 2 2 732 10 4 the best fit values are determined for the constants appearing in either law our strategy involves calculating the fractional difference between the measured computed mean velocity and that calculated using either of the two respective laws this fractional difference is bracketed in the region 0 5 so that an accurate objective measure of the boundary and extent of either law is determined it is found that although the extent of the power law region in outer variables is nearly constant over a wide range of reynolds numbers the log region extent increases monotonically with reynolds number the log law and the power law do not cover the same portion of the velocity profile a very small zone directly above the buffer layer is not represented by the power law on the other hand the inner region of the wake zone is covered by it in the region where both laws show comparable fractional differences the mean and variance were calculated from both measures it is concluded that the examined data do not indicate any statistically significant preference toward either law i the opening arguments t he reynolds numbers encountered in many practical situations are typically several orders of magnitude higher than those studied computationally or even experimentally high reynoldsnumber research facilities are expensive to build and operate and the few existing are heavily scheduled with mostly developmental work for traditional wind tunnels additional complications are introduced at high speeds due to compressibility effects and probe resolution limitations near walls likewise full computational simulation of high reynolds number flows is beyond the reach of current capabilities understanding of turbulence and modeling will therefore continue to play a vital role in the computation of high reynolds number practical flows using the reynolds averaged navier stokes equations because the existing knowledge base accumulated mostly through physical as well as numerical experiments is skewed toward the low reynolds numbers the key question in modeling high reynolds number flows is what the reynolds number effects are on the mean and statistical turbulence quantities one of the fundamental tenets of boundary layer research is the idea that for a given geometry any statistical turbulence quantity mean rms reynolds stress etc measured at different facilities and at different reynolds numbers will collapse to a single universal profile when nondimensionalized using the proper length and velocity scales different scales are used near the wall and away from it this is termed self similarity or self preservation and allows convenient extrapolation from the low reynolds number laboratory experiments to the much higher reynolds number situations encountered in typical field applications the universal log profile

ignorability for general longitudinal data likelihood factors that can be disregarded for inference are termed ignorable we demonstrate that close ties exist between ignorability and identification of causal effects by covariate adjustment a graphical condition stability plays a role analogous to that of missingness at random but is applicable to general longitudinal data our formulation of ignorability does not depend on any notion of missing data so is appealing in situations where missing data may not actually exist several examples illustrate how stability may be assessed

empirical bayes selection of wavelet thresholds this paper explores a class of empirical bayes methods for level dependent threshold selection in wavelet shrinkage the prior considered for each wavelet coefficient is a mixture of an atom of probability at zero and a heavy tailed density the mixing weight or sparsity parameter for each level of the transform is chosen by marginal maximum likelihood if estimation is carried out using the posterior median this is a random thresholding procedure the estimation can also be carried out using other thresholding rules with the same threshold details of the calculations needed for implementing the procedure are included in practice the estimates are quick to compute and there is software available simulations on the standard model functions show excellent performance and applications to data drawn from various fields of application are used to explore the practical performance of the approach by using a general result on the risk of the corresponding marginal maximum likelihood approach for a single sequence overall bounds on the risk of the method are found subject to membership of the unknown function in one of a wide range of besov classes covering also the case of f of bounded variation the rates obtained are optimal for any value of the parameter p in 0 simultaneously for a wide range of loss functions each dominating the lq norm of the th derivative with 0 and 0 q 2 attention is paid to the distinction between sampling the unknown function within white noise and sampling at discrete points and between placing constraints on the function itself and on the discrete wavelet transform of its sequence of values at the observation points results for all relevant combinations of these scenarios are obtained in some cases a key feature of the theory is a particular boundary corrected wavelet basis details of which are discussed overall the approach described seems so far unique in combining the properties of fast computation good theoretical properties and good performance in simulations and in practice a key feature appears to be that the estimate of sparsity adapts to three different zones of estimation first where the signal is not sparse enough for thresholding to be of benefit second where an appropriately chosen threshold results in substantially improved estimation and third where the signal is so sparse that the zero estimate gives the optimum accuracy rate

