A finiteness theorem for Markov bases of hierarchical models

Betti numbers of Stanley-Reisner rings determine hierarchical Markov degrees

There are two seemingly unrelated ideals associated with a simplicial complex Δ. One is the Stanley-Reisner ideal I_Δ, the monomial ideal generated by minimal non-faces of Δ, well-known in combinatorial commutative algebra. The other is the toric ideal I_{M(Δ)} of the facet subring of Δ, whose generators give a Markov basis for the hierarchical model defined by Δ, playing a prominent role in algebraic statistics. In this note we show that the complexity of the generators of I_{M(Δ)} is determined by the Betti numbers of I_Δ. The unexpected connection between the syzygies of the Stanley-Reisner…

等变格基

Equivariant lattice bases

本文研究了无限对称群作用下无限秩自由阿贝尔群中格的等变基有限性，提出代数统计中证明有限性的新框架，并证明了特定不变格具有有限等变 Graver basis，推广了相关结论。

马尔可夫基：25年更新

Markov bases: a 25 year update

本文回顾了马尔可夫基方法在条件分布采样中的挑战与最佳实践，并在其基本定理发表25年后提供新见解，同时证明了关于层次模型复杂性、对数线性模型纤维松弛及部分移动集局限性的三个新结果。