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Measurement Invariance, Entropy, and Probability
Department of Ecology and Evolutionary Biology, University of California, Irvine, CA 92697–2525, USA
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
* Author to whom correspondence should be addressed.
Received: 18 January 2010; in revised form: 16 February 2010 / Accepted: 23 February 2010 / Published: 26 February 2010
Abstract: We show that the natural scaling of measurement for a particular problem defines the most likely probability distribution of observations taken from that measurement scale. Our approach extends the method of maximum entropy to use measurement scale as a type of information constraint. We argue that a very common measurement scale is linear at small magnitudes grading into logarithmic at large magnitudes, leading to observations that often follow Student’s probability distribution which has a Gaussian shape for small fluctuations from the mean and a power law shape for large fluctuations from the mean. An inverse scaling often arises in which measures naturally grade from logarithmic to linear as one moves from small to large magnitudes, leading to observations that often follow a gamma probability distribution. A gamma distribution has a power law shape for small magnitudes and an exponential shape for large magnitudes. The two measurement scales are natural inverses connected by the Laplace integral transform. This inversion connects the two major scaling patterns commonly found in nature. We also show that superstatistics is a special case of an integral transform, and thus can be understood as a particular way in which to change the scale of measurement. Incorporating information about measurement scale into maximum entropy provides a general approach to the relations between measurement, information and probability.
Keywords: maximum entropy; information theory; superstatistics; power law; Student’s distribution; gamma distribution
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MDPI and ACS Style
Frank, S.A.; Smith, D.E. Measurement Invariance, Entropy, and Probability. Entropy 2010, 12, 289-303.
Frank SA, Smith DE. Measurement Invariance, Entropy, and Probability. Entropy. 2010; 12(3):289-303.
Frank, Steven A.; Smith, D. Eric. 2010. "Measurement Invariance, Entropy, and Probability." Entropy 12, no. 3: 289-303.