Quick Bayesian Probability Question

Hey all,

I need to check my understanding of this. Let have some prior on it and observe , which are independent conditional on , such that . I'm asked to find the posterior mean of given . I'll spare the extra details.

My reasoning is

(from conditional independnce)

(since EX|theta = theta)

So the posterior mean of is the posterior mean of . 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 and 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?