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On the maximum likelihood degree for Gaussian graphical models

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arxiv 2410.07007 v1 pith:646W3CPU submitted 2024-10-09 math.ST math.AGstat.TH

classification math.STmath.AGstat.TH
keywords gaussiangraphicallikelihoodml-degreemodelscyclemaximumargument
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abstract

In this paper we revisit the likelihood geometry of Gaussian graphical models. We give a detailed proof that the ML-degree behaves monotonically on induced subgraphs. Furthermore, we complete a missing argument that the ML-degree of the $n$-th cycle is larger than one for any $n\geq 4$, therefore completing the characterization that the only Gaussian graphical models with rational maximum likelihood estimator are the ones corresponding to chordal (decomposable) graphs. Finally, we prove that the formula for the ML-degree of a cycle conjectured by Drton, Sturmfels and Sullivant provides a correct lower bound.

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  1. The Maximum Likelihood Degree of Toric Models is Monotonic

    math.AG 2025-07 accept novelty 8.0 of 10

    The ML degree is monotone with respect to faces of the defining polytope for scaled toric models, settling a conjecture of Coons and Sullivant.

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