Systems, methods, and apparatus, including computer program products, for collaborative filtering are provided. In one implementation, a computer-implemented method is provided. The method includes receiving a shard of data representing a subset of a set of entities and a subset of a set of items, generating an iteration of a maximum likelihood estimate of a probability distribution model of a relationship between the set of entities and the set of items, the probability distribution model comprising a probability distribution of the set of items with respect to latent variables and a probability distribution of the latent variables with respect to the set of users, and generating statistics from results from the generating step which are passed to different shards for use in a next iteration of the maximum likelihood estimate.