Tropical Geometry of Statistical Models

This article presents a unified mathematical framework for inference in graphical models, building on the observation that graphical models are algebraic varieties. From this geometric viewpoint, observations generated from a model are coordinates of a point in the variety, and the sum-product algor...

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Bibliographic Details
Published in:Proceedings of the National Academy of Sciences - PNAS 2004-11, Vol.101 (46), p.16132-16137
Main Authors: Pachter, Lior, Sturmfels, Bernd, Fienberg, Stephen E.
Format: Article
Language:English
Subjects:
Algebra
Algorithms
Coordinate systems
Geometry
Inference
Markov analysis
Markov Chains
Markov models
Mathematics
Models, Statistical
Parametric models
Physical Sciences
Polynomials
Polytopes
Random variables
Statistical analysis
Statistical models
Vertices
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Summary:This article presents a unified mathematical framework for inference in graphical models, building on the observation that graphical models are algebraic varieties. From this geometric viewpoint, observations generated from a model are coordinates of a point in the variety, and the sum-product algorithm is an efficient tool for evaluating specific coordinates. Here, we address the question of how the solutions to various inference problems depend on the model parameters. The proposed answer is expressed in terms of tropical algebraic geometry. The Newton polytope of a statistical model plays a key role. Our results are applied to the hidden Markov model and the general Markov model on a binary tree.
ISSN:0027-8424
1091-6490
DOI:10.1073/pnas.0406010101