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

2020/12/31 by Carlos Améndola, Kathlén Kohn, Philipp Reichenbach +1 · 9 citations
Computer Science · Mathematics · #Applied mathematics #Bayesian Modeling and Causal Inference #Estimation theory #Geometry #Invariant (physics) #Markov Chains and Monte Carlo Methods #Mathematical optimization #Mathematics #Maximum likelihood #Maximum likelihood sequence estimation #Minification #Norm (philosophy) #Scaling #Statistical Methods and Inference #Statistics #Torus #math.AG #math.ST #msc:14L24 #msc:14P05 #msc:20G45 #msc:62F10 #msc:62H22 #msc:62R01 #stat.TH

paper · pdf · doi:10.2140/astat.2021.12.187

published in Algebraic Statistics 12(2), 187-211 (Mathematical Sciences Publishers) · This is a companion paper to arXiv:2003.13662. v2: referee comments worked in, added appendices A and B

openalex created_date 2020/12/21 · arxiv created 2021/10/04 · openalex publication_date 2021/12/13 · arxiv updated 2021/12/15 · openalex updated_date 2026/08/05

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.

Citations

• Invariant theory and scaling algorithms for maximum likelihood estimation
• Lectures on Invariant Theory
• An Iterative Procedure for Estimation in Contingency Tables
• A Relationship Between Arbitrary Positive Matrices and Doubly Stochastic Matrices
• Generalized Iterative Scaling for Log-Linear Models
• Interior-point methods for unconstrained geometric programming and scaling problems
• Ueber die Theorie der algebraischen Formen
• Convexity properties of the moment mapping