May have gotten confused because log is monotonically increasing e.g. log likelihood maximized at the same spot as likelihood. So log E(X) is maximized at the same spot as E(X). But log and E do not commute (Jensen’s inequality is not called Jensen’s equality, after all).
Yep, I was wrong. Now I need to figure out why I thought I was right..
May have gotten confused because log is monotonically increasing e.g. log likelihood maximized at the same spot as likelihood. So log E(X) is maximized at the same spot as E(X). But log and E do not commute (Jensen’s inequality is not called Jensen’s equality, after all).
Was probably part of it—I think the internal cheering for the wrong position included the words “But log likelihood!” :-/