Minimal and minimal invariant Markov bases of decomposable models for contingency tables

Abstract

We study Markov bases of decomposable graphical models consisting of primitive moves (i.e., square-free moves of degree two) by determining the structure of fibers of sample size two. We show that the number of elements of fibers of sample size two are powers of two and we characterize primitive moves in Markov bases in terms of connected components of induced subgraphs of the independence graph of a hierarchical model. This allows us to derive a complete description of minimal Markov bases and minimal invariant Markov bases for decomposable models.

Published in at http://dx.doi.org/10.3150/09-BEJ207 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (4)

• On the toric algebra of graphical models
• Markov bases for two-way subtable sum problems
• Boundary cliques, clique trees and perfect sequences of maximal cliques of a chordal graph
• Connecting tables with zero-one entries by a subset of a Markov basis

Cited by in corpus (4)

• A Markov Basis for Conditional Test of Common Diagonal Effect in Quasi-Independence Model for Square Contingency Tables
• Betti numbers of Stanley-Reisner rings determine hierarchical Markov degrees
• Markov Bases for Typical Block Effect Models of Two-way Contingency Tables
• Finiteness theorems and algorithms for permutation invariant chains of Laurent lattice ideals