Empirical Bayes - Index

Leaf index for the Empirical Bayes folder, covering Efron’s Large-Scale Inference Chapter 1 — “Empirical Bayes and the James-Stein Estimator.”

Routing Summary

• Need the big picture / how the two EB branches fit together? → Empirical Bayes - Overview
• Need the nonparametric posterior-mean formula $(x+1)f(x+1)/f(x)$? → Robbins Formula and Poisson Empirical Bayes
• Need the shrinkage estimator and its grand-mean form? → James-Stein Estimator
• Need why the MLE is inadmissible for $N\ge 3$ (the proof + risk)? → Stein’s Paradox and Risk Dominance
• Need “shrinkage = estimated prior” and the link to hierarchical Bayes / regularization? → Empirical Bayes Interpretation of Shrinkage
• Need the baseball worked example with real numbers? → James-Stein Estimator (Table 1.1) and Stein’s Paradox and Risk Dominance (Table 1.2)

Concept Map

Concept	Note	Type	Depends On	Key Result
EB program & two branches	Empirical Bayes - Overview	overview	(the 4 below)	Prior estimable from the marginal of $N$ parallel cases
Nonparametric (Poisson) EB	Robbins Formula and Poisson Empirical Bayes	theorem	Overview	$E[\theta \mid x]=(x+1)f(x+1)/f(x)$
James-Stein estimator	James-Stein Estimator	theorem	Overview, Robbins	${\hat{\mu }}^{(JS)}=(1-\frac{N-2}{S})\mathbf{z}$; grand-mean form (1.35)
Stein’s paradox / risk dominance	Stein’s Paradox and Risk Dominance	theorem	James-Stein, Overview	JS dominates MLE for $N\ge 3$; risk $N-E\{(N-2{)}^2/S\}$
Shrinkage as estimated prior	Empirical Bayes Interpretation of Shrinkage	concept	James-Stein, Robbins, Overview	Parametric EB; link to hierarchical Bayes & regularization

Notes

• Empirical Bayes - Overview — CONTAINS: Bayes rule & marginal density (1.2-1.3); the EB principle; the two historical branches (Robbins, Stein); baseball and kidney examples at a glance.
• Robbins Formula and Poisson Empirical Bayes — CONTAINS: Poisson marginal $f(x)$; Robbins’ formula $(x+1)f(x+1)/f(x)$ with proof idea; empirical-frequency plug-in estimator; insurance-claims example.
• James-Stein Estimator — CONTAINS: model (1.7); Bayes rule $B=A/(A+1)$ (1.16); marginal estimate $E\{(N-2)/S\}=1/(A+1)$ (1.22); JS estimator (1.23); grand-mean form (1.35); risk ratio (1.25); limited-translation (1.37); baseball Table 1.1; regression form (1.39).
• Stein’s Paradox and Risk Dominance — CONTAINS: dominance theorem (1.26); Stein’s unbiased risk identity (1.27-1.30); exact JS risk (1.31); total-vs-individual caveat; simulation Table 1.2; Clemente/Munson “paradox.”
• Empirical Bayes Interpretation of Shrinkage — CONTAINS: parametric EB (1.32-1.35); regression shrinkage (1.38-1.39); links to hierarchical Bayes, regularization/ridge, bias-variance; Fig. 1.1 “learning from others”; limited-translation as prior-protection.

Sources

• Efron - Empirical Bayes and the James-Stein Estimator (LSI Ch1).pdf — Efron, Large-Scale Inference, Ch. 1, pp. 1-12.