Challenging computations of Hilbert bases of cones associated with algebraic statistics

TL;DR

This paper presents new computational proofs confirming the normality of certain monoids related to contingency tables and computes Hilbert bases for complex cones in algebraic statistics, advancing classification efforts.

Contribution

It provides independent computational proofs of normality for specific monoids and computes the full Hilbert basis for a non-normal monoid in algebraic statistics, extending existing computational tools.

Findings

Confirmed normality of monoid from 5x5x3 tables

Identified unique minimal vector disproving normality for 6x4x3 tables

Computed the full Hilbert basis for the semi-graphoid cone at |N|=5

Abstract

In this paper we present two independent computational proofs that the monoid derived from $5\times 5\times 3$ contingency tables is normal, completing the classification by Hibi and Ohsugi. We show that Vlach's vector disproving normality for the monoid derived from $6\times 4\times 3$ contingency tables is the unique minimal such vector up to symmetry. Finally, we compute the full Hilbert basis of the cone associated with the non-normal monoid of the semi-graphoid for $|N|=5$. The computations are based on extensions of the packages LattE-4ti2 and Normaliz.