Showing papers in "Statistical Science in 2008"
TL;DR: In this paper, the authors discuss the analysis of fMRI data, from the initial acquisition of the raw data to its use in locating brain activity, making inference about brain connectivity and predictions about psychological or disease states.
Abstract: In recent years there has been explosive growth in the number of neuroimaging studies performed using functional Magnetic Resonance Imaging (fMRI). The field that has grown around the acquisition and analysis of fMRI data is intrinsically interdisciplinary in nature and involves contributions from researchers in neuroscience, psychology, physics and statistics, among others. A standard fMRI study gives rise to massive amounts of noisy data with a complicated spatio-temporal correlation structure. Statistics plays a crucial role in understanding the nature of the data and obtaining relevant results that can be used and interpreted by neuroscientists. In this paper we discuss the analysis of fMRI data, from the initial acquisition of the raw data to its use in locating brain activity, making inference about brain connectivity and predictions about psychological or disease states. Along the way, we illustrate interesting and important issues where statistics already plays a crucial role. We also seek to illustrate areas where statistics has perhaps been underutilized and will have an increased role in the future.
434 citations
TL;DR: In randomized experiments, treatment and control groups should be roughly the same in their distributions of pre-treatment variables as mentioned in this paper. But how nearly so? Can descriptive comparisons meaningfully be paired with significance tests? If so, should there be several such tests, one for each pretreatment variable, or should a single, omnibus test?
Abstract: In randomized experiments, treatment and control groups should be roughly the same—balanced—in their distributions of pre- treatment variables. But how nearly so? Can descriptive comparisons meaningfully be paired with significance tests? If so, should there be several such tests, one for each pretreatment variable, or should there be a single, omnibus test? Could such a test be engineered to give eas- ily computed p-values that are reliable in samples of moderate size, or would simulation be needed for reliable calibration? What new con- cerns are introduced by random assignment of clusters? Which tests of balance would be optimal? To address these questions, Fisher's randomization inference is ap- plied to the question of balance. Its application suggests the reversal of published conclusions about two studies, one clinical and the other a field experiment in political participation.
404 citations
TL;DR: In this article, the interplay of Bayesian and frequentist ideas in the two-groups setting, with particular attention focussed on Benjamini and Hochberg's false discovery rate method, is discussed.
Abstract: The classic frequentist theory of hypothesis testing developed by Neyman, Pearson, and Fisher has a claim to being the Twentieth Century’s most influential piece of applied mathematics. Something new is happening in the Twenty-First Century: high throughput devices, such as microarrays, routinely require simultaneous hypothesis tests for thousands of individual cases, not at all what the classical theory had in mind. In these situations empirical Bayes information begins to force itself upon frequentists and Bayesians alike. The two-groups model is a simple Bayesian construction that facilitates empirical Bayes analysis. This article concerns the interplay of Bayesian and frequentist ideas in the two-groups setting, with particular attention focussed on Benjamini and Hochberg’s False Discovery Rate method. Topics include the choice and meaning of the null hypothesis in large-scale testing situations, power considerations, the limitations of permutation methods, significance testing for groups of cases (such as pathways in microarray studies), correlation effects, multiple confidence intervals, and Bayesian competitors to the two-groups model.
396 citations
TL;DR: In this paper, the authors focus on high-breakdown methods, which can deal with a substantial fraction of outliers in the data, and give an overview of recent high breakdown robust methods for multivariate settings such as covariance estimation, multiple and multivariate regression, discriminant analysis, principal components and multiivariate calibration.
Abstract: When applying a statistical method in practice it often occurs that some observations deviate from the usual assumptions. However, many classical methods are sensitive to outliers. The goal of robust statistics is to develop methods that are robust against the possibility that one or several unannounced outliers may occur anywhere in the data. These methods then allow to detect outlying observations by their residuals from a robust fit. We focus on high-breakdown methods, which can deal with a substantial fraction of outliers in the data. We give an overview of recent high-breakdown robust methods for multivariate settings such as covariance estimation, multiple and multivariate regression, discriminant analysis, principal components and multivariate calibration.
