Mixtures and products in two graphical models

  • Anna Seigal Department of Mathematics, University of California, Berkeley
  • Guido Montufar Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany and Departments of Mathematics and Statistics, University of California, Los Angeles, USA


We compare two statistical models of three binary random variables. One is a mixture model and the other is a product of mixtures model called a restricted Boltzmann machine. Although the two models we study look different from their parametrizations, we show that they represent the same set of distributions on the interior of the probability simplex, and are equal up to closure. We give a semi-algebraic description of the model in terms of six binomial inequalities and obtain closed form expressions for the maximum likelihood estimates. We briefly discuss extensions to larger models.