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Parametric Empirical Bayes Inference: Theory and Applications
Abstract
This article reviews the state of multiparameter shrinkage estimators with emphasis on the empirical Bayes viewpoint, particularly in the case of parametric prior distributions. Some successful applications of major importance are considered. Recent results concerning estimates of error and confidence intervals are described and illustrated with data.
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“The highly parallel nature of gene expression experiments lends itself to a particular class of statistical methods, called parametric empirical Bayes, that borrow information between genes in a dynamic way (14,15).”
From limma powers differential expression analyses for RNA-sequencing and microarray studies (2015) · cites this paper as [15]
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Estimating the health value added by nursing homes
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“…home specific effects given that both the prior and likelihood function come from a normal distribution (Angrist et al., 2017; Chetty and Hendren, 2018; Kane and Staiger, 2008; Morris, 1983). performance in Section 5.1 and when correlating performance on other quality indicators in Section 6.”
BackgroundMethodInfluentialDistributed estimation of binary event probabilities via hierarchical Bayes and dual decomposition
Coluccia, A., & Notarstefano, G. (2013) · IEEE Conference on Decision and Control (CDC) · doi:10.1109/cdc.2013.6760959 · Cited by 11
“To overcome this limitation, we consider the Empirical Bayes approach [11], which requires only that the prior be in a family of distributions whose parameters, called hyperparameters, are estimated from the observations mi, i ∈ 1, .”
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“Like the similarities between the parametric empirical Bayes procedure of Morris (1983) and a fully Bayesian parametric method, our approach has a close relationship with a full semiparametric Bayesian analysis because we can regard the latter as a special case of empirical Bayes when the class ∏…”
MethodShrinkage Estimation of Log-odds Ratios for Comparing Mobility Tables
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Villa, C. (2016) · Test · doi:10.1007/s11749-016-0501-7 · Cited by 11
“Examples can be found in Morris (1983), Berger (1985), Carlin and Gelfand (1990) and Czado et al. (2005).”
BackgroundBayesian analysis of depth resolved OCT attenuation coefficients
Fiske, L. D., Aalders, M. C. G., Almasian, M., van Leeuwen, T. G. J. M., Katsaggelos, A. K., Cossairt, O. S., & Faber, D. J. (2021) · Scientific Reports · doi:10.1038/s41598-021-81713-7 · Cited by 11
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“…were used: Hedges (HE) (Hedges and Olkin 1985), Hunter–Schmidt (HS) (Hunter and Schmidt 2004), DerSimonian–Laird (DL) (DerSimonian and Laird 1986), Sidik–Jonkman (SJ) (Sidik and Jonkman 2005), (restricted) maximum likelihood (ML) (REML) (Viechtbauer 2005), and Empirical Bayes (EB) (Morris 1983).”
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Samaniego, F. J., & Neath, A. (1996) · Journal of the American Statistical Association · doi:10.1080/01621459.1996.10476941 · Cited by 13
Controlling error rates using prior information and marginal totals to select tumor sites
Louis, T. A., & Bailey, J. K. (1990) · Journal of Statistical Planning and Inference · doi:10.1016/0378-3758(90)90050-5 · Cited by 11
“The prior induces a constant under- lying tumor rate from experiment to experiment, since the estimated posterior distribution is degenerate at m. Morris (1983), and Laird and Louis (1986) discuss this issue.”
BackgroundMethodThe Kullback-Leibler risk of the Stein estimator and the conditional MLE
Yanagimoto, T. (1994) · Annals of the Institute of Statistical Mathematics · doi:10.1007/bf00773590 · Cited by 11
Protection against outliers in empirical bayes estimation
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Gaver, D. P., Jacobs, P. A., & O’Muircheartaigh, I. G. (1990) · Communication in Statistics- Theory and Methods · doi:10.1080/03610929008830413 · Cited by 11
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Abadie, A., & Kasy, M. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1703.10935 · Cited by 13
“The empirical Bayes framework was developed further by Efron and Morris (1973) and Morris (1983), among others.”
BackgroundMethodInfluentialSolving the Empirical Bayes Normal Means Problem with Correlated NoisePreprint
Sun, L., & Stephens, M. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1812.07488 · Cited by 12
“First named by Robbins (1956), EB methods have seen extensive theoretical study (e.g. Robbins, 1964; Morris, 1983; Efron, 1996; Jiang and Zhang, 2009; Brown and Greenshtein, 2009; Scott and Berger, 2010; Petrone et al., 2014; Rousseau and Szabo, 2017; Efron, 2018), and are becoming widely used in…”
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Begenau, J., & SIRIWARDANE, E. N. (2020) · SSRN Electronic Journal · doi:10.2139/ssrn.3526469 · Cited by 11
“However, 22See Morris (1983) for a formal description of Empirical Bayes inference.”
