When Does a Mixture of Products Contain a Product of Mixtures?

We prove results on the relative representational power of mixtures of product distributions and restricted Boltzmann machines (products of mixtures of pairs of product distributions). Tools of independent interest are mode-based polyhedral approximations sensitive enough to compare full-dimensional models, and characterizations of possible modes and support sets of multivariate probability distributions that can be represented by both model classes. We find, in particular, that an exponentially larger mixture model, requiring an exponentially larger number of parameters, is required to represent probability distributions that can be represented by the restricted Boltzmann machines. The title question is intimately related to questions in coding theory and point configurations in hyperplane arrangements.