Gaussian likelihood geometry of projective varieties

Abstract

We explore the maximum likelihood degree of a homogeneous polynomial $F$ on a projective variety $X$, ${\mathrm{MLD}}_F(X)$, which generalizes the concept of Gaussian maximum likelihood degree. We show that ${\mathrm{MLD}}_F(X)$ is equal to the count of critical points of a rational function on $X$, and give different geometric characterizations of it via topological Euler characteristic, dual varieties, and Chern classes.

arXiv comment25 pages, 1 figure. Final revised version. The main results were extended to the non-smooth case. To appear in SIAM Journal on Applied Algebra and Geometry