Grِbner Bases and Polyhedral Geometry of Reducible and Cyclic Models

This article studies the polyhedral structure and combinatorics of polytopes that arise from hierarchical models in statistics, and shows how to construct Gröbner bases of toric ideals associated to a subset of such models. We study the polytopes for cyclic models, and we give a complete polyhedral description of these polytopes in the binary cyclic case. Further, we show how to build Gröbner bases of a reducible model from the Gröbner bases of its pieces. This result also gives a different proof that decomposable models have quadratic Gröbner bases. Finally, we present the solution of a problem posed by Vlach (Discrete Appl. Math.13 (1986) 61–78) concerning the dimension of fibers coming from models corresponding to the boundary of a simplex.