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

hierarchical parcel swapping representation of turbulent mixing part 1 formulation and scaling properties an economical representation of effects of turbulence on the time evolving structure of diffusive scalar fields is obtained by introducing a hierarchical tree network connecting fluid parcels with effects of turbulent advection represented by swapping pairs of sub trees at rates determined by turbulence time scales associated with the sub trees the fluid parcels reside at the base of the tree the tree structure partitions the fluid parcels into adjacent pairs or more generally p tuples adjacent parcels intermix at rates governed by diffusion time scales based on molecular diffusivities and parcel sizes this simple procedure efficiently accomplishes long standing objectives of turbulent mixing model development such as generating physically based time histories of fluid parcel nearest neighbor encounters and the associated spatial structure of turbulent scalar fields correspondences between features of the hierarchical formulation and turbulent mixing phenomenology both generic and case specific are noted

bayesian model selection using test statistics existing bayesian model selection procedures require the specification of prior distributions on the parameters appearing in every model in the selection set in practice this requirement limits the application of bayesian model selection methodology to overcome this limitation we propose a new approach towards bayesian model selection that uses classical test statistics to compute bayes factors between possible models in several test cases our approach produces results that are similar to previously proposed bayesian model selection and model averaging techniques in which prior distributions were carefully chosen in addition to eliminating the requirement to specify complicated prior distributions this method offers important computational and algorithmic advantages over existing simulation based methods because it is easy to evaluate the operating characteristics of this procedure for a given sample size and specified number of covariates our method facilitates the selection of hyperparameter values through prior predictive simulation

exact resummations in the theory of hydrodynamic turbulence iii scenarios for anomalous scaling and intermittency elements of the analytic structure of anomalous scaling and intermittency in fully developed hydrodynamic turbulence are described we focus here on the structure functions of velocity differences that satisfy inertial range scaling laws s n r sim r zeta n and the correlation of energy dissipation k epsilon epsilon r sim r mu the goal is to understand the exponents zeta n and mu from first principles in paper ii of this series it was shown that the existence of an ultraviolet scale the dissipation scale eta is associated with a spectrum of anomalous exponents that characterize the ultraviolet divergences of correlations of gradient fields the leading scaling exponent in this family was denoted delta the exact resummation of ladder diagrams resulted in the calculation of delta which satisfies the scaling relation delta 2 zeta 2 in this paper we continue our analysis and show that nonperturbative effects may introduce multiscaling i e zeta n not being linear in n with the renormalization scale being the infrared outer scale of turbulence l it is shown that deviations from k41 scaling of s n r zeta n neq n 3 must appear if the correlation of dissipation is mixing i e mu 0 we derive an exact scaling relation mu 2 zeta 2 zeta 4 we present analytic expressions for zeta n for all n and discuss their relation to experimental data one surprising prediction is that the time decay constant tau n r propto r z n of s n r scales independently of n the dynamic scaling exponent z n is the same for all n order quantities z n zeta 2

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

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

seeking efficient data augmentation schemes via conditional and marginal augmentation data augmentation sometimes known as the method of auxiliary variables is a powerful tool for constructing optimisation and simulation algorithms in the context of optimisation meng van dyk 1997 1998 reported several successes of the working parameter approach for constructing efficient data augmentation schemes for fast and simple em type algorithms this paper investigates the use of working parameters in the context of markov chain monte carlo in particular in the context of tanner wong s 1987 data augmentation algorithm via a theoretical study of two working parameter approaches the conditional augmentation approach and the marginal augmentation approach posterior sampling under the univariate t model is used as a running example which particularly illustrates how the marginal augmentation approach obtains a fast mixing positive recurrent markov chain by first constructing a nonpositive recurrent markov chain in a larger space

