The problem of ranking Optimality Theoretic constraints in a fashion that is consistent with a given set of (input, output) pairs has been solved with a variety of algorithms (cf. Tesar and Smolensky 2002, Boersma and Hayes 1999). The real-world problem of learning from outputs alone, however, still presents a host of challenges. Chief among these is the fact that there are often several (possible-input, possible-grammar) pairs that are consistent with a given set of surface forms. This paper explores strategies for learning from surface forms that use information-theoretic measures of the randomness (entropy) of the input set associated with each grammar hypothesis as a heuristic to select the grammars that maximally encode any observed patterns. This represents a straightforward use of the Richness of the Base Hypotheses (Smolensky 1996) to avoid encoding observed patterns as accidental patterns in the lexicon.