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Miruna-Stefana Sorea: Exact Solutions in Log-Concave Maximum Likelihood Estimation

Time: Tue 2023-04-11 10.15

Location: 3721, Lindstedtsvägen 25, and Zoom

Video link: Meeting ID: 621 8808 6001

Participating: Miruna-Stefana Sorea (SISSA Trieste)

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We study probability density functions that are log-concave. Despite the space of all such densities being infinite-dimensional, the maximum likelihood estimate is the exponential of a piecewise linear function determined by finitely many quantities, namely the function values, or heights, at the data points. We explore in what sense exact solutions to this problem are possible. First, we show that the heights given by the maximum likelihood estimate are generically transcendental. For a cell in one dimension, the maximum likelihood estimator is expressed in closed form using the generalized W-Lambert function. Even more, we show that finding the log-concave maximum likelihood estimate is equivalent to solving a collection of polynomial-exponential systems of a special form. Even in the case of two equations, very little is known about solutions to these systems. As an alternative, we use Smale's α-theory to refine approximate numerical solutions and to certify solutions to log-concave density estimation. This is joint work with Alexandros Grosdos, Alexander Heaton, Kaie Kubjas, Olga Kuznetsova, Georgy Scholten.