Both adults and young children possess powerful statistical computation capabilities--they can infer the referent of a word from highly ambiguous contexts involving many words and many referents by aggregating cross-situational statistical information across contexts. This ability has been explained by models of hypothesis testing and by models of associative learning. This article describes a series of simulation studies and analyses designed to understand the different learning mechanisms posited by the 2 classes of models and their relation to each other. Variants of a hypothesis-testing model and a simple or dumb associative mechanism were examined under different specifications of information selection, computation, and decision. Critically, these 3 components of the models interact in complex ways. The models illustrate a fundamental tradeoff between amount of data input and powerful computations: With the selection of more information, dumb associative models can mimic the powerful learning that is accomplished by hypothesis-testing models with fewer data. However, because of the interactions among the component parts of the models, the associative model can mimic various hypothesis-testing models, producing the same learning patterns but through different internal components. The simulations argue for the importance of a compositional approach to human statistical learning: the experimental decomposition of the processes that contribute to statistical learning in human learners and models with the internal components that can be evaluated independently and together. (Contains 1 footnote, 1 table, and 8 figures.)