In empirical bayes methods, we can estimate hyper-parameter from using data.
In hierachical model, random vector $X = (X_1,X_2)^T \ where \ X_i=(X_{11},\cdots,X_{1k})^T, i\in{1,2} \\ X|p_{i} \sim binomial(n,p_i) \\ p_i|\lambda \sim beta(\alpha,\beta)$
Then Marginal distribution $ p_i|\alpha,\beta \sim^{iid} betabinomial(\alpha,\beta) $
I want to calculate the method of moments for hyper parameter $\alpha, \beta$
Since IID, I can have first method of moment by calculating $EX = \bar X $
here, I'm confusing.
In empirical bayes, estimated hyperparameter can incorporate the information about $p_1,p_2$
But, $EX_1 = \bar X_1, EX_2 = \bar X_2$
Although $EX_1 = EX_2 $, I think, $ \bar X_1 \ne \bar X_2 $ for observed values.
I don't know how informations are incoporated.