# Markov bases: a 25 year update

arXiv: https://arxiv.org/abs/2306.06270 · HTML: https://mlsanity.com/paper/2306.06270/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Jun 2023.

## Built on by (most cited first, 2 of 2)

- [Learning to sample fibers for goodness-of-fit testing](https://mlsanity.com/paper/2405.13950/) (2024)
- [Learning Fast Monomial Orders for Gr"obner Basis Computations](https://mlsanity.com/paper/2602.02972/) (2026)

## Abstract (arXiv)

In this paper, we evaluate the challenges and best practices associated with the Markov bases approach to sampling from conditional distributions. We provide insights and clarifications after 25 years of the publication of the fundamental theorem for Markov bases by Diaconis and Sturmfels. In addition to a literature review we prove three new results on the complexity of Markov bases in hierarchical models, relaxations of the fibers in log-linear models, and limitations of partial sets of moves in providing an irreducible Markov chain.