Much previous work has investigated weak supervision with HMMs and tag dictionaries for part-of-speech tagging, but there have been no similar investigations for the harder problem of supertagging. Here, I show that weak supervision for supertagging does work, but that it is subject to severe performance degradation when the tag dictionary is highly ambiguous. I show that lexical category complexity and information about how supertags may combine syntactically can be used to initialize the transition distributions of a first-order Hidden Markov Model for weakly supervised learning. This initialization proves more effective than starting with uniform transitions, especially when the tag dictionary is highly ambiguous.