Is there a connection between empirical Bayes and random effects? I recently happened to read about empirical Bayes (Casella, 1985, An introduction to empirical Bayes data analysis) and it looked a lot like random effects model; in that both have estimates shrunken to global mean. But I have not read it throughly...
Does anyone have any insight about the similarity and differences between them?
 A: There is a really great article in JASA back in the mid 1970s on the James-Stein estimator and empirical Bayes estimation  with a particular application to predicting baseball players batting averages.  The insight I can give on this is the result of James and Stein who showed to the surprise of the statistical world that for a multivariate normal distribution in three dimensions or more the MLE, which is the vector of coordinate averages, is inadmissible.  
The proof was achieved by showing that an estimator that shrinks the mean vector toward the origin is uniformly better based on mean square error as a loss function.  Efron and Morris showed that in a multivariate regression problem using an empirical Bayes approach the estimators they arrive at are shrinkage estimators of the James-Stein type.  They use this methodology to predict the final season batting averages of major league baseball players based on their early season result.  The estimate moves everyone's individual average to the grand average of all the players. 
I think this explains how such estimators can arise in multivariate linear models.  It doesn't completely connect it to any particular mixed effects model but may be a good lead in that direction.
Some references:


*

*B. Efron and C. Morris (1975), Data analysis using Stein's estimator and its generalizations, J. Amer. Stat. Assoc., vol. 70, no. 350, 311–319.

*B. Efron and C. Morris (1973), Stein's estimation rule and its competitors–An empirical Bayes approach, J. Amer. Stat. Assoc., vol. 68, no. 341, 117–130.

*B. Efron and C. Morris (1977), Stein's paradox in statistics, Scientific American, vol. 236, no. 5, 119–127.

*G. Casella (1985), An introduction to empirical Bayes data analysis, Amer. Statistician, vol. 39, no. 2, 83–87.

