Common Probability Patterns Arise from Simple Invariances
Shift and stretch invariance lead to the exponential-Boltzmann probability distribution. Rotational invariance generates the Gaussian distribution. Particular scaling relations transform the canonical exponential and Gaussian patterns into the variety of commonly observed patterns. The scaling relat...
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| Format: | Article |
| Language: | English |
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MDPI AG
2016-05-01
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| Series: | Entropy |
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| Online Access: | http://www.mdpi.com/1099-4300/18/5/192 |
| _version_ | 1798042150908198912 |
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| author | Steven A. Frank |
| author_facet | Steven A. Frank |
| author_sort | Steven A. Frank |
| collection | DOAJ |
| description | Shift and stretch invariance lead to the exponential-Boltzmann probability distribution. Rotational invariance generates the Gaussian distribution. Particular scaling relations transform the canonical exponential and Gaussian patterns into the variety of commonly observed patterns. The scaling relations themselves arise from the fundamental invariances of shift, stretch and rotation, plus a few additional invariances. Prior work described the three fundamental invariances as a consequence of the equilibrium canonical ensemble of statistical mechanics or the Jaynesian maximization of information entropy. By contrast, I emphasize the primacy and sufficiency of invariance alone to explain the commonly observed patterns. Primary invariance naturally creates the array of commonly observed scaling relations and associated probability patterns, whereas the classical approaches derived from statistical mechanics or information theory require special assumptions to derive commonly observed scales. |
| first_indexed | 2024-04-11T22:31:30Z |
| format | Article |
| id | doaj.art-d6cff393be8c45a68ce228c302153871 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-11T22:31:30Z |
| publishDate | 2016-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-d6cff393be8c45a68ce228c3021538712022-12-22T03:59:24ZengMDPI AGEntropy1099-43002016-05-0118519210.3390/e18050192e18050192Common Probability Patterns Arise from Simple InvariancesSteven A. Frank0Department of Ecology & Evolutionary Biology, University of California, Irvine, CA 92697, USAShift and stretch invariance lead to the exponential-Boltzmann probability distribution. Rotational invariance generates the Gaussian distribution. Particular scaling relations transform the canonical exponential and Gaussian patterns into the variety of commonly observed patterns. The scaling relations themselves arise from the fundamental invariances of shift, stretch and rotation, plus a few additional invariances. Prior work described the three fundamental invariances as a consequence of the equilibrium canonical ensemble of statistical mechanics or the Jaynesian maximization of information entropy. By contrast, I emphasize the primacy and sufficiency of invariance alone to explain the commonly observed patterns. Primary invariance naturally creates the array of commonly observed scaling relations and associated probability patterns, whereas the classical approaches derived from statistical mechanics or information theory require special assumptions to derive commonly observed scales.http://www.mdpi.com/1099-4300/18/5/192measurementmaximum entropyinformation theorystatistical mechanicsextreme value distributions |
| spellingShingle | Steven A. Frank Common Probability Patterns Arise from Simple Invariances Entropy measurement maximum entropy information theory statistical mechanics extreme value distributions |
| title | Common Probability Patterns Arise from Simple Invariances |
| title_full | Common Probability Patterns Arise from Simple Invariances |
| title_fullStr | Common Probability Patterns Arise from Simple Invariances |
| title_full_unstemmed | Common Probability Patterns Arise from Simple Invariances |
| title_short | Common Probability Patterns Arise from Simple Invariances |
| title_sort | common probability patterns arise from simple invariances |
| topic | measurement maximum entropy information theory statistical mechanics extreme value distributions |
| url | http://www.mdpi.com/1099-4300/18/5/192 |
| work_keys_str_mv | AT stevenafrank commonprobabilitypatternsarisefromsimpleinvariances |