The Maximum Likelihood Degree

Details The Maximum Likelihood Degree The Maximum Likelihood Degree

2004-06-25

arxiv.org/abs/math/0406533v1

Mathematics

Algebraic Geometry

32 pages, latex

Scientific paper

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.

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