Methods for measuring school effectiveness
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Generalized Empirical Bayes Modeling via Frequentist Goodness of Fit
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“…likelihood (ML) estimator 8 (Hardy and Thompson, 1996; Thompson and Sharp, 1999) and the restricted maximum likelihood (REML) estimator 9 (Morris, 1983; Berkey et al., 1995; Thompson and Sharp, 1999) are usually adopted to conduct the parameter estimation 10 in random effects…”
BackgroundTraffic safety meta-analysis of reversible lanes
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First-Time Failure Rates of Candidates for Board Certification
O’Day, D. M. (2008) · Archives of Ophthalmology · doi:10.1001/archopht.126.4.548 · Cited by 10
“Alternatively, in an empirical Bayes method, the 118 i’s are assumed to come from a common distribution, beta ( , ), in which the parameters and control the shape of the distribution and may be estimated using the data.(1,2) For our data, the parameters were estimated to be =2.”
BackgroundMethodRevisiting Beal's Confidence Intervals for the Difference of Two Binomial Proportions
Roths, S. A., & Tebbs, J. M. (2006) · Communication in Statistics- Theory and Methods · doi:10.1080/03610920600683622 · Cited by 10
“4, we propose “data-driven” Beal (1987) procedures, whereby prior hyperparameters are estimated using a parametric empirical-Bayes approach (Morris, 1983).”
MethodThe Need for More Emphasis on Prediction: A “Nondenominational” Model-Based Approach
Harville, D. A. (2013) · The American Statistician · doi:10.1080/00031305.2013.836987 · Cited by 10
“The partially hierarchical approach was introduced by Efron and Morris as an alternative to a fully Bayesian approach and, as such, is often referred to as the (parametric) empirical Bayesian approach—refer, for example, to Morris (1983).”
MethodNoma, H., & Matsui, S. (2012) · Statistics in Medicine · doi:10.1002/sim.5718 · Cited by 10
“…parametric empirical Bayes methods can suffer from lack of robustness when the true prior is not from the assumed parametric family as pointed out by Morris [45], such as a failure in incorporating the mixture structure or in specifying the form of the effect-size distribution ( G 1 and G 2 / .”
MethodEmpirical Bayes sequential estimation for exponential families: The untruncated component
Karunamuni, R. J. (1996) · Annals of the Institute of Statistical Mathematics · doi:10.1007/bf00052329 · Cited by 10
“Parametric models are becoming increasingly popular in empirical Bayes methods, due in large part to their compromising behavior between frequentist modelling and Bayesian modelling; see, e.g., Morris (1983a, 1983b) .”
Meirelles, O., Ding, J., Tanaka, T., Sanna, S., Yang, H., Dudekula, D. B., Cucca, F., Ferrucci, L., Abecasis, G. R., & Schlessinger, D. (2012) · European Journal of Human Genetics · doi:10.1038/ejhg.2012.215 · Cited by 10
Model Selection and Multiple Testing - A Bayesian and Empirical Bayes Overview and some New Results
Dutta, R., Bogdan, M., & Ghosh, J. K. (2015) · Warwick Research Archive Portal (University of Warwick) · doi:10.48550/arxiv.1510.00547 · Cited by 10
“Alternatively one gets Best Unbiased Estimators as in Morris (1983).”
BackgroundShinozaki, N., & Chang, Y.-T. (1993) · Communication in Statistics- Theory and Methods · doi:10.1080/03610929308831140 · Cited by 10
Improved Confidence Statements for the Usual Multivariate Normal Confidence Set
Robert, C., & Casella, G. (1994) · Springer eBooks · doi:10.1007/978-1-4612-2618-5_26 · Cited by 9
Monotone Empirical Bayes Estimators Based on More Informative Samples
van Houwelingen, H. C., & Stijnen, T. (1993) · Journal of the American Statistical Association · doi:10.1080/01621459.1993.10476430 · Cited by 10
Mantón, K. G., Holzer, C. E., MacKenzie, E. J., Spitznagel, E. L., Forsythe, A. B., & Jackson, D. (1985) · Elsevier eBooks · doi:10.1016/b978-0-08-091798-6.50021-x · Cited by 10
Bayes and Admissible Set Estimation
Meeden, G. D., & Vardeman, S. B. (1985) · Journal of the American Statistical Association · doi:10.1080/01621459.1985.10478142 · Cited by 10
Blacklisting Social Science Departments with Poor Ph.D. Submission Rates
Colombo, R. A., & Morrison, D. G. (1988) · Management Science · doi:10.1287/mnsc.34.6.696 · Cited by 10
Skills, Productivity and the Evaluation of Teacher PerformancePreprint
Sass, T. R., & Harris, D. (2012) · SSRN Electronic Journal · doi:10.2139/ssrn.2020717 · Cited by 10
“The empirical Bayes method “shrinks” teacher effect estimates toward the population mean, with the degree of shrinkage proportional to the standard error of the teacher effect estimate (see Morris (1983)).”
