On a Problem of Robbins

Gu, Jiaying; Koenker, Roger

doi:10.1111/insr.12098

Cited by 19 publications

(12 citation statements)

References 33 publications

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“…In fact, if the Bayesian assumption is true, is the optimal method of integrating

T_{j}

and

scriptD

. Even under the present frequentist setting described in Section , procedures motivated by Bayesian formalisms can still have excellent performance, for example, in frequentist compound decision problems (Robbins, ; Robbins et al., ; Zhang, ; Brown and Greenshtein, ; Jiang et al., ; Gu and Koenker, ).…”

Section: Methodsmentioning

confidence: 99%

“…The form of (4) is the key to the proposed approach. Unlike the optimal classifier δ (2), in which neither T nor D appear, (4) provides a sensible way for integrating the T j with the D. In fact, if the Bayesian assumption (3) is true, (4) is the optimal method of integrating T j and D. Even under the present frequentist setting described in Section 2.1, procedures motivated by Bayesian formalisms can still have excellent performance, for example, in frequentist compound decision problems Zhang, 2003;Brown and Greenshtein, 2009;Jiang et al, 2009;Gu and Koenker, 2015).…”

Section: Integrative Classification Via Bayesian Modelingmentioning

confidence: 99%

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Integrative Genetic Risk Prediction Using Non-Parametric Empirical Bayes Classification

Zhao

2016

Biometrics

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Genetic risk prediction is an important component of individualized medicine, but prediction accuracies remain low for many complex diseases. A fundamental limitation is the sample sizes of the studies on which the prediction algorithms are trained. One way to increase the effective sample size is to integrate information from previously existing studies. However, it can be difficult to find existing data that examine the target disease of interest, especially if that disease is rare or poorly studied. Furthermore, individual-level genotype data from these auxiliary studies are typically difficult to obtain. This paper proposes a new approach to integrative genetic risk prediction of complex diseases with binary phenotypes. It accommodates possible heterogeneity in the genetic etiologies of the target and auxiliary diseases using a tuning parameterfree nonparametric empirical Bayes procedure, and can be trained using only auxiliary summary statistics. Simulation studies show that the proposed method can provide superior predictive accuracy relative to non-integrative as well as integrative classifiers. The method is applied to a recent study of pediatric autoimmune diseases, where it substantially reduces prediction error for certain target/auxiliary disease combinations. The proposed method is implemented in the R package ssa.

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“…In fact, if the Bayesian assumption is true, is the optimal method of integrating

T_{j}

and

scriptD

Section: Methodsmentioning

confidence: 99%

Section: Integrative Classification Via Bayesian Modelingmentioning

confidence: 99%

Integrative Genetic Risk Prediction Using Non-Parametric Empirical Bayes Classification

Zhao

2016

Biometrics

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“…We consider the problem of estimating a vector of Poisson intensity parameters θ = ( θ 1 , …, θ k ) from a sample of Y i | θ i ~ Poisson( θ i ), where the Bayes estimate is given by: Two primary approaches for estimating ( 3.3 ): Parametric Culture 37 , 38 : If one assumes π ( θ ) to be the parametric conjugate Gamma distribution , then it is straightforward to show that Stein’s estimate takes the following analytical form , weighted average of the MLE y i and the prior mean αβ . Nonparametric Culture 4 , 7 , 39 : This was born out of Herbert Robbins’ ingenious observation that ( 3.3 ) can alternatively be written in terms of marginal distribution , and thus can be estimated non-parametrically by substituting empirical frequencies. This remarkable “prior-free” representation, however, does not hold in general for other distributions.…”

Section: Inferencementioning

confidence: 99%

Generalized Empirical Bayes Modeling via Frequentist Goodness of Fit

Mukhopadhyay

Fletcher

2018

Sci Rep

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The two key issues of modern Bayesian statistics are: (i) establishing principled approach for distilling statistical prior that is consistent with the given data from an initial believable scientific prior; and (ii) development of a consolidated Bayes-frequentist data analysis workflow that is more effective than either of the two separately. In this paper, we propose the idea of “Bayes via goodness-of-fit” as a framework for exploring these fundamental questions, in a way that is general enough to embrace almost all of the familiar probability models. Several examples, spanning application areas such as clinical trials, metrology, insurance, medicine, and ecology show the unique benefit of this new point of view as a practical data science tool.

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“…In Gu and Koenker (2016) we explore some extensions of this simple setting to several other multiple testing problems. We first consider a grouped setting in which we have…”

Section: Gaussian Mixtures and Multiple Testingmentioning

confidence: 99%

REBayes: An R Package for Empirical Bayes Mixture Methods

Gu¹,

Koenker²

2017

J. Stat. Soft.

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Models of unobserved heterogeneity, or frailty as it is commonly known in survival analysis, can often be formulated as semiparametric mixture models and estimated by maximum likelihood as proposed by Robbins (1950) and elaborated by Kiefer and Wolfowitz (1956). Recent developments in convex optimization, as noted by Koenker and Mizera (2014b), have led to dramatic improvements in computational methods for such models. In this vignette we describe an implementation contained in the R package REBayes with applications to a wide variety of mixture settings: Gaussian location and scale, Poisson and binomial mixtures for discrete data, Weibull and Gompertz models for survival data, and several Gaussian models intended for longitudinal data. While the dimension of the nonparametric heterogeneity of these models is inherently limited by our present gridding strategy, we describe how additional fixed parameters can be relatively easily accommodated via profile likelihood. We also describe some nonparametric maximum likelihood methods for shape and norm constrained density estimation that employ related computational methods.

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On a Problem of Robbins

Cited by 19 publications

References 33 publications

Integrative Genetic Risk Prediction Using Non-Parametric Empirical Bayes Classification

Integrative Genetic Risk Prediction Using Non-Parametric Empirical Bayes Classification

Generalized Empirical Bayes Modeling via Frequentist Goodness of Fit

REBayes: An R Package for Empirical Bayes Mixture Methods

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