Being able to accurately predict what a customer will purchase next is of paramount importance to successful online retailing. However, in an online context the scalability of the prediction method is just as important as its accuracy. In this paper we compare several purchase prediction methods that are able to overcome the limitations of the prediction methods currently used in practice. We consider mixtures of Dirichlet-Multinomials (MDM), which have a rich history in brand choice modeling, and introduce a new class of models, the latent motivation models, that extends latent Dirichlet allocation (LDA). The fundamental idea underlying this class of models is that product purchases are driven by latent motivations. Furthermore, the model-based approaches are contrasted against collaborative filtering algorithms. The methods are compared on their heterogeneity assumptions and scalability, the latter in terms of estimation complexity and required memory. We apply the methods to purchase data from an online retailer and show that the model-based approaches consistently outperform the collaborative filters. Moreover, the latent motivation models attain performance similar to that of MDM, while being more scalable in terms of assortment size, rendering them a promising alternative to purchase prediction in large assortments.