Mixed Membership Distributions with Applications to Modeling Multiple Strategy Usage.

This dissertation examines two related questions. How do mixed membership models work? and Can mixed membership be used to model how students use multiple strategies to solve problems? Mixed membership models have been used in thousands of applications from text and image processing to genetic microarray analysis. Yet these models are crafted on a case-by-case basis because we do not yet understand the larger class of mixed membership models. The work presented here addresses this gap and examines two different aspects of the general class of models. First I establish that categorical data is a special case, and allows for a different interpretation of mixed membership than in the general case. Second, I present a new identifiability result that characterizes equivalence classes of mixed membership models which produce the same distribution of data. These results provide a strong foundation for building a model that captures how students use multiple strategies. How to assess which strategies students use, is an open question. Most psychometric models either do not model strategies at all, or they assume that each student uses a single strategy on all problems, even if they allow different students to use different strategies. The problem is, that’s not what students do. Students switch strategies. Even on the very simplest of arithmetic problems, students use