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“Under certain conditions, EB is consistent with JamesStein estimators (e.g. Morris 1983).”
MethodInfluentialEmpirical Bayes When Estimation Precision Predicts Parameters
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Closed-form Bayesian inferences for the logit model via polynomial expansions
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“(19) Assume all xijt,p are integers; this is not a terribly restrictive assumption (1), and can be simply accomplished by changing the scale we use to measure the xijt,p’s.”
MethodCalibration Concordance for Astronomical Instruments via Multiplicative Shrinkage
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“To communicate our model aims clearly, when σ(2) is known, we express the MAP estimators of θ in terms of the usual linear shrinkage estimators (Efron and Morris, 1975; Morris, 1983) of B and G.”
BackgroundMethodOn general maximum likelihood empirical Bayes estimation of heteroscedastic IID normal means
Jiang, W. (2020) · Electronic Journal of Statistics · doi:10.1214/20-ejs1717 · Cited by 9
“However, while the existing studies are typically based on the shrinkage approach, our focus is on the general empirical Bayes [13, 15], or equivalently nonparametric empirical Bayes [12].”
Global robust Bayesian analysis in large models
Ho, P. (2022) · Journal of Econometrics · doi:10.1016/j.jeconom.2022.06.004 · Cited by 9
An Empirical Bayesian Approach to Limb Darkening in Modeling WASP-121b Transit Light Curves
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“The empirical Bayes method (Morris 1983; Casella 1985) solves this issue, in a general way, obtaining the prior from both the new data and the former knowledge.”
MethodSatisficing credibility for heterogeneous risks
Cheung, K. C., Yam, S. C. P., & Zhang, Y. (2021) · European Journal of Operational Research · doi:10.1016/j.ejor.2021.07.020 · Cited by 9
Using hidden Markov chains and empirical Bayes change-point estimation for transect data
VERHOEF, J. M., & Cressie, N. (1997) · Environmental and Ecological Statistics · doi:10.1023/a:1018582614125 · Cited by 9
“In this article, we unite these ideas for estimating the location of change-points in m. Philosophically, we adopt a parametric empirical Bayes approach (Morris, 1983).”
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Stout, R. L. (2007) · Addiction · doi:10.1111/j.1360-0443.2007.02036.x · Cited by 9
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“An empirical Bayes approach is also possible, in which (for the simplest case) the parameters are assumed to be independent and identically distributed from a (univariate) distribution which is estimated form the data (Morris, 1983; Efron, 2005).”
Empirical Bayes estimation and its superiority for two-way classification model
Wei, L., & Chen, J. (2003) · Statistics & Probability Letters · doi:10.1016/s0167-7152(03)00079-8 · Cited by 9
On Application of the Empirical Bayes Shrinkage in Epidemiological Settings
Zhao, Y., Lee, A. H., & Barnes, T. (2010) · International Journal of Environmental Research and Public Health · doi:10.3390/ijerph7020380 · Cited by 9
“Dating back to Efron and Morris in the early 1970s, the high risk of EBS estimation has been recognized for individual parameters far from the mean of the prior distribution [22,23].”
