Seminar Algebra and Geometry: Jan Draisma (Technische Universiteit Eindhoven)

Maximum-Likelihood duality for determinantal varieties

The maximum-likelihood degree of a smooth, locally closed subvariety X of a torus (C*)ⁿ is the number of critical points on X of any monomial function
    (p₁,...,p_n) --> p₁^u₁... p_n^u_n
with u in Nⁿ sufficiently general. This notion is motivated by statistics, where X is a family of probability distributions on {1,...,n}, u records observed data, and the maximum-likelihood estimate for p given u is one of the critical points of the function above.

In recent work by Hauenstein, Rodriguez, and Sturmfels the maximum-likelihood degree of determinantal varieties was studied. Extensive computations using numerical algebraic geometry led to the conjecture that the maximum-likelihood degree of the variety of rank-r matrices whose entries add up to 1 equals that of the variety of corank-(r-1) matrices whose entries add up to 1. I will present a proof of that conjecture, and variations of it for symmetric and skew-symmetric matrices.  Joint work with Jose Rodriguez.

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