The Likelihood Correspondence
Thomas Kahle, Hal Schenck, Bernd Sturmfels, Maximilian Wiesmann
TL;DR
This work addresses the likelihood geometry of SNC hypersurface arrangements by providing a concrete determinantal description of the likelihood ideal: under SNC, $I_{\mathcal{A}}$ is generated by the linear form $\sum_{i=1}^m d_i s_i$ together with the maximal minors of $Q_{\backslash 1}^s$, and the ideal is Cohen-Macaulay of codimension $n$. It establishes a precise matrix-based representation via $Q$ and its specialization, proves the equality $I_{\mathcal{A}} = I_{m+1}(Q_{\backslash 1}^s) + \langle \sum_i d_i s_i \rangle$, and computes the multidegree using Giambelli-Thom-Porteous. The paper then links likelihood geometry to topological and algebraic discriminants by showing the Euler discriminant $\Delta = \prod_{|I|\le n} \Delta_I$ detects SNC (nonvanishing iff SNC) and equals the topological Euler discriminant $\nabla_\chi$, with the ML degree matching the Euler characteristic of the arrangement complement. Finally, it connects the likelihood framework to the module of derivations $\mathrm{Der}(\mathcal{A})$, describing how circuit syzygies generate $\mathrm{Der}(\mathcal{A})$ and discussing tame and gentle properties that yield computational advantages and structural insight for SNC arrangements.
Abstract
An arrangement of hypersurfaces in projective space is strict normal crossing (SNC) if and only if its Euler discriminant is nonzero. We study the critical loci of arbitrary Laurent monomials in the equations of the smooth hypersurfaces. The family of these loci forms an irreducible variety in the product of two projective spaces, known in algebraic statistics as the likelihood correspondence and in particle physics as the scattering correspondence. We establish an explicit determinantal representation for the minimal generators of the bihomogeneous prime ideal that defines this variety.