MethodInfluentialA Bayesian approach to flexible modeling of multivariable response functions
Steinberg, D. M. (1990) · Journal of Multivariate Analysis · doi:10.1016/0047-259x(90)90033-e · Cited by 9
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An Empirical Bayes Approach to Estimating Loss Ratios
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“In this way, it provides the best estimate of the true difference in quality of secondary prevention across health markets.(19,20) To understand whether the quality of secondary prevention provided was related to health care capacity, we estimated the odds of a patient undergoing on-treatment follow-up testing relativ…”
BackgroundSteinized Empirical Bayes Estimation for Heteroscedastic Data
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“To mitigate the difficulty of prior specification, an empirical Bayes approach based on (1.1) and (1.2) is to estimate (γ, β) as unknown parameters in the marginal distribution, Yj ∼ N(xTj β, dj + γ), and then substitute the estimates of (β, γ) into the Bayes rule for estimating θ (Efron and Morris (1973); Morris (198…”
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Pathak, M., Dwivedi, S. N., & Thakur, B. (2019) · Clinical Epidemiology and Global Health · doi:10.1016/j.cegh.2019.06.011 · Cited by 9
“= k T T ˆ 1 1 w ( ˆ ) SJ i 2 iSJ SJ 2 (19) Where”
Inference for Welfare Quality Control Programs
Fairley, W., Izenman, A., & Bagchi, P. (1990) · Journal of the American Statistical Association · doi:10.1080/01621459.1990.10474957 · Cited by 9
“Morris (1983) developed approximations to posterior expectations and variances that take such errors into consideration.”
MethodHigh cost patients for cardiac surgery and hospital quality in Turkey
Başer, O., BÜrkan, A., Baser, E., Koselerli, R., Ertugay, E., & Altınbaş, A. (2012) · Health Policy · doi:10.1016/j.healthpol.2012.09.015 · Cited by 9
Wang, H., Higgins, J. J., & Blasi, D. A. (2009) · Statistics & Probability Letters · doi:10.1016/j.spl.2009.11.016 · Cited by 9
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“We develop estimation strategies for the hyper-parameters (~α,B) within the empirical Bayes framework (Morris, 1983; Carlin and Louis, 2005).”
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Siegel, C., Jones, K., LASKA, E. M., Meisner, M., & Lin, S. (1992) · Journal of Health Economics · doi:10.1016/0167-6296(92)90023-t · Cited by 9
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Bayesian Inference for Stochastic Epidemic Models using Markov chain Monte Carlo Methods
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“Empirical Bayes (EB) methods ( Morris, 1983 ; Snijders & Bosker, 1999 , pp.”
MethodSun, J., Zhang, Y., & Sun, Y. Z. (2020) · Journal of Statistical Computation and Simulation · doi:10.1080/00949655.2020.1858299 · Cited by 8
Meta-analysis methods for risk difference: A comparison of different models
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“8 EB estimator The EB estimator is a multi-parameter shrinkage estimator of τ.(53) If δk is the true effect size for study k and δ̂IV is the effect estimate from the IV method, then”
MethodCan Human Capital Explain Income-Based Disparities in Financial Services?
Huang, R., Linck, J. S., Mayer, E. J., & Parsons, C. A. (2024) · Review of Financial Studies · doi:10.1093/rfs/hhae004 · Cited by 8
Producing Health: Measuring Value Added of Nursing Homes
Liran Einav, B., Finkelstein, A. N., & Mahoney, N. (2025) · Econometrica · doi:10.3982/ecta21016 · Cited by 8
Simultaneous estimation of parameters of Poisson distributions with unbalanced sample sizes
Hamura, Y., & Kubokawa, T. (2019) · Japanese Journal of Statistics and Data Science · doi:10.1007/s42081-019-00039-x · Cited by 8
“For a related discussion, see, for example, Morris (1983).”
BackgroundLegal Judgment Prediction in the Context of Energy Market using Gradient Boosting
França, J. V. F., Boaro, J. M. C., dos Santos, P. T. C., Henrique, F., Garcia, V., Manfredini, C., Dias, D., de Oliveira, F. Y. C., Castro, C. E. P., Bráz, G., et al. (2020) · Conference proceedings/Conference proceedings - IEEE International Conference on Systems, Man, and Cybernetics · doi:10.1109/smc42975.2020.9283297 · Cited by 8
“It consists of calculating the weighted average of the mean target value for the observed class of the categorical variable p i and the overall mean target value p all [16], [17].”
