Lawrence Lifts, Matroids, and Maximum Likelihood Degrees

TL;DR

The paper provides a combinatorial framework linking the degree and maximum likelihood degree of Lawrence-lifted toric varieties to Tutte polynomial evaluations of their column matroids. Under total unimodularity with even circuits, it shows $\deg(X_{\Lambda(A)}) = \tau_{\mathcal{M}(A)}(1,1)$ and $\mathrm{mldeg}(X_{\Lambda(A)}) = \tau_{\mathcal{M}(A)}(1,0)$, with powerful graph-theoretic instantiations via incidence matrices of graphs (notably bipartite graphs). It yields explicit formulas for important models in algebraic statistics, such as no-three-way interaction models, three-dimensional quasi-independence, and hierarchical models, by reducing ML-degrees to Möbius/Tutte invariants of associated matroids. The work connects geometric properties of toric varieties to combinatorial invariants, enabling closed expressions for ML-degrees across several statistically relevant families and clarifying when the ML-estimator is a rational function (degree one).

Abstract

We express the maximum likelihood (ML) degrees of a family toric varieties in terms of Mobius invariants of matroids. The family of interest are those parametrized by monomial maps given by Lawrence lifts of totally unimodular matrices with even circuits. Specifying these matrices to be vertex-edge incidence matrices of bipartite graphs gives the ML degrees of some hierarchical models and three dimensional quasi-independence models. Included in this list are the no-three-way interaction models with one binary random variable, for which, we give closed formulae.

Figures (5)

• Figure 1: A three cycle, a four cycle, a tree, and the complete bipartite graph $K(2,3)$.
• Figure 2: All spanning trees of the graphs in \ref{['fig:graphs']}. The spanning trees in magenta have external activity zero.
• Figure 3: A directed bipartite graph and its incidence matrix. Up to sign, the matrix has only one circuit $\mathcal{C}_{A_{\vec{G}}}=\{(1,-1,1,1)\}$.
• Figure 4: A directed bipartite graph and its incidence matrix. Up to sign, the matrix has one circuit $\mathcal{C}_{A_{G^{\pm}}}=\pm \{(1,-1,-1,1)\}$.
• Figure 5: Lawrence lifts of simplicial complexes with facet sets $\{\{1\},\{2\}\}$, $\{\{1\},\{2,3\}\}$ and $\{\{1,2\},\{2,3\}\}$, respectively.

Theorems & Definitions (63)

• Theorem 1
• Corollary 2
• Remark 3
• Proposition 4
• Proposition 5
• Lemma 6
• proof
• Theorem 7
• proof
• Remark 8