Book contents
- Frontmatter
- Contents
- 0 Introduction
- 1 What is Fisher information?
- 2 Fisher information in a vector world
- 3 Extreme physical information
- 4 Derivation of relativistic quantum mechanics
- 5 Classical electrodynamics
- 6 The Einstein field equation of general relativity
- 7 Classical statistical physics
- 8 Power spectral 1/f noise
- 9 Physical constants and the 1/x probability law
- 10 Constrained-likelihood quantum measurement theory
- 11 Research topics
- 12 Summing up
- Appendix A Solutions common to entropy and Fisher I-extremization
- Appendix B Cramer–Rao inequalities for vector data
- Appendix C Cramer–Rao inequality for an imaginary parameter
- Appendix D Simplified derivation of the Schroedinger wave equation
- Appendix E Factorization of the Klein–Gordon information
- Appendix F Evaluation of certain integrals
- Appendix G Schroedinger wave equation as a non-relativistic limit
- Appendix H Non-uniqueness of potential A for finite boundaries
- References
- Index
2 - Fisher information in a vector world
Published online by Cambridge University Press: 30 January 2010
- Frontmatter
- Contents
- 0 Introduction
- 1 What is Fisher information?
- 2 Fisher information in a vector world
- 3 Extreme physical information
- 4 Derivation of relativistic quantum mechanics
- 5 Classical electrodynamics
- 6 The Einstein field equation of general relativity
- 7 Classical statistical physics
- 8 Power spectral 1/f noise
- 9 Physical constants and the 1/x probability law
- 10 Constrained-likelihood quantum measurement theory
- 11 Research topics
- 12 Summing up
- Appendix A Solutions common to entropy and Fisher I-extremization
- Appendix B Cramer–Rao inequalities for vector data
- Appendix C Cramer–Rao inequality for an imaginary parameter
- Appendix D Simplified derivation of the Schroedinger wave equation
- Appendix E Factorization of the Klein–Gordon information
- Appendix F Evaluation of certain integrals
- Appendix G Schroedinger wave equation as a non-relativistic limit
- Appendix H Non-uniqueness of potential A for finite boundaries
- References
- Index
Summary
Classical measurement of four-vectors
In the preceding chapter, we found that the accuracy in an estimate of a single parameter θ is determined by an information I that has some useful physical properties. It provides new definitions of disorder, time and temperature, and a variational approach to finding a single-component PDF law p(x) of a single variable x. However, many physical phenomena are describable only by multiple- component PDFs, as in quantum mechanics, and for vector variables x since worldviews are usually four-dimensional (as required by covariance). Our aim in this chapter, then, is to form a new, scalar information I that is appropriate to this multi-component, vector scenario. The information should be intrinsic to the phenomenon under measurement and not depend, e.g., upon exterior effects such as the noise of the measuring device.
The ‘intrinsic’ measurement scenario
In Bayesian statistics, a prior scenario is often used to define an otherwise unknown prior probability law; see, e.g., Good (1976), Jaynes (1985) or Frieden (1991). This is a model scenario that permits the prior probability law to be computed on the basis of some ideal conditions, such as independence of data, and/or ‘maximum ignorance’ (see below), etc. We will use the concept to define our unknown information expression.
- Type
- Chapter
- Information
- Physics from Fisher InformationA Unification, pp. 51 - 62Publisher: Cambridge University PressPrint publication year: 1998
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