Background2 A national center for statistical ecology and environmental statistics: A center without walls
Patil, G. P. (1994) · Handbook of statistics · doi:10.1016/s0169-7161(05)80004-5 · Cited by 8
Assessing the Risk of an Aquaculture Development on Shorebirds Using a Bayesian Belief Model
Gibbs, M. T. (2007) · Human and Ecological Risk Assessment An International Journal · doi:10.1080/10807030601105423 · Cited by 8
Empirical Bayes estimation in directional data
Bagchi, P. (1994) · Journal of Applied Statistics · doi:10.1080/757583874 · Cited by 8
Robustness to Parametric Assumptions in Missing Data Models
Graham, B. S., & Hirano, K. (2011) · American Economic Review · doi:10.1257/aer.101.3.538 · Cited by 8
Empirical Bayes analysis for a hierarchical Poisson generalized linear model
Tu, W., & Piegorsch, W. (2003) · Journal of Statistical Planning and Inference · doi:10.1016/s0378-3758(02)00305-1 · Cited by 8
Using a Short Screening Scale for Small-Area Estimation of Mental Illness Prevalence for Schools
Li, F., & Zaslavsky, A. M. (2010) · Journal of the American Statistical Association · doi:10.1198/jasa.2010.ap09185 · Cited by 8
The optimal confidence region for a random parameter
Uno, H., Tian, L., & Wei, L. (2005) · Biometrika · doi:10.1093/biomet/92.4.957 · Cited by 8
Liang, L.-J. F., Weiss, R. E., Redelings, B. D., & Suchard, M. A. (2009) · Bioinformatics · doi:10.1093/bioinformatics/btp473 · Cited by 8
Using an empirical Bayes model to estimate currency exchange rate
Okafor, R. (1999) · Journal of Applied Statistics · doi:10.1080/02664769921972 · Cited by 8
Influence diagnostics in meta‐regression model
Ши, Л., Zuo, S., Yu, D., & Zhou, X. (2017) · Research Synthesis Methods · doi:10.1002/jrsm.1247 · Cited by 8
Highway improvement project selection by the joint consideration of cost-benefit and risk criteria
Kelle, P., Schneider, H., Raschke, C., & Shirazi, H. (2012) · Journal of the Operational Research Society · doi:10.1057/jors.2012.55 · Cited by 8
“Morris (1983) showed that the mean squared error for the empirical Bayes estimate is smaller than the variance of either the estimate of the overall mean or the estimate of the individual mean.”
BackgroundConfidence Intervals for Ridge Regression Parameters
Vinod, H. D. (1987) · Springer eBooks · doi:10.1007/978-94-009-4790-0_19 · Cited by 8
Econometric Models and Spatial Parametric Instability: Relevant Concepts and an Instability Index
Casetti, E., & Poon, J. (1995) · Advances in spatial science · doi:10.1007/978-3-642-79877-1_13 · Cited by 8
Monitoring Project Performance with Post-Audit Information: Cash Flow Control Charts
Prueitt, G. C., & Park, C. S. (1991) · The Engineering Economist · doi:10.1080/00137919108903052 · Cited by 8
Empirical Bayes estimation of gene-specific effects in micro-array research
Edwards, J. W., Page, G. P., Gadbury, G. L., Heo, M., Kayo, T., Weindruch, R. H., & Allison, D. B. (2004) · Functional & Integrative Genomics · doi:10.1007/s10142-004-0123-0 · Cited by 8
“An ensemble of estimators is obtained by application of Bayes law to obtain estimators of the θi conditional on the data (Morris 1983).”
BackgroundMethodInfluentialShrinkage estimation in the two-way multivariate normal model
Sun, L. (1996) · The Annals of Statistics · doi:10.1214/aos/1032894468 · Cited by 8
Lui, K., Martinez, B., & Mercy, J. A. (1990) · Suicide and Life-Threatening Behavior · doi:10.1111/j.1943-278x.1990.tb00108.x · Cited by 8
Metadata
- Authors
- Morris, C. N.
- Affiliations
- The University of Texas at Austin (US)
- Corresponding author
- Morris, Carl N.
- Venue
- Journal of the American Statistical Association
- Publisher
- Informa UK Limited
- Published
- 1983-03
- Volume
- 78
- Issue
- 381
- Pages
- 47-55
- ISSN
- 0162-1459 (ISSN-L)
- 1537-274X
- OpenAlex
- W1983607152
- Cited by
- 1,448
- References
- 36
- Citations by year
- 2026: 32 · 2025: 30 · 2024: 52 · 2023: 42 · 2022: 40 · 2021: 51 · 2020: 62 · 2019: 44 · 2018: 49 · 2017: 43 · 2016: 54 · 2015: 63 · 2014: 66 · 2013: 49 · 2012: 48
- FWCI
- 24.44
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Journal of the American Statistical Association (publishedVersion)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Distribution Estimation and Applications
- Keywords
- parametric empirical Bayes, confidence intervals, shrinkage estimation
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