329 citations
TL;DR: Why Monte Carlo standard errors are important, how they can be easily calculated in Markov chain Monte Carlo and how they are used to decide when to stop the simulation are discussed.
Abstract: Current reporting of results based on Markov chain Monte Carlo computations could be improved. In particular, a measure of the accuracy of the resulting estimates is rarely reported. Thus we have little ability to objectively assess the quality of the reported estimates. We address this issue in that we discuss why Monte Carlo standard errors are important, how they can be easily calculated in Markov chain Monte Carlo and how they can be used to decide when to stop the simulation. We compare their use to a popular alternative in the context of two examples.
320 citations
TL;DR: A new class of procedures, covariate-adjusted response adaptive (CARA) randomization procedures that attempt to optimize both efficiency and ethical considerations, while maintaining randomization are advocated.
Abstract: There has been a split in the statistics community about the need for taking covariates into account in the design phase of a clinical trial. There are many advocates of using stratification and covariate-adaptive randomization to promote balance on certain known covariates. However, balance does not always promote efficiency or ensure more patients are assigned to the better treatment. We describe these procedures, including model-based procedures, for incorporating covariates into the design of clinical trials, and give examples where balance, efficiency and ethical considerations may be in conflict. We advocate a new class of procedures, covariate-adjusted response-adaptive (CARA) randomization procedures that attempt to optimize both efficiency and ethical considerations, while maintaining randomization. We review all these procedures, present a few new simulation studies, and conclude with our philosophy.
147 citations
TL;DR: The logit model is often used to analyze experimental data, but randomization does not justify the model, so the usual estimators can be inconsistent, so a consistent estimator is proposed.
Abstract: The logit model is often used to analyze experimental data. However, randomization does not justify the model, so the usual estimators can be inconsistent. A consistent estimator is proposed. Neyman’s non-parametric setup is used as a benchmark. In this setup, each subject has two potential responses, one if treated and the other if untreated; only one of the two responses can be observed. Beside the mathematics, there are simulation results, a brief review of the literature, and some recommendations for practice.
146 citations
TL;DR: By generalizing current approaches to analyze multiple causes, it is shown how most of the difficult assumptions underlying existing methods can be dropped and make physician review, expert algorithms, and parametric statistical assumptions unnecessary.
Abstract: Verbal autopsy procedures are widely used for estimating cause-specific mortality in areas without medical death certification. Data on symptoms reported by caregivers along with the cause of death are collected from a medical facility, and the cause-of-death distribution is estimated in the population where only symptom data are available. Current approaches analyze only one cause at a time, involve assump- tions judged difficult or impossible to satisfy, and require e time-consuming, or unreliable physician reviews, expert algorithms, or parametric statistical models. By generalizing current approaches to analyze multiple causes, we show how most of the difficult assumptions underlying existing methods can be dropped. These generalizations also make physician review, expert algorithms and parametric statistical as- sumptions unnecessary. With theoretical results, and empirical analy- ses in data from China and Tanzania, we illustrate the accuracy of this approach. While no method of analyzing verbal autopsy data, includ- ing the more computationally intensive approach offered here, can give accurate estimates in all circumstances, the procedure offered is concep- tually simpler, less expensive, more general, as or more replicable, and easier to use in practice than existing approaches. We also show how our focus on estimating aggregate proportions, which are the quantities of primary interest in verbal autopsy studies, may also greatly reduce the assumptions necessary for, and thus improve the performance of, many individual classifiers in this and other areas. As a companion to this paper, we also offer easy-to-use software that implements the methods discussed herein.
110 citations
TL;DR: In this paper, the authors give families of examples where sharp rates of convergence to stationarity of the widely used Gibbs sampler are available, where the transition operator is explicitly diagonalizable with classical orthogonal polynomials as eigenfunctions.
