Revisiting Prior distributions, Part I: Priors based on a physical invariance principle
 Rafi Baker,
 George Christakos
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Determination of uninformative prior distributions is essential in many branches of knowledge integration and system processing. The conceptual difficulties of this determination are due to lack of uniqueness and consequential lack of objectivity associated with the state of complete ignorance. The present work overcomes the above difficulty by considering a class of priors that are consistent with a physical invariance principle, namely, invariance with respect to a change in the system of dimensional units. These priors do not represent total ignorance and they do not suffer from the aforementioned conceptual difficulties. This Dimensional Invariance Requirement (DIR) leads to a class of prior densities, which constitute a generalization of Jeffrey’s proposal concerning priors of inherently positive variables. This generalization possesses certain important features, from a formal as well as an interpretive viewpoint, which involve the notion of a knowledgebased natural reference point of physical random variables (RV). Conceptual difficulties associated with uninformative priors are resolved, whereas wellestablished results are derived as special cases of the DIR. Application of this requirement to a system of RV yields the familiar result that at the prior knowledge stage these variables should be considered as independent. Prior distributions for nondimensional physical quantities are obtained by defining these variables in terms of dimensional quantities. A logarithmic transformation carries the physical prior into a uniform (flat) density that is convenient in certain applications. In a companion paper we examine the improvements gained in the maximum entropy context by means of the proposed class of physical prior densities.
Inside
Within this Article
 Introduction
 The DIR functional equation and its solutions
 Natural reference points of dimentional physical variables
 System of random variables
 Physical priors for nondimensional physical variables
 Summary and conclusions
 References
 References
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 Title
 Revisiting Prior distributions, Part I: Priors based on a physical invariance principle
 Journal

Stochastic Environmental Research and Risk Assessment
Volume 21, Issue 4 , pp 427434
 Cover Date
 20070401
 DOI
 10.1007/s0047700600750
 Print ISSN
 14363240
 Online ISSN
 14363259
 Publisher
 SpringerVerlag
 Additional Links
 Topics

 Waste Water Technology / Water Pollution Control / Water Management / Aquatic Pollution
 Numerical and Computational Methods in Engineering
 Statistics for Engineering, Physics, Computer Science, Chemistry & Geosciences
 Probability Theory and Stochastic Processes
 Math. Applications in Geosciences
 Math. Appl. in Environmental Science
 Keywords

 Random variables
 Prior probability
 Invariance requirements
 Knowledge integration
 Industry Sectors
 Authors

 Rafi Baker ^{(1)}
 George Christakos ^{(2)}
 Author Affiliations

 1. Faculty of Civil and Environmental Engineering, TechnionIsrael Institute of Technology, Technion City, Haifa, 32 000, Israel
 2. Department of Geography, San Diego State University, San Diego, CA, 981824493, USA