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Abstract
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We study the Gaussian statistical models whose log-likelihood function has a unique
complex critical point, i.e., has maximum likelihood degree 1. We exploit
the connection developed by Améndola et al. between the models having
maximum likelihood degree 1 and homaloidal polynomials. We study the
spanning tree generating function of a graph and show this polynomial is
homaloidal when the graph is chordal. When the graph is a cycle on
vertices,
, we prove the
polynomial is not homaloidal, and show that the maximum likelihood degree of the resulting
model is the
-th
Eulerian number. These results support our conjecture that the spanning tree
generating function is a homaloidal polynomial if and only if the graph is chordal. We
also provide an algebraic formulation for the defining equations of these models.
Using existing results, we provide a computational study on constructing new
families of homaloidal polynomials. In the end, we analyze the symmetric
determinantal representation of such polynomials and provide an upper bound on the
size of the matrices involved.
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Keywords
homaloidal polynomial, maximum likelihood degree,
multivariate normal distribution, Gaussian graphical model,
symmetric determinantal representation
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Mathematical Subject Classification
Primary: 14E05, 62H22, 62R01
Secondary: 14M99
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Milestones
Received: 8 February 2024
Revised: 24 September 2024
Accepted: 12 October 2024
Published: 3 December 2024
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© 2024 MSP (Mathematical Sciences
Publishers). |
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