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

the local structure of turbulence in incompressible viscous fluid for very large reynolds numbers 1 we shall denote by u p u x 1 x 2 x 3 t 1 2 3 the components of velocity at the moment t at the point with rectangular cartesian coordinates x 1 x 2 x 3 in considering the turbulence it is natural to assume the components of the velocity u p at every point p x 1 x 2 x 3 t of the considered domain g of the four dimensional space x 1 x 2 x 3 t are random variables in the sense of the theory of probabilities cf for this approach to the problem millionshtchikov 1939 denoting by the mathematical expectation of the random variable a we suppose that 2 and d u d x 2 are finite and bounded in every bounded subdomain of the domain g

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

flexible parsimonious smoothing and additive modeling a simple method is presented for fitting regression models that are nonlinear in the explanatory variables despite its simplicity or perhaps because of it the method has some powerful characteristics that cause it to be competitive with and often superior to more sophisticated techniques especially for small data sets in the presence of high noise

scalable bayesian model averaging through local information propagation this article shows that a probabilistic version of the classical forward stepwise variable inclusion procedure can serve as a general data augmentation scheme for model space distributions in generalized linear models this latent variable representation takes the form of a markov process thereby allowing information propagation algorithms to be applied for sampling from model space posteriors in particular we propose a sequential monte carlo method for achieving effective unbiased bayesian model averaging in high dimensional problems using proposal distributions constructed using local information propagation the method called lips for local information propagation based sampling is illustrated using real and simulated examples with dimensionality ranging from 15 to 1000 and its performance in estimating posterior inclusion probabilities and in out of sample prediction is compared to those of several other methods namely mcmc bas ibma and lasso in addition it is shown that the latent variab

posterior odds ratios for regression hypotheses general considerations and some specific results abstract much work is econometrics and statistics has been concerned with comparing bayesian and non bayesian estimation results while much less has involved comparisons of bayesian and non bayesian analyses of hypotheses some issues arising in this latter area that are mentioned and discussed in the paper are 1 is it meaningful to associate probabilities with hypotheses 2 what concept of probability is to be employed in analyzing hypotheses 3 is a separate theory of testing needed 4 must a theory of testing be capable of treating both sharp and non sharp hypotheses 5 how is prior information incorporated in testing 6 does the use of power functions in practice necessitate the use of prior information 7 how are significance levels determined when sample sizes are large and what are the interpretations of p values and tail areas 8 how are conflicting results provided by asymptotically equivalent testing procedures to be reconciled 9 what is the rationale for the 5 accept reject syndrome that afflicts econometrics and applied statistics 10 does it make sense to test a null hypothesis with no alternative hypothesis present and 11 how are the results of analyses of hypotheses to be combined with estimation and prediction procedures brief discussions of these issues with references to the literature are provided since there is much controversy concerning how hypotheses are actually analyzed in applied work the results of a small survey relating to 22 articles employing empirical data published in leading economic and econometric journals in 1978 are presented the major results of this survey indicate that there is wide spread use of the 1 and 5 levels of significance in non bayesian testing with no systematic relation between choice of significance level and sample size also power considerations are not generally discussed in empirical studies in fact there was a discussion of power in only one of the articles surveyed further there was very little formal or informal use of prior information employed in testing hypotheses and practically no attention was given to the effects of tests or pre tests on the properties of subsequent tests or estimation results these results indicate that there is much room for improvement in applied analyses of hypotheses given the findings of the survey of applied studies it is suggested that bayesian procedures for analyzing hypotheses may be helpful in improving applied analyses in this connection the paper presents a review of some bayesian procedures and results for analyzing sharp and non sharp hypotheses with explicit use of prior information in general bayesian procedures have good sampling properties and enable investigators to compute posterior probabilities and posterior odds ratios associated with alternative hypotheses quite readily the relationships of several posterior odds ratios to usual non bayesian testing procedures is clearly demonstrated also a relation between the p value or tail area and a posterior odds ratio is described in detail in the important case of hypotheses about a mean of a normal distribution other examples covered in the paper include posterior odds ratios for the hypotheses that 1 i and i where i is a regression coefficient 2 data are drawn from either of two alternative distributions 3 0 and where is the mean of a normal distribution 4 0 and 0 where is a vector of regression coefficients 5 2 0 vs 2 0 where 1 2 is a vector regression coefficients and 1 s value is unrestricted in several cases is a vector of regression coefficients and 1 s value is unrestricted in several cases tabulations of odds ratios are provided bayesian versions of the chow test for equality of regression coefficients and of the goldfeld quandt test for equality of disturbance variances are given also an application of bayesian posterior odds ratios to a regression model selection problem utilizing the hald data is reported in summary the results reported in the paper indicate that operational bayesian procedures for analyzing many hypotheses encountered in model selection problems are available these procedures yield posterior odds ratios and posterior probabilities for competing hypotheses these posterior odds ratios represent the weight of the evidence supporting one model or hypothesis relative to another given a loss structure as is well known one can choose among hypotheses so as to minimize expected loss also with posterior probabilities available and an estimation or prediction loss function it is possible to choose a point estimate or prediction that minimizes expected loss by averaging over alternative hypotheses or models thus it is seen that the bayesian approach for analyzing competing models or hypotheses provides a unified framework that is extremely useful in solving a number of model selection problems

