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Entropy 2016, 18(5), 192;

Common Probability Patterns Arise from Simple Invariances

Department of Ecology & Evolutionary Biology, University of California, Irvine, CA 92697, USA
Academic Editor: Antonio M. Scarfone
Received: 26 January 2016 / Revised: 29 April 2016 / Accepted: 14 May 2016 / Published: 19 May 2016
(This article belongs to the Collection Advances in Applied Statistical Mechanics)
Full-Text   |   PDF [290 KB, uploaded 19 May 2016]


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. View Full-Text
Keywords: measurement; maximum entropy; information theory; statistical mechanics; extreme value distributions measurement; maximum entropy; information theory; statistical mechanics; extreme value distributions
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Frank, S.A. Common Probability Patterns Arise from Simple Invariances. Entropy 2016, 18, 192.

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