The Experts below are selected from a list of 16251 Experts worldwide ranked by ideXlab platform
Bradley Efron - One of the best experts on this subject based on the ideXlab platform.
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Bayes oracle Bayes and Empirical Bayes
Statistical Science, 2019Co-Authors: Bradley EfronAbstract:This article concerns the Bayes and frequentist aspects of Empirical Bayes inference. Some of the ideas explored go back to Robbins in the 1950s, while others are current. Several examples are discussed, real and artificial, illustrating the two faces of Empirical Bayes methodology: “oracle Bayes” shows Empirical Bayes in its most frequentist mode, while “finite Bayes inference” is a fundamentally Bayesian application. In either case, modern theory and computation allow us to present a sharp finite-sample picture of what is at stake in an Empirical Bayes analysis.
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two modeling strategies for Empirical Bayes estimation
arXiv: Methodology, 2014Co-Authors: Bradley EfronAbstract:Empirical Bayes methods use the data from parallel experiments, for instance, observations $X_k\sim\mathcal{N}(\Theta_k,1)$ for $k=1,2,\ldots,N$, to estimate the conditional distributions $\Theta_k|X_k$. There are two main estimation strategies: modeling on the $\theta$ space, called "$g$-modeling" here, and modeling on the $x$ space, called "$f$-modeling." The two approaches are described and compared. A series of computational formulas are developed to assess their frequentist accuracy. Several examples, both contrived and genuine, show the strengths and limitations of the two strategies.
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two modeling strategies for Empirical Bayes estimation
Statistical Science, 2014Co-Authors: Bradley EfronAbstract:Empirical Bayes methods use the data from parallel experiments, for instance, observations Xk i« N(Θk, 1) for k = 1, 2, . . . , N, to estimate the conditional distributions Θk|Xk. There are two main estimation strategies: modeling on the ¦E space, called i°g-modelingi± here, and modeling on the x space, called i°f -modeling.i± The two approaches are described and compared. A series of computational formulas are developed to assess their frequentist accuracy. Several examples, both contrived and genuine, show the strengths and limitations of the two strategies.
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Empirical Bayes estimates for large scale prediction problems
Journal of the American Statistical Association, 2009Co-Authors: Bradley EfronAbstract:Classical prediction methods, such as Fisher’s linear discriminant function, were designed for small-scale problems in which the number of predictors N is much smaller than the number of observations n. Modern scientific devices often reverse this situation. A microarray analysis, for example, might include n=100 subjects measured on N=10,000 genes, each of which is a potential predictor. This article proposes an Empirical Bayes approach to large-scale prediction, where the optimum Bayes prediction rule is estimated employing the data from all of the predictors. Microarray examples are used to illustrate the method. The results demonstrate a close connection with the shrunken centroids algorithm of Tibshirani et al. (2002), a frequentist regularization approach to large-scale prediction, and also with false discovery rate theory.
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microarrays Empirical Bayes and the two groups model
Statistical Science, 2008Co-Authors: Bradley EfronAbstract: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.
Cunhui Zhang - One of the best experts on this subject based on the ideXlab platform.
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group linear Empirical Bayes estimates for a heteroscedastic normal mean
Journal of the American Statistical Association, 2018Co-Authors: Asaf Weinstein, Lawrence D Brown, Cunhui ZhangAbstract:ABSTRACTThe problem of estimating the mean of a normal vector with known but unequal variances introduces substantial difficulties that impair the adequacy of traditional Empirical Bayes estimators. By taking a different approach that treats the known variances as part of the random observations, we restore symmetry and thus the effectiveness of such methods. We suggest a group-linear Empirical Bayes estimator, which collects observations with similar variances and applies a spherically symmetric estimator to each group separately. The proposed estimator is motivated by a new oracle rule which is stronger than the best linear rule, and thus provides a more ambitious benchmark than that considered in the previous literature. Our estimator asymptotically achieves the new oracle risk (under appropriate conditions) and at the same time is minimax. The group-linear estimator is particularly advantageous in situations where the true means and observed variances are Empirically dependent. To demonstrate the meri...