Abstract: We give families of examples where sharp rates of convergence to stationarity of the widely used Gibbs sampler are available. The examples involve standard exponential families and their conjugate priors. In each case, the transition operator is explicitly diagonalizable with classical orthogonal polynomials as eigenfunctions.
107 citations
TL;DR: This article traces the origins of this period in terms of the infrastructure required to produce this explosive growth: recognition of the importance of systematic data collection by the state; the rise of statistical theory and statistical thinking; enabling developments of technology; and inventions of novel methods to portray statistical data.
Abstract: Statistical graphics and data visualization have long histories, but their modern forms began only in the early 1800s. Between roughly 1850 and 1900 (±10), an explosive growth occurred in both the general use of graphic methods and the range of topics to which they were applied. Innovations were prodigious and some of the most exquisite graphics ever produced appeared, resulting in what may be called the “Golden Age of Statistical Graphics.” In this article I trace the origins of this period in terms of the infrastructure required to produce this explosive growth: recognition of the importance of systematic data collection by the state; the rise of statistical theory and statistical thinking; enabling developments of technology; and inventions of novel methods to portray statistical data. To illustrate, I describe some specific contributions that give rise to the appellation “Golden Age.”
103 citations
TL;DR: In this article, Cook et al. provide a remedy for two concerns that have dogged the use of principal components in regression: principal components are computed from the predictors alone and do not make apparent use of the response.
Abstract: We provide a remedy for two concerns that have dogged the use of principal components in regression: (i) principal components are computed from the predictors alone and do not make apparent use of the response, and (ii) principal components are not invariant or equivariant under full rank linear transformation of the predictors. The development begins with principal fitted components [Cook, R. D. (2007). Fisher lecture: Dimension reduction in regression (with discussion). Statist. Sci. 22 1–26] and uses normal models for the inverse regression of the predictors on the response to gain reductive information for the forward regression of interest. This approach includes methodology for testing hypotheses about the number of components and about conditional independencies among the predictors.
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TL;DR: A few procedures for deriving priors under a submodel, namely marginalization, conditioning, and Kullback--Leibler projection are reviewed, and comparisons with existing standard approaches are provided.
Abstract: Bayesian model comparison requires the specification of a prior distribution on the parameter space of each candidate model. In this connection two concerns arise: on the one hand the elicitation task rapidly becomes prohibitive as the number of models increases; on the other hand numerous prior specifications can only exacerbate the well-known sensitivity to prior assignments, thus producing less dependable conclusions. Within the subjective framework, both difficulties can be counteracted by linking priors across models in order to achieve simplification and compatibility; we discuss links with related objective approaches. Given an encompassing, or full, model together with a prior on its parameter space, we review and summarize a few procedures for deriving priors under a submodel, namely marginalization, conditioning, and Kullback–Leibler projection. These techniques are illustrated and discussed with reference to variable selection in linear models adopting a conventional g-prior; comparisons with existing standard approaches are provided. Finally, the relative merits of each procedure are evaluated through simulated and real data sets.
TL;DR: In this paper, a margin-based perspective on multiway spectral clustering is presented, which illuminates both the relaxation and rounding aspects of clustering, providing a unified analysis of existing algorithms and guiding the design of new algorithms.
Abstract: Spectral clustering is a broad class of clustering procedures in which an intractable combinatorial optimization formulation of clustering is “relaxed” into a tractable eigenvector problem, and in which the relaxed solution is subsequently “rounded” into an approximate discrete solution to the original problem. In this paper we present a novel margin-based perspective on multiway spectral clustering. We show that the margin-based perspective illuminates both the relaxation and rounding aspects of spectral clustering, providing a unified analysis of existing algorithms and guiding the design of new algorithms. We also present connections between spectral clustering and several other topics in statistics, specifically minimum-variance clustering, Procrustes analysis and Gaussian intrinsic autoregression.
TL;DR: In this article, a special case of stochastic approximation (SA) is studied for estimating the mixing distribution of a mixture model, and a modification of Newton's algorithm that allows for estimation of an unknown parameter in the model is presented.
