Mixtures of Discrete Decomposable Graphical Models

In a discrete graphical model, an underlying graph encodes conditional independences among a set of discrete random variables that label the vertices of the graph. Mixtures of these models allow us to incorporate the assumption that the population is split into subpopulations, each of which may follow a different distribution in the graphical model. From the perspective of algebraic statistics, a discrete graphical model is the real positive piece of a toric variety and the mixture model is part of one of its secant variety. In this talk, we use discrete geometry to investigate the dimensions of the second mixtures of decomposable discrete graphical models. We show that when the underlying graph is not a "clique star", the mixture model has the maximal dimension. This in turn allows us to prove results on the local identifiability of the parameters.