Information Geometry and its Applications III

Abstract Noburu Murata

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Noburu Murata  (Waseda University, Japan)
Friday, August 06, 2010, room Hörsaal 2
A geometrical extension of the Bradley-Terry model

The Bradley-Terry (BT) model is a basic probability model for item ranking or user preference from paired comparison data. For example, it can be used for estimating intrinsic strength of football teams based on results in the league. Also it can be applied for solving multi-class discriminant problem with binary classifiers.

In the BT model, each item (or user) is given a positive value, and comparison result of two items is modeled by a Bernoulli distribution parametrized by the ratio of values given to these two items. So far, estimation methods of this model have been discussed from several contexts, and most of them are based on the maximum likelihood method, i.e. minimizing the sum of weighted Kullback-Leibler (KL) divergences between Bernoulli distributions and paired comparison data.

We focus on the following two important facts: 1) a set of normalized values given to items (sum up to 1) can be identified with a categorical distribution, 2) observations, i.e. paired comparison data, can be regarded as incomplete data from categorical distributions, and each observation constructs an m-flat submanifold in the space of categorical distributions represented as a probability simplex. Based on these notions, we construct an objective function with the sum of KL divergences between a categorical distribution and a submanifold, and derive an em-like algorithm, which is an iterative estimation method with e-projections and m-projections. Moreover, by considering the geometrical relationship between empirical influence functions and m-flat submanifolds, we can introduce a natural estimation method of confidence in each observation. We will demonstrate effectiveness and advantages of our proposal with synthetic and real-world data.

This work was done in collaboration with my colleague, Yu Fujimoto and Hideitsu Hino.


Date and Location

August 02 - 06, 2010
University of Leipzig
04103 Leipzig

Scientific Organizers

Nihat Ay
Max Planck Institute for Mathematics in the Sciences
Information Theory of Cognitive Systems Group
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Paolo Gibilisco
Università degli Studi di Roma "Tor Vergata"
Facoltà di Economia
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František Matúš
Academy of Sciences of the Czech Republic
Institute of Information Theory and Automation
Czech Republic
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Scientific Committee

Shun-ichi Amari
Brain Science Institute, Mathematical Neuroscience Laboratory
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Imre Csiszár
Hungarian Academy of Sciences
Alfréd Rényi Institute of Mathematics
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Dénes Petz
Budapest University of Technology and Economics
Department for Mathematical Analysis
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Giovanni Pistone
Collegio Carlo Alberto, Moncalieri
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Administrative Contact

Antje Vandenberg
Max Planck Institute for Mathematics in the Sciences
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Phone: (++49)-(0)341-9959-552
Fax: (++49)-(0)341-9959-555

05.04.2017, 12:42