# Measurement Invariance, Entropy, and Probability

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Measurement, Information Invariance, and Probability

#### 2.1. Maximum entropy

#### 2.2. Measurement and transformation

#### 2.3. Example: ratio and scale invariance

## 3. The linear to Logarithmic Measurement Scale

#### 3.1. Measurement

#### 3.2. Probability

#### 3.3. Transition between linear and logarithmic scales

#### 3.4. Linear-log exponential distribution

#### 3.5. Student’s distribution

## 4. The Inverse Logarithmic to Linear Measurement Scale

#### 4.1. Measurement

#### 4.2. Probability

## 5. Integral Transforms and Superstatistics

## 6. Connections and Caveats

#### 6.1. Discrete versus continuous variables

#### 6.2. General measure invariance versus particular linear-log scales

## 7. Discussion

## 8. Conclusions

## Acknowledgements

## References

- Hand, D. Measurement Theory and Practice; Arnold: London, UK, 2004. [Google Scholar]
- Aparicio, F.; Estrada, J. Empirical distributions of stock returns: European securities markets, 1990-95. Eur. J. Financ.
**2001**, 7, 1–21. [Google Scholar] [CrossRef] - Dragulescu, A.A.; Yakovenko, V.M. Exponential and power-law probability distributions of wealth and income in the United Kingdom and the United States. Physica A
**2001**, 299, 213–221. [Google Scholar] [CrossRef] - Jaynes, E.T. Probability Theory: The Logic of Science; Cambridge University Press: New York, NY, USA, 2003. [Google Scholar]
- Beck, C.; Cohen, E. Superstatistics. Physica A: Statist. Mech. Appl.
**2003**, 322, 267–275. [Google Scholar] [CrossRef] - Tsallis, C. Introduction to Nonextensive Statistical Mechanics; Springer: New York, NY, USA, 2009. [Google Scholar]
- Jaynes, E.T. Information theory and statistical mechanics. Phys. Rev.
**1957**, 106, 620–630. [Google Scholar] [CrossRef] - Jaynes, E.T. Information theory and statistical mechanics, II. Phys. Rev.
**1957**, 108, 171–190. [Google Scholar] [CrossRef] - Frank, S.A. The common patterns of nature. J. Evol. Biol.
**2009**, 22, 1563–1585. [Google Scholar] [CrossRef] [PubMed] - Luce, R.D.; Narens, L. Measurement, theory of. In The New Palgrave Dictionary of Economics; Durlauf, S.N., Blume, L.E., Eds.; Palgrave Macmillan: Basingstoke, UK, 2008. [Google Scholar]
- Narens, L.; Luce, R.D. Meaningfulness and invariance. In The New Palgrave Dictionary of Economics; Durlauf, S.N., Blume, L.E., Eds.; Palgrave Macmillan: Basingstoke, UK, 2008. [Google Scholar]
- Johnson, N.L.; Kotz, S.; Balakrishnan, N. Continuous Univariate Distributions, 2nd ed.; Wiley: New York, NY, USA, 1994. [Google Scholar]
- Sivia, D.S.; Skilling, J. Data Analysis: A Bayesian Tutorial; Oxford University Press: New York, NY, USA, 2006. [Google Scholar]

© 2010 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license http://creativecommons.org/licenses/by/3.0/.

## Share and Cite

**MDPI and ACS Style**

Frank, S.A.; Smith, D.E. Measurement Invariance, Entropy, and Probability. *Entropy* **2010**, *12*, 289-303.
https://doi.org/10.3390/e12030289

**AMA Style**

Frank SA, Smith DE. Measurement Invariance, Entropy, and Probability. *Entropy*. 2010; 12(3):289-303.
https://doi.org/10.3390/e12030289

**Chicago/Turabian Style**

Frank, Steven A., and D. Eric Smith. 2010. "Measurement Invariance, Entropy, and Probability" *Entropy* 12, no. 3: 289-303.
https://doi.org/10.3390/e12030289