# Empirical Bayes Estimation

Suppose $X_i$ conditioned on $\mu$ is iid $N(\mu, \sigma^2)$ and $\mu$ is distributed as $N(\mu_0, \tau^2)$. Is there a way to estimate $\tau^2$?

Using a method of moments approximation and the unbiased sample variance approximation for the population variance:

$\operatorname{Var}[X] = \sigma^2 + \tau^2$

Then just solve for $\tau^2$

For proof:

For any compound distribution:

$\operatorname{Var}(X) = \operatorname{E}_G\bigl[\operatorname{Var}_F(X|\theta)\bigr] + \operatorname{Var}_G\bigl(\operatorname{E}_F[X|\theta]\bigr)$

Therefore

\begin{align} \operatorname{Var}(X) &= \operatorname{E}[\sigma^2] + \operatorname{Var}[\mu]\\ &= \sigma^2 + \tau^2 \end{align}

• what if you don't know $\sigma^2$? – user795305 Sep 29 '17 at 2:05