Hey all,

I need to check my understanding of this. Let $\displaystyle \theta$ have some prior on it and observe $\displaystyle X_1, ..., X_n, X_{n + 1}$, which are independent conditional on $\displaystyle \theta$, such that $\displaystyle EX_i|\theta = \theta$. I'm asked to find the posterior mean of $\displaystyle X_{n + 1}$ given $\displaystyle X_1, ..., X_n$. I'll spare the extra details.

My reasoning is

$\displaystyle EX_{n + 1} | X_1, ..., X_n = E\left(EX_{n + 1} | \theta, X_1, ..., X_n \right) | X_1, ..., X_n $

$\displaystyle = E\left(EX_{n + 1} | \theta \right) | X_1, ..., X_n$ (from conditional independnce)

$\displaystyle = E\theta | X_1, ..., X_n$ (since EX|theta = theta)

So the posterior mean of $\displaystyle X_{n + 1}$ is the posterior mean of $\displaystyle \theta$. This seems a little off to me for some reason I can't explain. Does this look okay? It seems like this is working out to an expectation over the joint distribution of $\displaystyle \theta|X_1, ..., X_n$ and $\displaystyle X_{n + 1} | \theta$ but I guess I'm not sure if that's what is meant by asking for the posterior mean.

More broadly, if I'm asked for anything posterior, should I get something that is free of theta?