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group linear Empirical Bayes estimates for a heteroscedastic normal mean
arXiv: Methodology, 2015Co-Authors: Asaf Weinstein, Lawrence D Brown, Cunhui ZhangAbstract:The problem of estimating the mean of a normal vector with known but unequal variances introduces substantial difficulties that impair the adequacy of traditional Empirical Bayes estimators. By taking a different approach, that treats the known variances as part of the random observations, we restore symmetry and thus the effectiveness of such methods. We suggest a group-linear Empirical Bayes estimator, which collects observations with similar variances and applies a spherically symmetric estimator to each group separately. The proposed estimator is motivated by a new oracle rule which is stronger than the best linear rule, and thus provides a more ambitious benchmark than that considered in previous literature. Our estimator asymptotically achieves the new oracle risk (under appropriate conditions) and at the same time is minimax. The group-linear estimator is particularly advantageous in situations where the true means and observed variances are Empirically dependent. To demonstrate the merits of the proposed methods in real applications, we analyze the baseball data used in Brown (2008), where the group-linear methods achieved the prediction error of the best nonparametric estimates that have been applied to the dataset, and significantly lower error than other parametric and semi-parametric Empirical Bayes estimators.
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general Empirical Bayes wavelet methods and exactly adaptive minimax estimation
arXiv: Statistics Theory, 2005Co-Authors: Cunhui ZhangAbstract:In many statistical problems, stochastic signals can be represented as a sequence of noisy wavelet coefficients. In this paper, we develop general Empirical Bayes methods for the estimation of true signal. Our estimators approximate certain oracle separable rules and achieve adaptation to ideal risks and exact minimax risks in broad collections of classes of signals. In particular, our estimators are uniformly adaptive to the minimum risk of separable estimators and the exact minimax risks simultaneously in Besov balls of all smoothness and shape indices, and they are uniformly superefficient in convergence rates in all compact sets in Besov spaces with a finite secondary shape parameter. Furthermore, in classes nested between Besov balls of the same smoothness index, our estimators dominate threshold and James-Stein estimators within an infinitesimal fraction of the minimax risks. More general block Empirical Bayes estimators are developed. Both white noise with drift and nonparametric regression are considered.
Tachen Liang - One of the best experts on this subject based on the ideXlab platform.
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Empirical Bayes testing for success probability of bernoulli process with negative binomial sampling nonidentical components case
Journal of Applied Mathematics and Computing, 2009Co-Authors: Tachen LiangAbstract:We study an Empirical Bayes testing problem for the success probability of Bernoulli process with negative binomial sampling under nonidentical components situation. An Empirical Bayes test δ +1, * is constructed based on an estimator c n of the critical point c G of the Bayes test δ n+1,G . The Empirical Bayes test δ +1, * possesses the asymptotic optimality, and its associated regret converges to zero at an exponential decay rate O(exp (−n α)) for some positive value α, depending on the unknown prior distribution G. This result extends the exponential type decay rate of convergence obtained in Liang (Ann. Stat. 16:1635–1642, 1988; Stat. Probab. Lett. 44:241–249, 1999) to the nonidentical components case.
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on an improved Empirical Bayes estimator for positive exponential families
Journal of Nonparametric Statistics, 2005Co-Authors: Tachen LiangAbstract:This paper deals with the Empirical Bayes estimation of the parameter θ in positive exponential families having probability density function (pdf) f (x| θ) = u(x)c(θ) exp(−x/θ). A new Empirical Bayes estimator is studied. It is proved that under certain regularity conditions, is asymptotically optimal at a rate (ln2 n/n)(λ s − 2)/2s , where s > 2 and (2/s) < λ ≤ 2(1 − (1/2)). Examples are given to illustrate the performance of . It is shown that is superior to the Empirical Bayes estimator of Singh and Wei [Annals of the Institute of Statistical Mathematics, 44 (1992), 435–449] in the sense that possesses a faster rate of convergence under weaker conditions.
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convergence rates for Empirical Bayes estimation of the scale parameter in a pareto distribution
Computational Statistics & Data Analysis, 1993Co-Authors: Tachen LiangAbstract:Abstract Let f(χ∣θ) = αθ α χ α+1 I (θ,∞) (χ) be the pdf of a Pareto distribution with known shape scale parameter α > 0 and unknown scale parameter θ. We study the problem of estimating the scale parameter θ under a squared-error loss through the nonparametric Empirical Bayes approach. An Empirical Bayes estimator is proposed and the corresponding asymptotic optimality is also investigated. It is shown that under certain weak conditions the proposed Empirical Bayes estimator is asymptotically optimal and the associated rate of convergence is of order O(n − 2 3 ) .
Robert Tibshirani - One of the best experts on this subject based on the ideXlab platform.
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Empirical Bayes methods and false discovery rates for microarrays
Genetic Epidemiology, 2002Co-Authors: Bradley Efron, Robert TibshiraniAbstract:In a classic two-sample problem, one might use Wilcoxon's statistic to test for a difference between treatment and control subjects. The analogous microarray experiment yields thousands of Wilcoxon statistics, one for each gene on the array, and confronts the statistician with a difficult simultaneous inference situation. We will discuss two inferential approaches to this problem: an Empirical Bayes method that requires very little a priori Bayesian modeling, and the frequentist method of “false discovery rates” proposed by Benjamini and Hochberg in 1995. It turns out that the two methods are closely related and can be used together to produce sensible simultaneous inferences. Genet. Epidemiol. 23:70–86, 2002. © 2002 Wiley-Liss, Inc.
