Well, p definitely is the solution in the case I mentioned. It is indeed the maximum likelihood solution. You could incorporate prior info about theta via a regularization term, if so inclined. What does not make sense in this?

Not sure what the OP meant, but I though it might be useful to mention how estimators may be interpreted as anything probabilistic at all. Often, arbitrary numbers between 0 and 1 are termed "probabilities", but in this case there actually is some proportion or probability to which f(x) should ideally correspond.