Classifying Gaussian models with rational maximum-likelihood estimators
For an integer $m \ge 2$ let $V = \operatorname{Sym}_m(\mathbb{C})$ be the space of complex symmetric $m \times m$ matrices $S$, with coordinates $s_{ij}$, $i \le j$. For a rational function $f$ on $V$ let $\nabla_S f$ be its gradient with respect to the trace pairing: the symmetric matrix with $$(\nabla_S f)_{ij} = \frac{1}{2 - \delta_{ij}} \frac{\partial f}{\partial s_{ij}} \qquad (i \le j).$$ Consider the nonzero rational functions $\Phi$ on $V$ with $$\Phi(tS) = t^{-m} \Phi(S) \quad (t \in \mathbb{C} \setminus \{0\}), \qquad \Phi(S) = \det\Bigl(-\frac{\nabla_S \Phi(S)}{\Phi(S)}\Bigr),$$ both as identities of rational functions; two solutions are the same only if they are equal as rational functions, not merely up to a scalar factor. Améndola, Gustafsson, Kohn, Marigliano and Seigal proved that these solutions are in bijection with the projective varieties $X \subseteq \mathbb{P}(\operatorname{Sym}_m(\mathbb{C}))$ of Gaussian maximum likelihood degree one, those on which the log-likelihood $\log\det K - \operatorname{tr}(KS)$ has a single complex critical point for general $S$: $\Phi$ goes to the projectivized Zariski closure of the image of $-\nabla_S \Phi/\Phi$, which is the maximum likelihood estimator of the model. The full model has $\Phi(S) = (\det S)^{-1}$; they give the solutions for chordal undirected and for directed acyclic Gaussian graphical models, and show that a linear concentration model has maximum likelihood degree one exactly when the determinant restricted to it is a homaloidal polynomial. Cox, Misra and Semnani proved that the spanning tree generating polynomial of a chordal graph is homaloidal and constructed further families of homaloidal polynomials.
Problem. Give an explicit structural parametrization of all such $\Phi$: a construction of solutions from specified algebraic or combinatorial data, together with a proof that every solution arises this way. A restatement of the differential equation, a test of a given candidate, or a classification under a fixed bound on the degrees of numerator and denominator does not suffice.
Source: Améndola, Gustafsson, Kohn, Marigliano and Seigal (2024), introduction, for the homaloidal PDE of any homogeneous polynomial. Stated by AIM — Open Applied Problems for the determinant on symmetric matrices.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- ST — Statistics Theory
- Secondary
- AG
- Keywords
- AIM Open Applied Problems, Gaussian models, homaloidal polynomials, maximum likelihood estimators
References
@article{amendola2024,
author = {Am{\'e}ndola, Carlos and Gustafsson, Lukas and Kohn, Kathl{\'e}n and Marigliano, Orlando and Seigal, Anna},
title = {Differential equations for {G}aussian statistical models with rational maximum likelihood estimator},
journal = {SIAM J. Appl. Algebra Geom.},
volume = {8},
number = {3},
year = {2024},
pages = {465--492},
doi = {10.1137/23M1569228},
eprint = {2304.12054},
archivePrefix = {arXiv},
}
@article{cox2024,
author = {Cox, Shelby and Misra, Pratik and Semnani, Pardis},
title = {Homaloidal polynomials and {G}aussian models of maximum likelihood degree 1},
journal = {Algebr. Stat.},
volume = {15},
number = {2},
year = {2024},
pages = {167--198},
doi = {10.2140/astat.2024.15.167},
eprint = {2402.06090},
archivePrefix = {arXiv},
}
@misc{aim-rational-gaussian-mle-classification,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Classifying Gaussian models with rational maximum-likelihood estimators}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/530-rational-gaussian-mle-classification.md},
note = {Entry 530 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00302, "Classifying Gaussian models with rational maximum-likelihood estimators", clopen.solutions.