Geometry of q-Exponential Family of Probability Distributions
The Gibbs distribution of statistical physics is an exponential family of probability distributions, which has a mathematical basis of duality in the form of the Legendre transformation. Recent studies of complex systems have found lots of distributions obeying the power law rather than the standard...
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| Format: | Article |
| Language: | English |
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MDPI AG
2011-06-01
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| Series: | Entropy |
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| Online Access: | http://www.mdpi.com/1099-4300/13/6/1170/ |
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| author | Shun-ichi Amari Atsumi Ohara |
| author_facet | Shun-ichi Amari Atsumi Ohara |
| author_sort | Shun-ichi Amari |
| collection | DOAJ |
| description | The Gibbs distribution of statistical physics is an exponential family of probability distributions, which has a mathematical basis of duality in the form of the Legendre transformation. Recent studies of complex systems have found lots of distributions obeying the power law rather than the standard Gibbs type distributions. The Tsallis q-entropy is a typical example capturing such phenomena. We treat the q-Gibbs distribution or the q-exponential family by generalizing the exponential function to the q-family of power functions, which is useful for studying various complex or non-standard physical phenomena. We give a new mathematical structure to the q-exponential family different from those previously given. It has a dually flat geometrical structure derived from the Legendre transformation and the conformal geometry is useful for understanding it. The q-version of the maximum entropy theorem is naturally induced from the q-Pythagorean theorem. We also show that the maximizer of the q-escort distribution is a Bayesian MAP (Maximum A posteriori Probability) estimator. |
| first_indexed | 2024-04-11T22:50:08Z |
| format | Article |
| id | doaj.art-7a608bf6862f4323b6f5bc1b2e9e9d41 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-11T22:50:08Z |
| publishDate | 2011-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-7a608bf6862f4323b6f5bc1b2e9e9d412022-12-22T03:58:36ZengMDPI AGEntropy1099-43002011-06-011361170118510.3390/e13061170Geometry of q-Exponential Family of Probability DistributionsShun-ichi AmariAtsumi OharaThe Gibbs distribution of statistical physics is an exponential family of probability distributions, which has a mathematical basis of duality in the form of the Legendre transformation. Recent studies of complex systems have found lots of distributions obeying the power law rather than the standard Gibbs type distributions. The Tsallis q-entropy is a typical example capturing such phenomena. We treat the q-Gibbs distribution or the q-exponential family by generalizing the exponential function to the q-family of power functions, which is useful for studying various complex or non-standard physical phenomena. We give a new mathematical structure to the q-exponential family different from those previously given. It has a dually flat geometrical structure derived from the Legendre transformation and the conformal geometry is useful for understanding it. The q-version of the maximum entropy theorem is naturally induced from the q-Pythagorean theorem. We also show that the maximizer of the q-escort distribution is a Bayesian MAP (Maximum A posteriori Probability) estimator.http://www.mdpi.com/1099-4300/13/6/1170/q-exponential familyq-entropyinformation geometryq-Pythagorean theoremq-Max-Ent theoremconformal transformation |
| spellingShingle | Shun-ichi Amari Atsumi Ohara Geometry of q-Exponential Family of Probability Distributions Entropy q-exponential family q-entropy information geometry q-Pythagorean theorem q-Max-Ent theorem conformal transformation |
| title | Geometry of q-Exponential Family of Probability Distributions |
| title_full | Geometry of q-Exponential Family of Probability Distributions |
| title_fullStr | Geometry of q-Exponential Family of Probability Distributions |
| title_full_unstemmed | Geometry of q-Exponential Family of Probability Distributions |
| title_short | Geometry of q-Exponential Family of Probability Distributions |
| title_sort | geometry of q exponential family of probability distributions |
| topic | q-exponential family q-entropy information geometry q-Pythagorean theorem q-Max-Ent theorem conformal transformation |
| url | http://www.mdpi.com/1099-4300/13/6/1170/ |
| work_keys_str_mv | AT shunichiamari geometryofqexponentialfamilyofprobabilitydistributions AT atsumiohara geometryofqexponentialfamilyofprobabilitydistributions |