on comparing restricted least squares estimators abstract this paper examines the conditions under which it is possible to make comparisons between restricted least squares estimators under a generalized mean square error criterion our results demonstrate the rather extreme limitations on such comparisons in many cases it is not possible to compare two restricted least squares estimators using the generalized mean square error criterion

boundary layer turbulence as a kangaroo process a nonlocal mixing length theory of turbulence transport by finite size eddies is developed by means of a novel evaluation of the reynolds stress the analysis involves the contruct of a sample path space and a stochastic closure hypothesis the simplifying property of exhange strong eddies is satisfied by an analytical sampling rate model a nonlinear scaling relation maps the path space onto the semi infinite boundary layer the underlying near wall behavior of fluctuating velocities perfectly agrees with recent direct numerical simulations the resulting integro differential equation for the mixing of scalar densities represents fully developed boundary layer turbulence as a nondiffusive kubo anderson or kangaroo type of stochastic process the model involves a scaling exponent with in the diffusion limit for the partly analytical solution for the mean velocity profile excellent agreement with the experimental data yields 0 58 1995 the american physical society

armitage lecture 2010 understanding treatment effects the value of integrating longitudinal data and survival analysis there is a single minded focus on events in survival analysis and we often ignore longitudinal data that are collected together with the event data this is due to a lack of methodology but also a result of the artificial distinction between survival and longitudinal data analyses understanding the dynamics of such processes is important but has been hampered by a lack of appreciation of the difference between confirmatory and exploratory causal inferences the latter represents an attempt at elucidating mechanisms by applying mediation analysis to statistical data and will usually be of a more tentative character than a confirmatory analysis the concept of local independence and the associated graphs are useful this is related to granger causality an important method from econometrics that is generally undervalued by statisticians this causality concept is different from the counterfactual one since it lacks lacks the intervention aspect the notion that one can intervene at will in naturally occurring processes which seems to underly much of modern causal inference is problematic when studying mediation and mechanisms it is natural to assume a stochastic process point of view when analyzing dynamic relationships we present some examples to illustrate this it is not clear how survival analysis must be developed to handle the complex life history data that are increasingly being collected today we give some suggestions copyright 2012 john wiley sons ltd

testing a point null hypothesis the irreconcilability of p values and evidence abstract the problem of testing a point null hypothesis or a small interval null hypothesis is considered of interest is the relationship between the p value or observed significance level and conditional and bayesian measures of evidence against the null hypothesis although one might presume that a small p value indicates the presence of strong evidence against the null such is not necessarily the case expanding on earlier work especially edwards lindman and savage 1963 and dickey 1977 it is shown that actual evidence against a null as measured say by posterior probability or comparative likelihood can differ by an order of magnitude from the p value for instance data that yield a p value of 05 when testing a normal mean result in a posterior probability of the null of at least 30 for any objective prior distribution objective here means that equal prior weight is given the two hypotheses and that the prior is symmetric and nonincreasing away from the null other definiti