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Empirical Bayes methods and false discovery rates for microarrays
Genetic Epidemiology, 2002Co-Authors: Bradley Efron, Robert TibshiraniAbstract:In a classic two-sample problem, one might use Wilcoxon's statistic to test for a difference between treatment and control subjects. The analogous microarray experiment yields thousands of Wilcoxon statistics, one for each gene on the array, and confronts the statistician with a difficult simultaneous inference situation. We will discuss two inferential approaches to this problem: an Empirical Bayes method that requires very little a priori Bayesian modeling, and the frequentist method of "false discovery rates" proposed by Benjamini and Hochberg in 1995. It turns out that the two methods are closely related and can be used together to produce sensible simultaneous inferences.
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Empirical Bayes analysis of a microarray experiment
Journal of the American Statistical Association, 2001Co-Authors: Bradley Efron, Robert Tibshirani, John D Storey, Virginia Goss TusherAbstract:Microarrays are a novel technology that facilitates the simultaneous measurement of thousands of gene expression levels. A typical microarray experiment can produce millions of data points, raising serious problems of data reduction, and simultaneous inference. We consider one such experiment in which oligonucleotide arrays were employed to assess the genetic effects of ionizing radiation on seven thousand human genes. A simple nonparametric Empirical Bayes model is introduced, which is used to guide the efficient reduction of the data to a single summary statistic per gene, and also to make simultaneous inferences concerning which genes were affected by the radiation. Although our focus is on one specific experiment, the proposed methods can be applied quite generally. The Empirical Bayes inferences are closely related to the frequentist false discovery rate (FDR) criterion.
Sihai Dave Zhao - One of the best experts on this subject based on the ideXlab platform.
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a nonparametric Empirical Bayes approach to large scale multivariate regression
Computational Statistics & Data Analysis, 2021Co-Authors: Yihe Wang, Sihai Dave ZhaoAbstract:Abstract Multivariate regression has many applications, ranging from time series prediction to genomics. Borrowing information across the outcomes can improve prediction error, even when outcomes are statistically independent. Many methods exist to implement this strategy, for example the multiresponse lasso, but choosing the optimal method for a given dataset is difficult. These issues are addressed by establishing a connection between multivariate linear regression and compound decision problems. A nonparametric Empirical Bayes procedure that can learn the optimal regression method from the data itself is proposed. Furthermore, the proposed procedure is free of tuning parameters and performs well in simulations and in a multiple stock price prediction problem.
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a nonparametric Empirical Bayes approach to covariance matrix estimation
arXiv: Methodology, 2020Co-Authors: Huiqin Xin, Sihai Dave ZhaoAbstract:We propose an Empirical Bayes method to estimate high-dimensional covariance matrices. Our procedure centers on vectorizing the covariance matrix and treating matrix estimation as a vector estimation problem. Drawing from the compound decision theory literature, we introduce a new class of decision rules that generalizes several existing procedures. We then use a nonparametric Empirical Bayes g-modeling approach to estimate the oracle optimal rule in that class. This allows us to let the data itself determine how best to shrink the estimator, rather than shrinking in a pre-determined direction such as toward a diagonal matrix. Simulation results and a gene expression network analysis shows that our approach can outperform a number of state-of-the-art proposals in a wide range of settings, sometimes substantially.
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high dimensional classification via nonparametric Empirical Bayes and maximum likelihood inference
Biometrika, 2016Co-Authors: Lee H Dicker, Sihai Dave ZhaoAbstract:We propose new nonparametric Empirical Bayes methods for high-dimensional classification. Our classifiers are designed to approximate the Bayes classifier in a hypothesized hierarchical model, where the prior distributions for the model parameters are estimated nonparametrically from the training data. As is common with nonparametric Empirical Bayes, the proposed classifiers are effective in high-dimensional settings even when the underlying model parameters are in fact nonrandom. We use nonparametric maximum likelihood estimates of the prior distributions, following the elegant approach studied by Kiefer & Wolfowitz in the 1950s. However, our implementation is based on a recent convex optimization framework for approximating these estimates that is well-suited for large-scale problems. We derive new theoretical results on the accuracy of the approximate estimator, which help control the misclassification rate of one of our classifiers. We show that our methods outperform several existing methods in simulations and perform well when gene expression microarray data is used to classify cancer patients.
