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The maximum likelihood degree. (English) Zbl 1123.13019
Summary: Maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials. We study the algebraic degree of the critical equations of this optimization problem. This degree is related to the number of bounded regions in the corresponding arrangement of hypersurfaces, and to the Euler characteristic of the complexified complement. Under suitable hypotheses, the maximum likelihood degree equals the top Chern class of a sheaf of logarithmic differential forms. Exact formulae in terms of degrees and Newton polytopes are given for polynomials with generic coefficients.

MSC:
13P99 Computational aspects and applications of commutative rings
62F10 Point estimation
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
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