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

parameter expansion for data augmentation abstract viewing the observed data of a statistical model as incomplete and augmenting its missing parts are useful for clarifying concepts and central to the invention of two well known statistical algorithms expectation maximization em and data augmentation recently liu rubin and wu demonstrated that expanding the parameter space along with augmenting the missing data is useful for accelerating iterative computation in an em algorithm the main purpose of this article is to rigorously define a parameter expanded data augmentation px da algorithm and to study its theoretical properties the px da is a special way of using auxiliary variables to accelerate gibbs sampling algorithms and is closely related to reparameterization techniques we obtain theoretical results concerning the convergence rate of the px da algorithm and the choice of prior for the expansion parameter to understand the role of the expansion parameter we establish a new theory for iterative conditional sampling under the tra

estimation of direct effects for survival data by using the aalen additive hazards model summary we extend the definition of the controlled direct effect of a point exposure on a survival outcome other than through some given time fixed intermediate variable to the additive hazard scale we propose two stage estimators for this effect when the exposure is dichotomous and randomly assigned and when the association between the intermediate variable and the survival outcome is confounded only by measured factors which may themselves be affected by the exposure the first stage of the estimation procedure involves assessing the effect of the intermediate variable on the survival outcome via aalen s additive regression for the event time given exposure intermediate variable and confounders the second stage involves applying aalen s additive model given the exposure alone to a modified stochastic process i e a modification of the observed counting process based on the first stage estimates we give the large sample properties of the estimator proposed and investigate its small sample properties by monte carlo simulation a real data example is provided for illustration

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

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

the collapsed gibbs sampler in bayesian computations with applications to a gene regulation problem abstract this article describes a method of grouping and collapsing in using the gibbs sampler and proves from an operator theory viewpoint that the method is in general beneficial the norms of the forward operators associated with the corresponding nonreversible markov chains are used to discriminate among different simulation schemes when applied to bayesian missing data problems the idea of collapsing suggests skipping the steps of sampling parameter s values in standard data augmentation by doing this we obtain a predictive update version of the gibbs sampler a procedure of calculating the posterior odds ratio via the collapsed gibbs sampler when incomplete observations are involved is presented as an illustration of possible applications three examples along with a bayesian treatment for identifying common protein binding sites in unaligned dna sequences are provided

spike and slab variable selection frequentist and bayesian strategies variable selection in the linear regression model takes many apparent faces from both frequentist and bayesian standpoints in this paper we introduce a variable selection method referred to as a rescaled spike and slab model we study the importance of prior hierarchical specifications and draw connections to frequentist generalized ridge regression estimation specifically we study the usefulness of continuous bimodal priors to model hypervariance parameters and the effect scaling has on the posterior mean through its relationship to penalization several model selection strategies some frequentist and some bayesian in nature are developed and studied theoretically we demonstrate the importance of selective shrinkage for effective variable selection in terms of risk misclassification and show this is achieved using the posterior from a rescaled spike and slab model we also show how to verify a procedure s ability to reduce model uncertainty in finite samples using a specialized forward selection strategy using this tool we illustrate the effectiveness of rescaled spike and slab models in reducing model uncertainty

adaptive sampling for bayesian variable selection our paper proposes adaptive monte carlo sampling schemes for bayesian variable selection in linear regression that improve on standard markov chain methods we do so by considering metropolis hastings proposals that make use of accumulated information about the posterior distribution obtained during sampling adaptation needs to be done carefully to ensure that sampling is from the correct ergodic distribution we give conditions for the validity of an adaptive sampling scheme in this problem and for simulating from a distribution on a finite state space in general and suggest a class of adaptive proposal densities which uses best linear prediction to approximate the gibbs sampler our sampling scheme is computationally much faster per iteration than the gibbs sampler and when this is taken into account the efficiency gains when using our sampling scheme compared to alternative approaches are substantial in terms of precision of estimation of posterior quantities of interest for a given amount of computation time we compare our method with other sampling schemes for examples involving both real and simulated data the methodology developed in the paper can be extended to variable selection in more general problems