Abstract: Many statistical problems involve mixture models and the need for computationally efficient methods to estimate the mixing distribution has increased dramatically in recent years. Newton [Sankhyā Ser. A 64 (2002) 306–322] proposed a fast recursive algorithm for estimating the mixing distribution, which we study as a special case of stochastic approximation (SA). We begin with a review of SA, some recent statistical applications, and the theory necessary for analysis of a SA algorithm, which includes Lyapunov functions and ODE stability theory. Then standard SA results are used to prove consistency of Newton’s estimate in the case of a finite mixture. We also propose a modification of Newton’s algorithm that allows for estimation of an additional unknown parameter in the model, and prove its consistency.
TL;DR: In this article, the authors outline how modern likelihood theory, which provides essentially exact inferences in a variety of parametric statistical problems, may routinely be applied in practice, and argue that approximations are often more appropriate than "exact" procedures, even when these exist.
Abstract: We outline how modern likelihood theory, which provides essentially exact inferences in a variety of parametric statistical problems, may routinely be applied in practice. Although the likelihood procedures are based on analytical asymptotic approximations, the focus of this paper is not on theory but on implementation and applications. Numerical illustrations are given for logistic regression, nonlinear models, and linear non-normal models, and we describe a sampling approach for the third of these classes. In the case of logistic regression, we argue that approximations are often more appropriate than ‘exact’ procedures, even when these exist.
TL;DR: In this paper, the authors focus on two of Pearson's major errors that nonetheless have left a positive and lasting impression upon the statistical world and highlight the importance of these errors in the development of statistical research.
Abstract: Karl Pearson played an enormous role in determining the content and organization of statistical research in his day, through his research, his teaching, his establishment of laboratories, and his initiation of a vast publishing program. His technical contributions had initially and continue today to have a profound impact upon the work of both applied and theoretical statisticians, partly through their inadequately acknowledged influence upon Ronald A. Fisher. Particular attention is drawn to two of Pearson’s major errors that nonetheless have left a positive and lasting impression upon the statistical world.
TL;DR: This paper presents a meta-modelling framework for solving complex substantive questions with complex data problems and demonstrates the power of data-driven approaches to solve these problems.
Abstract: Model selection and assessment with incomplete data pose challenges in addition to the ones encountered with complete data. There are two main reasons for this. First, many models describe characteristics of the complete data, in spite of the fact that only an incomplete subset is observed. Direct comparison between model and data is then less than straightforward. Second, many commonly used models are more sensitive to assumptions than in the complete-data situation and some of their properties vanish when they are fitted to incomplete, unbalanced data. These and other issues are brought forward using two key examples, one of a continuous and one of a categorical nature. We argue that model assessment ought to consist of two parts: (i) assessment of a model’s fit to the observed data and (ii) assessment of the sensitivity of inferences to unverifiable assumptions, that is, to how a model described the unobserved data given the observed ones.
TL;DR: Efron has given us a comprehensive and thoughtful review of his approach to large-scale testing stemming from the challenges of analyzing microarray data as discussed by the authors, and discussed philosophy, motivation, methodologies and even practicalities, and along this process gave us a view of the field of statistics from an eagle's eye.
Abstract: Efron has given us a comprehensive and thoughtful review of his approach to large-scale testing stemming from the challenges of analyzing microarray data. Addressing the microarray challenge right from the emergence of the technology, and adapting the point of view on multiple testing that emphasizes the false discovery rate, Efron’s contributions in both fields have been immense. In the discussed paper he reviews philosophy, motivation, methodologies and even practicalities, and along this process gives us a view of the field of statistics from an eagle’s eye. A thorough discussion of such a work is a major undertaking. Instead, I shall first comment on five issues and then discuss new directions for research on largescale multiple inference that bear on Efron’s review. The scope of the review challenges a discussant to try and address some of the issues raised from a broader point of view. I shall give it a try.
