Likelihood geometry. (English) Zbl 1328.14004

Huh, June; Sturmfels, Bernd

Di Rocco, Sandra (ed.) et al., Combinatorial algebraic geometry. Lecture notes of the CIME-CIRM summer school, Levico Terme, Italy, June 10–15, 2013. Cham: Springer; Florence: Fondazione CIME (ISBN 978-3-319-04869-7/pbk; 978-3-319-04870-3/ebook). Lecture Notes in Mathematics 2108. CIME Foundation Subseries, 63-117 (2014).

Summary: We study the critical points of monomial functions over an algebraic subset of the probability simplex. The number of critical points on the Zariski closure is a topological invariant of that embedded projective variety, known as its maximum likelihood degree. We present an introduction to this theory and its statistical motivations. Many favorite objects from combinatorial algebraic geometry are featured: toric varieties, \(A\)-discriminants, hyperplane arrangements, Grassmannians, and determinantal varieties. Several new results are included, especially on the likelihood correspondence and its bidegree. This article represents the lectures given by the second author at the CIME-CIRM course on Combinatorial Algebraic Geometry at Levico Terme in June 2013.
For the entire collection see [Zbl 1290.14001].

MSC:

14-02	Research exposition (monographs, survey articles) pertaining to algebraic geometry
14N05	Projective techniques in algebraic geometry
14M12	Determinantal varieties
14M15	Grassmannians, Schubert varieties, flag manifolds
14M25	Toric varieties, Newton polyhedra, Okounkov bodies
13P25	Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.)