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Abstract
We study probability density functions that are logconcave. Despite the space of all such densities being infinitedimensional, 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 WLambert function. Even more, we show that finding the logconcave maximum likelihood estimate is equivalent to solving a collection of polynomialexponential 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 αtheory to refine approximate numerical solutions and to certify solutions to logconcave density estimation.
Original language  English 

Article number  102448 
Pages (fromto)  132 
Number of pages  32 
Journal  Advances in Applied Mathematics 
Volume  143 
DOIs  
Publication status  Published  Feb 2023 
MoE publication type  A1 Journal articlerefereed 
Keywords
 Certified solutions
 Lambert functions
 Logconcavity
 Maximum likelihood estimation
 Polyhedral subdivisions
 Polynomialexponential systems
 Smale's αtheory
 Transcendence theory
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Dive into the research topics of 'Exact solutions in logconcave maximum likelihood estimation'. Together they form a unique fingerprint.Projects
 1 Finished

: Algebraic geometry of hidden variable models in statistics
Kubjas, K. (Principal investigator), Ardiyansyah, M. (Project Member), Boege, T. (Project Member), Kuznetsova, O. (Project Member), Metsälampi, L. (Project Member), Lindy, E. (Project Member), Sodomaco, L. (Project Member) & Henriksson, O. (Project Member)
01/09/2019 → 31/08/2023
Project: Academy of Finland: Other research funding