The ML-degree conjecture for no-three-way interaction models

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For i∈{0,…,m−1}i\in\{0,\ldots,m-1\} and j,k∈{0,1}j,k\in\{0,1\}, let CmC_m be the no-three-way interaction model with one mm-ary variable and two binary variables, defined by

pijk=prob⁡(X=i,Y=j,Z=k)=aijbikcjk.p_{ijk}=\operatorname{prob}(X=i,Y=j,Z=k)=a_{ij}b_{ik}c_{jk}.

Let AmA_m be its associated configuration and Q=conv⁡(Am)Q=\operatorname{conv}(A_m). ML-degree conjecture. The ML degree of CmC_m is 2m−12^m-1. The source states this based on computations for m≤5m\leq5; the general formula remains unresolved there.

References

Primary source

Carlos Améndola, Nathan Bliss, Isaac Burke, Courtney R. Gibbons, Martin Helmer, Serkan Hoşten, Evan D. Nash, Jose Israel Rodriguez and Daniel Smolkin, “The Maximum Likelihood Degree of Toric Varieties”, arXiv:1703.02251 (2017).

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