### 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 language | English |
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Title of host publication | 2017 IEEE Information Theory Workshop, ITW 2017 |

Publisher | Institute of Electrical and Electronics Engineers Inc. |

Pages | 429-433 |

Number of pages | 5 |

Volume | 2018-January |

ISBN (Electronic) | 9781509030972 |

DOIs | |

Publication status | Published - Jan 31 2018 |

Event | 2017 IEEE Information Theory Workshop, ITW 2017 - Kaohsiung, Taiwan, Province of China Duration: Nov 6 2017 → Nov 10 2017 |

### Other

Other | 2017 IEEE Information Theory Workshop, ITW 2017 |
---|---|

Country | Taiwan, Province of China |

City | Kaohsiung |

Period | 11/6/17 → 11/10/17 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

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

### Cite this

*2017 IEEE Information Theory Workshop, ITW 2017*(Vol. 2018-January, pp. 429-433). Institute of Electrical and Electronics Engineers Inc.. https://doi.org/10.1109/ITW.2017.8278008

**Information geometry of the family of Markov kernels defined by a context tree.** / Takeuchi, Junnichi; Nagaoka, Hiroshi.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*2017 IEEE Information Theory Workshop, ITW 2017.*vol. 2018-January, Institute of Electrical and Electronics Engineers Inc., pp. 429-433, 2017 IEEE Information Theory Workshop, ITW 2017, Kaohsiung, Taiwan, Province of China, 11/6/17. https://doi.org/10.1109/ITW.2017.8278008

}

TY - GEN

T1 - Information geometry of the family of Markov kernels defined by a context tree

AU - Takeuchi, Junnichi

AU - Nagaoka, Hiroshi

PY - 2018/1/31

Y1 - 2018/1/31

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85046370905&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85046370905&partnerID=8YFLogxK

U2 - 10.1109/ITW.2017.8278008

DO - 10.1109/ITW.2017.8278008

M3 - Conference contribution

AN - SCOPUS:85046370905

VL - 2018-January

SP - 429

EP - 433

BT - 2017 IEEE Information Theory Workshop, ITW 2017

PB - Institute of Electrical and Electronics Engineers Inc.

ER -