Information geometry of the family of Markov kernels defined by a context tree

Jun'Ichi Takeuchi, Hiroshi Nagaoka

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

We prove that a tree model is an exponential family (e-family) of Markov kernels, if and only if it is an FSMX model. The notion of e-family of Markov kernels was first introduced by Nakagawa and Kanaya ('93) in the one-dimensional case. Then, Nagaoka ('05) gave its established form, and Hayashi & Watanabe ('16) discussed it. A tree model is the Markov model defined by a context tree. It is noted by Weinberger et al., ('95) that tree models are classified into two classes; FSMX models and non-FSMX models, depending on the shape of their context trees. The FSMX model is a tree model and a finite state machine. We further show that, for Markov models, the e-family of Markov kernels is equivalent to the asymptotic e-family, which was introduced by Takeuchi & Barron ('98). Note that Takeuchi & Kawabata ('07) proved that non-FSMX tree models are not asymptotic e-families for the binary alphabet case. This paper enhances their result and reveals the information geometrical properties of tree models.

Original languageEnglish
Title of host publication2017 IEEE Information Theory Workshop, ITW 2017
PublisherInstitute of Electrical and Electronics Engineers Inc.
Pages429-433
Number of pages5
ISBN (Electronic)9781509030972
DOIs
Publication statusPublished - Jan 31 2018
Event2017 IEEE Information Theory Workshop, ITW 2017 - Kaohsiung, Taiwan, Province of China
Duration: Nov 6 2017Nov 10 2017

Publication series

NameIEEE International Symposium on Information Theory - Proceedings
Volume2018-January
ISSN (Print)2157-8095

Other

Other2017 IEEE Information Theory Workshop, ITW 2017
CountryTaiwan, Province of China
CityKaohsiung
Period11/6/1711/10/17

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Information Systems
  • Modelling and Simulation
  • Applied Mathematics

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  • Cite this

    Takeuchi, JI., & Nagaoka, H. (2018). Information geometry of the family of Markov kernels defined by a context tree. In 2017 IEEE Information Theory Workshop, ITW 2017 (pp. 429-433). (IEEE International Symposium on Information Theory - Proceedings; Vol. 2018-January). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/ITW.2017.8278008