
\subsection{Empirical Bayes}

\subsubsection{Bayes rule}

We can calculate the posterior probability for \(\theta \), but we need a prior \(P(\theta )\).

\(P(\theta | X)=\dfrac{P(X|\theta )P(\theta )}{P(X)}\)

\subsubsection{Empirical Bayes}

With empirical Bayes we get our prior from the data.

We have \(P(X|\theta )\)

And \(P(\theta |\rho )\)

We observe \(X\) and want to estimate \(\theta \).

\(P(\theta |X)=\dfrac{P(X|\theta)P(\theta )}{P(X)}=\dfrac{P(X|\theta)}{P(X)}\int P(\theta | \rho )P(\rho )d\rho \)

