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

2 Citations (Scopus)


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.
Number of pages5
ISBN (Electronic)9781509030972
Publication statusPublished - Jul 2 2017
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
ISSN (Print)2157-8095


Other2017 IEEE Information Theory Workshop, ITW 2017
Country/TerritoryTaiwan, Province of China

All Science Journal Classification (ASJC) codes

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


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