Estimation of the Random Effects and Empirical Bayes Estimates EBEs

ESTIMATION OF THE RANDOM EFFECTS AND EMPIRICAL BAYES ESTIMATES (EBEs) 

The basic tool that Bayesians use derives from Bayes theorem, from which it is known that the 

In the case of normal prior and likelihood, the posterior distribution is a normal distribution with a weighted mean of the prior value mean and observed data mean with weights proportional to the inverse of their respective variances. The posterior variance is a weighted average of the individual variances. 

In the linear mixed model it is assumed that the marginal model for the ith subject is given by 

Another assumption is that s, is normally distributed with mean 0 and variance Rj and that Uj is normally distributed with mean 0 and variance G. The latter assumption is the prior distribution of the parameters Uj. Once data are collected, the posterior distribution of Uj can be generated conditional on Y . Estimation of U is given by the average of the values for the ith subject and the population mean. If the subject mean is higher than the population mean then this would suggest that Uj is positive. Given the assumptions for the model, the expected value of Uj given the subject mean is 

---