  We study the problem of learning from unlabeled samples very general
statistical mixture models on large finite sets. Specifically, the model to be
learned, $\vartheta$, is a probability distribution over probability
distributions $p$, where each such $p$ is a probability distribution over $[n]
= \{1,2,\dots,n\}$. When we sample from $\vartheta$, we do not observe $p$
directly, but only indirectly and in very noisy fashion, by sampling from $[n]$
repeatedly, independently $K$ times from the distribution $p$. The problem is
to infer $\vartheta$ to high accuracy in transportation (earthmover) distance.
  We give the first efficient algorithms for learning this mixture model
without making any restricting assumptions on the structure of the distribution
$\vartheta$. We bound the quality of the solution as a function of the size of
the samples $K$ and the number of samples used. Our model and results have
applications to a variety of unsupervised learning scenarios, including
learning topic models and collaborative filtering.
