# How close is $\operatorname{argmax}_p E[\log(f(p,\alpha)]$ to $\operatorname{argmax}_p \log(E[f(p,\alpha)])$?

Here $\alpha$ is a random variable and the expectation is taken with respect to that variable. I am wondering if it's the same in any case or there's a theorem quantifying how close both things are. I think I've come to proove that if $\dfrac{\partial f(p,\alpha)}{\partial p}$ never change sign in $\alpha$ the results are the same, according to the mean value theorem, but I'm looking for more general results.

Thank you

• What do you mean by $\partial_p f(p, \alpha)$ never change sign in $\alpha$? Also, do you notice that $E(\log(f(p,\alpha)))$ might not exist? – user251257 Jul 21 '15 at 20:57