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PreprintCC BY-SA 4.0arXiv

Maximum Likelihood Degrees via One-Edge Chordal Completion

  • Nilava Metya
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Abstract

The maximum likelihood degree of a Gaussian graphical model measures the algebraic complexity of its likelihood equations. This paper studies maximum likelihood degrees of Gaussian graphical models through chordal completion. When a graph becomes chordal after one missing edge is added, its maximum likelihood degree is determined by the maximal cliques and clique-tree separators containing that edge. This recovers Uhler's formula for complete bipartite graphs with one part of size two, and extends it to deleted-spine clique books, whose degree is unchanged when a page is thickened to a clique. For several fill edges, the same method gives a product formula when no clique or separator bag contains two fill edges. We also determine four classes of vertices at which clique blow-up preserves the degree. Two follow from known results, and two are established here.

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Cite

BibTeXRIS
BibTeX
@misc{metya2026maximum,
  title = {{Maximum Likelihood Degrees via One-Edge Chordal Completion}},
  author = {Nilava Metya},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.19669},
  eprint = {2610.19669},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.19669},
  note = {CC BY-SA 4.0},
}
RIS
TY  - UNPB
TI  - Maximum Likelihood Degrees via One-Edge Chordal Completion
AU  - Nilava Metya
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.19669
UR  - https://arxiv.org/abs/2610.19669
AB  - The maximum likelihood degree of a Gaussian graphical model measures the algebraic complexity of its likelihood equations. This paper studies maximum likelihood degrees of Gaussian graphical models through chordal completion. When a graph becomes chordal after one missing edge is added, its maximum likelihood degree is determined by the maximal cliques and clique-tree separators containing that edge. This recovers Uhler's formula for complete bipartite graphs with one part of size two, and extends it to deleted-spine clique books, whose degree is unchanged when a page is thickened to a clique. For several fill edges, the same method gives a product formula when no clique or separator bag contains two fill edges. We also determine four classes of vertices at which clique blow-up preserves the degree. Two follow from known results, and two are established here.
ER  -