a theoretical comparison of the data augmentation marginal augmentation and px da algorithms the data augmentation da algorithm is a widely used markov chain monte carlo mcmc algorithm that is based on a markov transition density of the form p x x fy f x y x y f y x y x dy where f x y and f y x are conditional densities the px da and marginal augmentation algorithms of liu and wu j amer statist assoc 94 1999 1264 1274 and meng and van dyk biometrika 86 1999 301 320 are alternatives to da that often converge much faster and are only slightly more computationally demanding the transition densities of these alternative algorithms can be written in the form p r x x f y y f x y x y r y dy f y x y x dy where r is a markov transition function on y we prove that when r satisfies certain conditions the mcmc algorithm driven by pr is at least as good as that driven by p in terms of performance in the central limit theorem and in the operator norm sense these results are brought to bear on a theoretical comparison of the da px da and marginal augmentation algorithms our focus is on situations where the group structure exploited by liu and wu is available we show that the px da algorithm based on haar measure is at least as good as any px da algorithm constructed using a proper prior on the group

identifying direct and indirect effects in a non counterfactual framework summary identifying direct and indirect effects is a common problem in the social science and medical literature and can be described as follows a treatment is administered and a response is recorded however another variable mediates the effect of the treatment on the response in some way channelling a part of the treatment effect the question is how to extricate the direct and channelled indirect effects from one another when it is not possible to intervene on the mediating variable the aim of the paper is to tackle this problem by using a model for direct and indirect effects based on the decision theoretic framework for causal inference

alternative tests of rational expectations models the case of the term structure a linearized version of the rational expectations models of the term structure is put forth in terms of a complete vector of equally spaced observations along the yield curve a data series on intermediate maturity yields which meets the specifications of the model is presented the model is tested against a specific and easily interpreted alternative earlier studies of rational expectations models which used volatility tests or likelihood ratio tests are discussed this abstract was borrowed from another version of this item

multiple shrinkage and subset selection in wavelets this paper discusses bayesian methods for multiple shrinkage estimation in wavelets wavelets are used in applications for data denoising via shrinkage of the coefficients towards zero and for data compression by shrinkage and setting small coefficients to zero we approach wavelet shrinkage by using bayesian hierarchical models assigning a positive prior probability to the wavelet coefficients being zero the resulting estimator for the wavelet coefficients is a multiple shrinkage estimator that exhibits a wide variety of nonlinear patterns we discuss fast computational implementations with a focus on easy to compute analytic approximations as well as importance sampling and markov chain monte carlo methods multiple shrinkage estimators prove to have excellent mean squared error performance in reconstructing standard test functions we demonstrate this in simulated test examples comparing various implementations of multiple shrinkage to commonly used shrinkage rules finally we illustrate our approach with an application to the so called glint data

a bayesian approach to testing the arbitrage pricing theory abstract we consider new methods of testing the restrictions imposed on asset returns implied by an equilibrium version of the arbitrage pricing theory in the competitive equilibrium version of the apt see for example connor and korajczyk 1988 excess returns on a collection of portfolios or individual assets are related to measured factors in a multivariate regression model without intercepts the traditional significance testing approach uses estimates of these intercept terms and a likelihood ratio procedure for statistical testing in our approach we quantify differences between the restricted and unrestricted models by computing posterior model probabilities these posterior model probabilities are combined to form posterior odds ratios which represent the weight of prior and sample evidence for or against the pricing restrictions a dramatic simplification of the odds ratio calculation is achieved by using the savage density ratio method the special structure of the apt model with factors extracted by principal components is exploited to produce a simple prior the sensitivity of the odds ratio to different prior specifications is explored we apply these new methodologies to tests of the apt with a collection of ten size based portfolios the factors are extracted by a principal components technique developed by connor and korajczyk using all listed amex nyse firms over the period 1964 1983 the odds ratios show that the sample evidence is weak and does not favor the apt pricing restrictions we are able to gauge the strength of the evidence in the data by examining the sensitivity of the odds ratio to the prior