TL;DR: It is shown that a satisfactory, likelihood-based general solution exists by using appropriate forms of the relative Kullback-Leibler information, and that the proposed measures are computationally inexpensive given the maximized likelihoods with the observed data.
Abstract: Many practical studies rely on hypothesis testing procedures ap- plied to data sets with missing information. An important part of the analysis is to determine the impact of the missing data on the performance of the test, and this can be done by properly quantifying the relative (to complete data) amount of available information. The problem is directly motivated by appli- cations to studies, such as linkage analyses and haplotype-based association projects, designed to identify genetic contributions to complex diseases. In the genetic studies the relative information measures are needed for the ex- perimental design, technology comparison, interpretation of the data, and for understanding the behavior of some of the inference tools. The central dif- ficulties in constructing such information measures arise from the multiple, and sometimes conflicting, aims in practice. For large samples, we show that a satisfactory, likelihood-based general solution exists by using appropriate forms of the relative Kullback-Leibler information, and that the proposed measures are computationally inexpensive given the maximized likelihoods with the observed data. Two measures are introduced, under the null and alternative hypothesis respectively. We exemplify the measures on data com- ing from mapping studies on the inflammatory bowel disease and diabetes. For small-sample problems, which appear rather frequently in practice and sometimes in disguised forms (e.g., measuring individual contributions to a large study), the robust Bayesian approach holds great promise, though the choice of a general-purpose "default prior" is a very challenging problem. We also report several intriguing connections encountered in our investigation, such as the connection with the fundamental identity for the EM algorithm, the connection with the second CR (Chapman-Robbins) lower information bound, the connection with entropy, and connections between likelihood ra- tios and Bayes factors. We hope that these seemingly unrelated connections, as well as our specific proposals, will stimulate a general discussion and re- search in this theoretically fascinating and practically needed area.
TL;DR: The two-groups model is used here to unearth the Bayesian roots of Benjamini and Hochberg’s originally frequentist construction, and to show Bayesian and frequentist ideas merging into a practical framework for largescale simultaneous testing.
Abstract: The Fisher–Neyman–Pearson theory of hypothesis testing was a triumph of mathematical elegance and practical utility. It was never designed, though, to handle 10,000 tests at once, and one can see contemporary statisticians struggling to develop theories appropriate to our new scientific environment. This paper is part of that effort: starting from just the two-groups model (2.1), it aims to show Bayesian and frequentist ideas merging into a practical framework for largescale simultaneous testing. False discovery rates, Benjamini and Hochberg’s influential contribution to modern statistical theory, is the main methodology featured in the paper, but I really was not trying to sell any specific technology as the final word. In fact, the discussants offer an attractive menu of alternatives. It is still early in the large-scale hypothesis testing story, and I expect, and hope for, major developments in both theory and practice. The central issue, as Carl Morris makes clear, is the combination of information from a collection of more or less similar sources, for example from the expression levels of different genes in a microarray study. Crucial questions revolve around the comparability and relevance of the various sources, as well as the proper choice of a null distribution. Technical issues such as the exact control of Type I errors are important as well, but, in my opinion, have played too big a role in the microarray literature. The discussions today are an appealing mixture of technical facility and big-picture thinking. They are substantial essays in their own right, and I will be able to respond here to only a few of the issues raised. I once wrote, about the jackknife, that good simple ideas are our most precious intellectual commodity. False discovery rates fall into that elite category. The two-groups model is used here to unearth the Bayesian roots of Benjamini and Hochberg’s originally frequentist construction. In a Bayesian framework it is natural
TL;DR: Howard Raiffa as mentioned in this paper has been a member of the faculty at Harvard University and is now the Frank P. Ramsey Chair in Managerial Economics (Emeritus) in the Graduate School of Business Administration and the Kennedy School of Government.
