Weyl Prior and Bayesian Statistics
When using Bayesian inference, one needs to choose a prior distribution for parameters. The well-known Jeffreys prior is based on the Riemann metric tensor on a statistical manifold. Takeuchi and Amari defined the <inline-formula> <math display="inline"> <semantics> <m...
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| Format: | Article |
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MDPI AG
2020-04-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/22/4/467 |
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| author | Ruichao Jiang Javad Tavakoli Yiqiang Zhao |
| author_facet | Ruichao Jiang Javad Tavakoli Yiqiang Zhao |
| author_sort | Ruichao Jiang |
| collection | DOAJ |
| description | When using Bayesian inference, one needs to choose a prior distribution for parameters. The well-known Jeffreys prior is based on the Riemann metric tensor on a statistical manifold. Takeuchi and Amari defined the <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula>-parallel prior, which generalized the Jeffreys prior by exploiting a higher-order geometric object, known as a Chentsov–Amari tensor. In this paper, we propose a new prior based on the Weyl structure on a statistical manifold. It turns out that our prior is a special case of the <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula>-parallel prior with the parameter <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula> equaling <inline-formula> <math display="inline"> <semantics> <mrow> <mo>−</mo> <mi>n</mi> </mrow> </semantics> </math> </inline-formula>, where <i>n</i> is the dimension of the underlying statistical manifold and the minus sign is a result of conventions used in the definition of <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula>-connections. This makes the choice for the parameter <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula> more canonical. We calculated the Weyl prior for univariate Gaussian and multivariate Gaussian distribution. The Weyl prior of the univariate Gaussian turns out to be the uniform prior. |
| first_indexed | 2024-03-10T20:21:10Z |
| format | Article |
| id | doaj.art-4ee835f9cd474d19897be174e9b633d0 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-10T20:21:10Z |
| publishDate | 2020-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-4ee835f9cd474d19897be174e9b633d02023-11-19T22:08:26ZengMDPI AGEntropy1099-43002020-04-0122446710.3390/e22040467Weyl Prior and Bayesian StatisticsRuichao Jiang0Javad Tavakoli1Yiqiang Zhao2Department of Mathematics, The University of British Columbia Okanagan, Kelowna, BC V1V 1V7, CanadaDepartment of Mathematics, The University of British Columbia Okanagan, Kelowna, BC V1V 1V7, CanadaSchool of Mathematics and Statistics, Carlton University, Ottawa, ON K1S 5B6, CanadaWhen using Bayesian inference, one needs to choose a prior distribution for parameters. The well-known Jeffreys prior is based on the Riemann metric tensor on a statistical manifold. Takeuchi and Amari defined the <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula>-parallel prior, which generalized the Jeffreys prior by exploiting a higher-order geometric object, known as a Chentsov–Amari tensor. In this paper, we propose a new prior based on the Weyl structure on a statistical manifold. It turns out that our prior is a special case of the <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula>-parallel prior with the parameter <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula> equaling <inline-formula> <math display="inline"> <semantics> <mrow> <mo>−</mo> <mi>n</mi> </mrow> </semantics> </math> </inline-formula>, where <i>n</i> is the dimension of the underlying statistical manifold and the minus sign is a result of conventions used in the definition of <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula>-connections. This makes the choice for the parameter <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula> more canonical. We calculated the Weyl prior for univariate Gaussian and multivariate Gaussian distribution. The Weyl prior of the univariate Gaussian turns out to be the uniform prior.https://www.mdpi.com/1099-4300/22/4/467information geometryBayesian statisticsprior distributionsconformal geometry |
| spellingShingle | Ruichao Jiang Javad Tavakoli Yiqiang Zhao Weyl Prior and Bayesian Statistics Entropy information geometry Bayesian statistics prior distributions conformal geometry |
| title | Weyl Prior and Bayesian Statistics |
| title_full | Weyl Prior and Bayesian Statistics |
| title_fullStr | Weyl Prior and Bayesian Statistics |
| title_full_unstemmed | Weyl Prior and Bayesian Statistics |
| title_short | Weyl Prior and Bayesian Statistics |
| title_sort | weyl prior and bayesian statistics |
| topic | information geometry Bayesian statistics prior distributions conformal geometry |
| url | https://www.mdpi.com/1099-4300/22/4/467 |
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