# UI STAT 4520 - Introduction to Empirical Bayes (5 pages)

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## Introduction to Empirical Bayes

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- University of Iowa
- Course:
- Stat 4520 - Bayesian Statistics

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1 22S 138 Introduction to Empirical Bayes 2 Empirical Bayes Bayesian analysis requires specifying fixed values for parameters of highest stage priors Lecture 22 Nov 16 2009 these values must come from source other than the current dataset goal of empirical Bayes analysis is to fit hierarchical models without introducing information external to the current dataset Kate Cowles 374 SH 335 0727 kcowles stat uiowa edu EB approach estimates final stage parameters using current data then proceeds as though prior were known requires adjustment to posterior standard deviations and credible sets 3 Compound sampling framework observed data conditionally independent given parameters Yi i fi yi i i 1 n family of prior distributions indexed by lowdimensional parameter i g i 4 Parametric EB PEB point estimation if were known fully Bayes f y g i p i yi i i i mi yi i e posterior for i depends on data only through yi mi is marginal likelihood of yi PEB used marginal distribution of all the data to estimate m y use maximum likelihood or method of moments to get estimate plug into above expression to get estimated posterior p i yi use estimated posterior for all inference e g point estimate of posterior mean this point estimate depends on all the data through y 5 6 estimated posterior distribution of i Example Normal Normal models p i yi N B 1 B yi 1 B 2 two stage model where Yi i N i 2 i 1 n i N 2 i 1 n 2 B 2 2 exactly the same as fully Bayesian posterior for this case except that known prior mean is replaced by sample mean computed from all the data PEB point estimate of i first assume both 2 and 2 known calculations we have seen in GCSR show that marginally Yis are i i d i e with is integrated out Yi N 2 2 so the marginal likelihood of all the Yis is 1 m y 2 2 2 n 2 1 n X 2 yi exp 2 2 2 i 1 EB analysis requires estimation of marginal MLE of i B y 1 B yi y 1 B yi y inference for single component borrows information from data on all components shrinkage estimator y 7 now suppose 2 as

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