Mrs. Gerber's Lemma lower bounds the entropy at the output of a binary symmetric channel in terms of the entropy of the input process. In this paper, we lower bound the output entropy via a different measure of input uncertainty, pertaining to the minimum mean squared error (MMSE) prediction cost of the input process. We show that in many cases our bound is tighter than the one obtained from Mrs. Gerber's Lemma. As an application, we evaluate the bound for binary hidden Markov processes, and obtain new estimates for the entropy rate.