# zbMATH — the first resource for mathematics

Likelihood geometry. (English) Zbl 1328.14004
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.)
Full Text: