Mixtures and products in two graphical models
This work clarifies theoretical relationships between mixture and product models in graphical models, but it is incremental as it focuses on a small-scale case.
The paper compares two statistical models for three binary random variables, showing they represent the same set of distributions up to closure, and provides a semi-algebraic description with six binomial inequalities and closed-form maximum likelihood estimates.
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.