Vol. 12, No. 2, 2021

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Toric invariant theory for maximum likelihood estimation in log-linear models

Carlos Améndola, Kathlén Kohn, Philipp Reichenbach and Anna Seigal

Vol. 12 (2021), No. 2, 187–211
DOI: 10.2140/astat.2021.12.187
Abstract

We establish connections between invariant theory and maximum likelihood estimation for discrete statistical models. We show that norm minimization over a torus orbit is equivalent to maximum likelihood estimation in log-linear models. We use notions of stability under a torus action to characterize the existence of the maximum likelihood estimate, and discuss connections to scaling algorithms.

Keywords
maximum likelihood estimation, log-linear models, graphical models, torus actions, null cone, scaling algorithms
Mathematical Subject Classification
Primary: 14L24, 20G45, 62F10, 62H22, 62R01
Milestones
Received: 29 January 2021
Revised: 28 July 2021
Accepted: 27 September 2021
Published: 13 December 2021
Authors
Carlos Améndola
Department of Mathematics
Technical University of Munich
Germany
Kathlén Kohn
Department of Mathematics
KTH Royal Institute of Technology
Stockholm
Sweden
Philipp Reichenbach
Institut für Mathematik
Technische Universität Berlin
Germany
Anna Seigal
Mathematical Institute
University of Oxford
United Kingdom
Department of Mathematics
Harvard University
Cambridge, MA
United States