turbulence large scale sweeping and the emergence of small scale kolmogorov spectra the dynamics of fully developed hydrodynamic turbulence still is a basically unsolved theoretical problem due to the strong coupling long range nonlinearities in the navier stokes equations the present analysis focuses on the small scale fluctuations in a turbulent boundary layer with one external length scale yo after taking a 2 1 d spatiotemporal spectral transform of the fluctuating vorticity fields care is taken of large scale sweeping which arises as a collective zero mode from the nonlinear flow terms the unswept small scale nonlinearities are then shown to be asymptotically locally isotropic i e for wave numbers k by internal consistency which allows to close the nonlinear hierarchy the navier stokes equations without external forcing are integrated to give the spectral response of the fluctuating small scale velocity fields on the presence of a locally isotropic blob of turbulence while it is being swept around over an arbitrary steady state mean velocity profile using viscous boundary conditions at y 0 averaging the response spectrum over all possible orientational configurations and sweep velocities results in a novel self consistency integral for the 4d energy spectrum function the distribution of turbulence sweep velocities is modeled by means of levy type densities having an algebraic tail with power p 1 the generic case which includes von karman s logarithmic mean velocity profile is found to correspond to 1 p 3 asymptotic analysis of the self consistency integral leads to a differential equation which fixes the scaling exponent of the unswept frequency and admits a nonempty integrable and positive definite airy type frequency spectrum ei k k k with so called normal kolmogorov scaling that is 7 3 and 2 3 anomalous scaling is possible for one special mean profile

rao blackwellization for bayesian variable selection and model averaging in linear and binary regression a novel data augmentation approach choosing the subset of covariates to use in regression or generalized linear models is a ubiquitous problem the bayesian paradigm addresses the problem of model uncertainty by considering models corresponding to all possible subsets of the covariates where the posterior distribution over models is used to select models or combine them via bayesian model averaging bma although conceptually straightforward bma is often difficult to implement in practice since either the number of covariates is too large for enumeration of all subsets calculations cannot be done analytically or both for orthogonal designs with the appropriate choice of prior the posterior probability of any model can be calculated without having to enumerate the entire model space and scales linearly with the number of predictors p in this article we extend this idea to a much broader class of nonorthogonal design matrices we propose a novel method which augments the observed nonorthogonal design by at most p new rows to obtain

lower bounds on bayes factors for interval null hypotheses abstract expressions are derived for lower bounds on bayes factors for interval null hypotheses these are then studied for varying interval lengths and are related to two interesting results one for point null hypotheses and another for one sided hypotheses these results are illustrated with two of the most frequently used hypothesis tests the normal test and student s t test

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

automatic bayesian curve fitting summary a method of estimating a variety of curves by a sequence of piecewise polynomials is proposed motivated by a bayesian model and an appropriate summary of the resulting posterior distribution a joint distribution is set up over both the number and the position of the knots defining the piecewise polynomials throughout we use reversible jump markov chain monte carlo methods to compute the posteriors the methodology has been successful in giving good estimates for smooth functions i e continuous and differentiable as well as functions which are not differentiable and perhaps not even continuous at a finite number of points the methodology is extended to deal with generalized additive models