Abstract: Howard Raiffa earned his bachelor’s degree in mathematics, his master’s degree in statistics and his Ph.D. in mathematics at the University of Michigan. Since 1957, Raiffa has been a member of the faculty at Harvard University, where he is now the Frank P. Ramsey Chair in Managerial Economics (Emeritus) in the Graduate School of Business Administration and the Kennedy School of Government. A pioneer in the creation of the field known as decision analysis, his research interests span statistical decision theory, game theory, behavioral decision theory, risk analysis and negotiation analysis. Raiffa has supervised more than 90 doctoral dissertations and written 11 books. His new book is Negotiation Analysis: The Science and Art of Collaborative Decision Making. Another book, Smart Choices, co-authored with his former doctoral students John Hammond and Ralph Keeney, was the CPR (formerly known as the Center for Public Resources) Institute for Dispute Resolution Book of the Year in 1998. Raiffa helped to create the International Institute for Applied Systems Analysis and he later became its first Director, serving in that capacity from 1972 to 1975. His many honors and awards include the Distinguished Contribution Award from the Society of Risk Analysis; the Frank P. Ramsey Medal for outstanding contributions to the field of decision analysis from the Operations Research Society of America; and the Melamed Prize from the University of Chicago Business School for The Art and Science of Negotiation. He earned a Gold Medal from the International Association for Conflict Management and a Lifetime Achievement Award from the CPR Institute for Dispute Resolution. He holds honorary doctor’s degrees from Carnegie Mellon University, the University of Michigan, Northwestern University, Ben Gurion University of the Negev and Harvard University. The latter was awarded in 2002.
TL;DR: Jerzy Neyman's life history and some of his contributions to applied statistics are reviewed and a number of data sets and corresponding substantive questions are addressed.
Abstract: Jerzy Neyman's life history and some of his contributions to applied statistics are reviewed. In a 1960 article he wrote: "Currently in the period of dynamic indeterminism in science, there is hardly a serious piece of research which, if treated realistically, does not involve operations on stochastic processes. The time has arrived for the theory of stochastic processes to become an item of usual equipment of every applied statistician." The emphasis in this article is on stochastic processes and on stochastic process data analysis. A number of data sets and corresponding substantive questions are addressed. The data sets concern sardine depletion, blowfly dynamics, weather modification, elk movement and seal journeying. Three of the examples are from Neyman's work and four from the author's joint work with collaborators.
TL;DR: In this paper, a simple probability model for particle physics experiments is discussed and both frequentist and non-informative Bayesian procedures for inference about the signal are derived, with the frequentist procedure having the edge for interval estimation, and the Bayesian procedure yielding slightly better point estimates.
Abstract: Particle physics experiments such as those run in the Large Hadron Collider result in huge quantities of data, which are boiled down to a few numbers from which it is hoped that a signal will be detected. We discuss a simple probability model for this and derive frequentist and noninformative Bayesian procedures for inference about the signal. Both are highly accurate in realistic cases, with the frequentist procedure having the edge for interval estimation, and the Bayesian procedure yielding slightly better point estimates. We also argue that the significance, or p-value, function based on the modified likelihood root provides a comprehensive presentation of the information in the data and should be used for inference.
TL;DR: The drawbacks of using p-value in large-scale multiple testing are discussed, the fundamental role played by the Lfdr is demonstrated and estimation of the null distribution and the proportion of the nonnulls are discussed.
Abstract: Professor Efron is to be congratulated for his innovative and valuable contributions to large-scale multiple testing. He has given us a very interesting article with much material for thought and exploration. The twogroup mixture model (2.1) provides a convenient and effective framework for multiple testing. The empirical Bayes approach leads naturally to the local false discovery rate (Lfdr) and gives the Lfdr a useful Bayesian interpretation. This and other recent papers of Efron raised several important issues in multiple testing such as theoretical null versus empirical null and the effects of correlation. Much research is needed to better understand these issues. Virtually all FDR controlling procedures in the literature are based on thresholding the ranked p-values. The difference among these methods is in the choice of the threshold. In multiple testing, typically one first uses a p-value based method such as the Benjamini– Hochberg procedure for global FDR control and then uses the Lfdr as a measure of significance for individual nonnull cases. See, for example, Efron (2004, 2005). In what follows I will first discuss the drawbacks of using p-value in large-scale multiple testing and demonstrate the fundamental role played by the Lfdr. I then discuss estimation of the null distribution and the proportion of the nonnulls. I will end with some comments about dealing with the dependency. In the discussion I shall use the notation given in Table 1 to summarize the outcomes of a multiple testing procedure. With the notation given in the table, the false discovery rate (FDR) is then defined as FDR = E(N10/R|R > 0)Pr(R > 0).
