Danai Deligeorgaki: On the marginal independence structure of DAG models

Abstract

We consider the problem of estimating the marginal independence structure of a DAG model from observational data. In order to so, we divide the space of directed acyclic graphs (DAGs) into certain equivalence classes, where each class can be represented by a unique undirected graph called the unconditional dependence graph. The unconditional dependence graphs satisfy certain graphical properties, namely having equal intersection and independence number. Using this observation, we can construct a Grobner basis for an associated toric ideal and define additional binomial relations to connect the space of unconditional dependence graphs. With these moves, we can implement a search algorithm, GrUES (Grobner-based Unconditional Equivalence Search), that estimates the conditional independence structure of the graphical model. The implementation shows that GrUES recovers the true marginal independence structure via a BIC-optimal or MAP estimate at a higher rate than simple independence tests while also yielding an estimate of the posterior. This is joint work with Alex Markham, Pratik Misra and Liam Solus.