bayes factors for nonlinear hypotheses and likelihood distributions summary new methods are proposed which allow bayes factors to be computed for hypotheses which impose nonlinear restrictions on the parameters projection methods are used to induce the prior distribution over the restricted parameter space which is required for computation of the bayes factor various distance metrics are introduced to define the projection including a utility based metric which gives kullback leibler divergence as a special case draws from the restricted and unrestricted prior distributions are used to construct marginal distributions of the likelihood which is shown to have additional diagnostic value over and above the bayes factor these methods are applied to hypotheses statisticians are frequently asked to entertain hypotheses which are nonlinear in the model parameters nonlinear hypotheses arise both from subject matter theory as well as from the process of model selection to address nonlinear hypotheses researchers use frequentist methods which are usually implemented using asymptotic approximations berger 1987 zellner 1971 1978 1984 and others have discussed the weaknesses of the classical significance testing approach and suggest bayesian alternatives such as odds ratios the odds ratio is the ratio of posterior probabilities of two competing hypotheses and is the product of the prior odds ratio and the bayes factor the odds ratio has the conceptual advantages of providing a natural probability metric for assessing the strength of data and prior beliefs because of the difficulties of assessing priors under the nonlinear restrictions and the computation of the required integrals few bayes factors have been computed for nonlinear hypotheses see lindley 1988 for an exception in the present paper we outline a general method for computing bayes factors which is flexible conceptually appealing and computationally tractable the literature on bayes factors starts with the work of jeffreys 1961 other references include berger delampady 1987 dickey 1971 good 1985 kass vaidyanathan 1992 smith spiegelhalter 1980 and zellner 1971 1978 1984 these papers deal with the testing of linear restrictions on the parameter space often in the context of linear models even the widely used schwarz 1978 model selection criterion considers only linear restrictions in models of the exponential family form the key difficulty which has kept researchers from developing bayes factors for nonlinear hypotheses is the

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

covariance structure of the gibbs sampler with applications to the comparisons of estimators and augmentation schemes summary we study the covariance structure of a markov chain generated by the gibbs sampler with emphasis on data augmentation when applied to a bayesian missing data problem the gibbs sampler produces two natural approximations for the posterior distribution of the parameter vector the empirical distribution based on the sampled values of the parameter vector and a mixture of complete data posteriors we prove that rao blackwellization causes a one lag delay for the autocovariances among dependent samples obtained from data augmentation and consequently the mixture approximation produces estimates with smaller variances than the empirical approximation the covariance structure results are used to compare different augmentation schemes it is shown that collapsing and grouping random components in a gibbs sampler with two or three components usually result in more efficient sampling schemes

the horseshoe estimator for sparse signals this paper proposes a new approach to sparsity called the horseshoe estimator the horseshoe is a close cousin of other widely used bayes rules arising from for example double exponential and cauchy priors in that it is a member of the same family of multivariate scale mixtures of normals but the horseshoe enjoys a number of advantages over existing approaches including its robustness its adaptivity to dierent sparsity patterns and its analytical tractability we prove two theorems that formally characterize both the horseshoe s adeptness at large outlying signals and its super ecient rate of convergence to the correct estimate of the sampling density in sparse situations finally using a combination of real and simulated data we show that the horseshoe estimator corresponds quite closely to the answers one would get by pursuing a full bayesian model averaging approach using a discrete mixture prior to model signals and noise

flexible empirical bayes estimation for wavelets summary wavelet shrinkage estimation is an increasingly popular method for signal denoising and compression although bayes estimators can provide excellent mean squared error mse properties the selection of an effective prior is a difficult task to address this problem we propose empirical bayes eb prior selection methods for various error distributions including the normal and the heavier tailed student t distributions under such eb prior distributions we obtain threshold shrinkage estimators based on model selection and multiple shrinkage estimators based on model averaging these eb estimators are seen to be computationally competitive with standard classical thresholding methods and to be robust to outliers in both the data and wavelet domains simulated and real examples are used to illustrate the flexibility and improved mse performance of these methods in a wide variety of settings

the calculation of posterior distributions by data augmentation abstract the idea of data augmentation arises naturally in missing value problems as exemplified by the standard ways of filling in missing cells in balanced two way tables thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze this device is used to great advantage by the em algorithm dempster laird and rubin 1977 in solving maximum likelihood problems in situations when the likelihood cannot be approximated closely by the normal likelihood maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements from the bayesian point of view one must now calculate the posterior distribution of parameters of interest if data augmentation can be used in the calculation of the maximum likelihood estimate then in the same cases one ought to be able to use it in the computation of the posterior distribution it is the purpose of this article to explain how this can be done the basic idea