Revisiting Prior distributions, Part I: Priors based on a physical invariance principle

Original Paper

Abstract

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 knowledge-based natural reference point of physical random variables (RV). Conceptual difficulties associated with uninformative priors are resolved, whereas well-established 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 non-dimensional 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.

Keywords

Random variables Prior probability Invariance requirements Knowledge integration 

References

  1. Baker R, Christakos G (2006) Revisiting prior distributions part, II: implications in maximum entropy analysis. J Stochastic Environ Res Risk Assess (in press)Google Scholar
  2. Chen WF, Saleeb AF (1982) Constitutive equations for engineering materials. Wily, New York, NYGoogle Scholar
  3. Christakos G (1990) A Bayesian/maximum entropy view to spatial estimation problem. Math Geol 22:763–776CrossRefGoogle Scholar
  4. Christakos G (2000) Modern spatiotemporal geostatistics. Oxford University Press, New York, NYGoogle Scholar
  5. Christakos G (2002a) On a deductive logic-based spatiotemporal random field theory. Probab Theory Math Stat (Teoriya Imovirnostey ta Matematychna Statystyka) 66:54–65Google Scholar
  6. Christakos G (2002b) On the assimilation of uncertain physical knowledge bases: Bayesian and non-Bayesian techniques. Adv Water Resour 25(8–12):1257–1274CrossRefGoogle Scholar
  7. Einstein A (1931) Relativity: the special and the general theory. Crown, New York, NYGoogle Scholar
  8. Hacking I (2001) Probability and inductive logic. Cambridge University Press, Cambridge, UKGoogle Scholar
  9. Hartigan J (1964) Invariant prior distributions. Ann Math Stat 35:836–845Google Scholar
  10. Jaynes ET (1965) Prior probabilities and transformation groups. Unpublished note. http://bayes.wustl.edu/etj/node2.html
  11. Jaynes ET (1968) Prior probabilities. IEEE Trans Syst Sci Cyber SSC-4(3):227–241CrossRefGoogle Scholar
  12. Jaynes ET (1978) Where do we stand on maximum entropy. In: Levin RD, Tribus M (eds) The maximum entropy formalism. MIT Press, Cambridge, MAGoogle Scholar
  13. Jaynes ET (2003) Probability theory: the logic of science. In: Bretthorst GL (ed) Cambridge University Press, UKGoogle Scholar
  14. Jeffreys H (1939) Theory of probability. Cambridge University Press, Cambridge, UKGoogle Scholar
  15. Kovitz J, Christakos G (2004) Assimilation of fuzzy data by the BME method. Stochastic Environ Res Risk Assess 18(2):79–90CrossRefGoogle Scholar
  16. Lakes R (1987) Foam structures with a negative Poisson’s ratio. Science 235:1038–1040CrossRefGoogle Scholar
  17. Papoulis A (1965) Probability, random variables. and stochastic processes. McGraw-Hill, New York, NYGoogle Scholar
  18. Stanley HE (1971) Introduction to phase transitions and critical phenomena. Oxford University Press, New YorkGoogle Scholar
  19. Synge JL, Griffith BA (1959) Principles of mechanics. McGraw-Hill, New YorkGoogle Scholar
  20. Tarantola A (1987) Inverse problem theory-methods for data fitting and model parameter estimation. Elsevier, Amsterdam. The NetherlandsGoogle Scholar
  21. Taylor EF, Wheeler JA (1966) Space–time physics. W.H. Freeman & Co., San Francisco, CAGoogle Scholar
  22. Trusdell C, Noll W (1965) The non-linear field theories of mechanics. In: Flugge S (ed), vol III/3. Encyclopedia of physics. Springer-Verlarg, New York, NYGoogle Scholar
  23. Villegas C (1977) On the representation of ignorance. J Am Stat Assoc 72:651–654CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Faculty of Civil and Environmental EngineeringTechnion-Israel Institute of TechnologyHaifaIsrael
  2. 2.Department of GeographySan Diego State UniversitySan DiegoUSA

Personalised recommendations