Lifting Markov Bases and Higher Codimension Toric Fiber Products

TL;DR

This paper introduces a lifting algorithm for Markov and Gröbner bases along lattice maps, with applications to toric fiber products, enabling computation of bases for complex models and establishing finiteness results.

Contribution

It presents a novel lifting algorithm for Markov bases in the context of toric fiber products, advancing computational methods in algebraic statistics.

Findings

Lifting algorithm works under normal semigroup conditions.

Computed Markov bases for multiple hierarchical models.

Established new finiteness results for iterated toric fiber products.

Abstract

We study how to lift Markov bases and Gr\"obner bases along linear maps of lattices. We give a lifting algorithm that allows to compute such bases iteratively provided a certain associated semigroup is normal. Our main application is the toric fiber product of toric ideals, where lifting gives Markov bases of the factor ideals that satisfy the compatible projection property. We illustrate the technique by computing Markov bases of various infinite families of hierarchical models. The methodology also implies new finiteness results for iterated toric fiber products.