Original title: On the maximum likelihood degree of Gaussian graphical models
Carlos Améndola, Rodica Dinu, Mateusz Michałek, Martin Vodička · September 2026
Scientists use networks of points and lines, called Gaussian graphical models, to map relationships between variables. To fit these models to real data, researchers solve equations to find the best settings. The number of possible answers to these equations is called the maximum likelihood degree, or ML-degree. When this degree equals 1, the best answer can be written as a simple fraction, which makes it much easier to calculate. The researchers studied the math behind these networks. They proved that smaller chunks inside a larger network never increase this degree. They also proved that loops always have a degree greater than 1. This allowed them to confirm that only chordal graphs, which lack simple unbridged loops, yield these simple fractional answers.
Imagine you are trying to find the shortest path through a network of roads. If the network has no loops, there is a simple rule to find the path. But if there are loops, you might have multiple equally short paths and need to compare them. Here, the network represents statistical dependencies between variables, and the 'shortest path' is the best estimate from data. The researchers prove that only networks without loops have a simple unique estimate; loops always create multiple possibilities.
Statisticians using Gaussian graphical models now know exactly which graphs allow a closed-form maximum likelihood estimator: only chordal graphs. For cycles and other non-chordal graphs, the estimator is algebraic and requires numerical methods. This completes a long-standing characterization and guides model choice in practice.
The abstract doesn't say. It focuses on theoretical proofs and does not discuss computational experiments or applications. The paper does not give an exact formula for the ML-degree of cycles, only a lower bound, and does not address other non-chordal graphs beyond cycles.
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On the maximum likelihood degree of Gaussian graphical models — Carlos Améndola, Rodica Dinu, Mateusz Michałek et al. — Bulletin of the London Mathematical Society — doi:10.1112/blms.70507
Abstract licence: CC BY
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