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TL;DR: Neyman's life history and some of his contributions to applied statistics are reviewed in this article, where a number of data sets and corresponding substantive questions are addressed, including sardine depletion, blowfly dynamics, weather modification, elk movement and seal journeying.
Abstract: Jerzy Neyman's life history and some of his contributions to applied statistics are reviewed. In a 1960 article he wrote: \"Currently in the period of dynamic indeterminism in science, there is hardly a serious piece of research which, if treated realistically, does not involve operations on stochastic processes. The time has arrived for the theory of stochastic processes to become an item of usual equipment of every applied statistician.\" The emphasis in this article is on stochastic processes and on stochastic process data analysis. A number of data sets and corresponding substantive questions are addressed. The data sets concern sardine depletion, blowfly dynamics, weather modification, elk movement and seal journeying. Three of the examples are from Neyman's work and four from the author's joint work with collaborators.
TL;DR: Efron et al. as discussed by the authors used a two-level model to test many hypotheses when exchangeabilityrestrictions are plausible, with the empirical null (null refers to a distribution, not to a hypothesis) extension of local fdr.
Abstract: . Brad Efron’s paper has inspired a return to the ideas be-hind Bayes, frequency and empirical Bayes. The latter preferably wouldnot be limited to exchangeable models for the data and hyperparam-eters. Parallels are revealed between microarray analyses and profilingof hospitals, with advances suggesting more decision modeling for geneidentification also. Then good multilevel and empirical Bayes modelsfor random effects should be sought when regression toward the meanis anticipated. Key words and phrases: Bayes, frequency, interval estimation, ex-changeable, general model, random effects.1. FREQUENCY, BAYES, EMPIRICAL BAYESAND A GENERAL MODELBrad Efron’s two-groups approach and the empir-ical null (“null” refers to a distribution, not to ahypothesis) extension of his local fdr addresses test-ing many hypotheses simultaneously, with model-ing enabled by the repeated presence of many simi-lar problems. He assumes two-level models for ran-dom effects, developing theory by drawing on andcombining ideas from frequency, Bayesian and em-pirical Bayesian perspectives. The last half-centuryin statistics has seen exciting developments frommany perspectives for simultaneous estimation ofrandom effects, but there has been little explicit par-allel work on the complementary problem of hypoth-esis testing. That changes in Brad’s paper,especiallyfor testing many hypotheses when exchangeabilityrestrictions are plausible.“Empirical Bayes” is in the paper’s title, said inSection 3 to be a “bipolar” methodology that draws
TL;DR: In this article, the authors set their work within the framework of Lancaster probabilities, a remoted corner of the theory of probability, now described in Section 6.1.1, and the reader is referred to Lancaster (1958, 1963, 1975) and the synthesis by Koudou (1995, 1996) for more details.
Abstract: It is a pleasure to congratulate the authors for this excellent, original and pedagogical paper. I read a preliminary draft at the end of 2006 and I then mentioned to the authors that their work should be set within the framework of Lancaster probabilities, a remoted corner of the theory of probability, now described in their Section 6.1. The reader is referred to Lancaster (1958, 1963, 1975) and the synthesis by Koudou (1995, 1996) for more details. Given probabilities μ(dx) and ν(dy) on spaces X and Y, and given orthonormal bases p = (pn(x)) and q = (qn(y)) of L2(μ) and L2(ν), a probability σ on X×Y is said to be of the Lancaster type if either there exists a sequence ρ = (ρn) in 2 such that σ(dx, dy)= [∑
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