Presentation | 2014-11-18 Gauge Theory on Measure Space : τ-Information Geometry Masaru TANAKA, |
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Abstract(in Japanese) | (See Japanese page) |
Abstract(in English) | A new formulation of information geometry is given by extending a translation operation on an affine space of probability density functions with up to scale from a logarithmic function to a τ-logarithmic function. In this formulation, this extended translation is called a τ-affine structure and an affine space with a τ-affine structure is called a τ-affine space. Its dual space is defined by taking a Holder conjugation. A contraction operation is defined on a τ-affine space and its conjugated space. This leads to Fisher metric for score functions. Then it is revealed that Fisher metric is independent of a τ-affine structure. The definition of an entropy requires a renormalization process. With a renormalization technique for a τ-log-likelihood, the contraction leads to a new definition of an expectation. This new expectation reveals that an escort distribution is nothing but a normalized and renormalized τ-log-likelihood. Therefore, the escort distribution is no longer necessary for us. By using this new expectation, a conformal entropy is also given. The conformal entropy is related to Tsallis entropy with a simple modification and gives two types of non-additivity according to formulae of τ-logarithmic function for multiplication. |
Keyword(in Japanese) | (See Japanese page) |
Keyword(in English) | τ-affine structure / contraction / τ-log-likelihood / escort distribution / renormalization / entropy / non-additivity |
Paper # | IBISML2014-72 |
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Committee | IBISML |
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Conference Date | 2014/11/10(1days) |
Place (in Japanese) | (See Japanese page) |
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Registration To | Information-Based Induction Sciences and Machine Learning (IBISML) |
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Language | JPN |
Title (in Japanese) | (See Japanese page) |
Sub Title (in Japanese) | (See Japanese page) |
Title (in English) | Gauge Theory on Measure Space : τ-Information Geometry |
Sub Title (in English) | |
Keyword(1) | τ-affine structure |
Keyword(2) | contraction |
Keyword(3) | τ-log-likelihood |
Keyword(4) | escort distribution |
Keyword(5) | renormalization |
Keyword(6) | entropy |
Keyword(7) | non-additivity |
1st Author's Name | Masaru TANAKA |
1st Author's Affiliation | Faculty of Science, Fukuoka University() |
Date | 2014-11-18 |
Paper # | IBISML2014-72 |
Volume (vol) | vol.114 |
Number (no) | 306 |
Page | pp.pp.- |
#Pages | 8 |
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