Subject: Re: True Randomness & The Law Of Large Numbers Date: 18 Apr 1999 00:00:00 GMT From: hrubin@odds.stat.purdue.edu (Herman Rubin) Organization: Purdue University Statistics Department Newsgroups: sci.crypt References: 1 , 2 , 3 , 4 In article <3719e4ad.1582104@nntp.ix.netcom.com>, R. Knauer wrote: >On 18 Apr 1999 08:19:25 -0500, hrubin@odds.stat.purdue.edu (Herman >Rubin) wrote: >>If one looks at statistics in a reasonably intelligent manner, >>the problem is not to decide whether or not the point null >>hypothesis is true, but to decide whether it is better to act >>as if it is than the alternative. This does affect testing. >It is interesting to note that in Billingsley's book where he >discusses Chernoff's Theorem, he points out in balancing the error of >rejecting one hypothesis over another for the value of p, that as p >approaches 1/2 it becomes increasingly difficult to discriminate >between the p = 1/2 hypothesis and an hypothesis for a slightly >different value near 1/2. Why should this be surprising? It is a little easier to see in the case of a normal translation parameter, and the problem is extremely similar; the sample size needed for a given amount of discrimination is proportional to 1/Kd^2, where K is the Fisher information in a single observation, and d is separation. -- This address is for information only. I do not claim that these views are those of the Statistics Department or of Purdue University. Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 http://www.optics.arizona.edu/News/Oscillations/July.htm In early April, the Center hosted the first international meeting of scientists who work with Fisher information in application to physics. Professor B. Roy Frieden, who founded the group, said the goal of the meeting was to promote research on the connection between Fisher information and basic physics. The charter members of the Fisher Information Interest Group are (standing) Optical Sciences Center Professors Arvind Marathay and B. Roy Frieden with Bernard Soffer of Hughes, Malibu and (seated) Roy Hughes of DoD, Australia. Many members of the group believe that all of thermodynamics can be derived from the standpoint of minimum Fisher information. Roy and his colleagues will be seriously attacking this problem in the fall when another group member, Professor Angelo Plastino of the Department of Physics of the University of La Plata in Argentina, visits the Center. He will also work with Roy on Fisher temperature-and time concepts. Fisher information is an old concept. It has been used since about 1922 to judge the quality of statistical estimates. Now it is the central concept in a new theory of measurement. This predicts that each physical phenomenon arises in response to a request for data about it. Roy explained, "The probe particle that initiates a measurement perturbs the measured particle's wave function. This perturbs the particle's Fisher information level, and initiates a variational principle whose solution and output is the probability law that produces the requested measurement. For example, the Schroedinger wave equation arises out of a request for the position of a particle. In this way, the phenomenon that is to be measured is produced 'on the spot.' The result is a kind of local creation of reality. This appears to be an effect that is new to both physics and metaphysics, resembling the 'participatory universe' of Professor J.A. Wheeler." Roy continued, "Virtually all of known physics, from relativistic quantum mechanics to statistical mechanics to quantum gravity, has been derived by this measurement approach. The local creation of reality is a microscopic effect. It arises in measuring and interacting with single elementary particles. It's reminiscent of the microscopic reversibility to time of the laws of physics. As with the latter, we don't yet know how and if 'reality on demand' translates into a macroscopic effect." A traditional measure of disorder, entropy, has provided the usual definitions of time and temperature. Said Roy, "Fisher information provides us with new definitions. They arise out of a newly discovered 'H-theorem' for the information: It can only decrease with time. This makes Fisher information a measure of disorder and means that Fisher information must provide its own definitions of time and temperature. Time is defined to increase when Fisher information decreases. Intriguingly, we find that Fisher time and entropy time do not agree about 1% 'of the time.' Temperature is defined as the resistance to a change in energy of the Fisher information of a system. The relationship of the Fisher temperature scale to the entropic, or conventional, temperature scale is currently not known." B. Roy Frieden's fall 1998 course offering, Opti 529, crosslisted as Phys 529, is titled "Physics from Fisher Information, a Unification." It will show that Fisher information provides a basis for nearly all physical laws, including quantum mechanics, classical e&m theory, special and general relativity, diffraction optics, the statistical gas laws, quantum gravity, and the ubiquitous 1/f-noise power law. The textbook for the course, Physics from Fisher Information, a Unification by B. Roy Frieden will be published in December 1998 by Cambridge University Press. B. Roy Frieden has been with the Center since 1966 and has worked with Fisher information since 1987. His research interests include digital methods of enhancing or restoring images, blind deconvolution, and the synthesis of physics using Fisher information. A.R. Plastino, A.Plastino, Fisher Information and Bounds to the Entropy Increase, The Physical Review A 52 (1995) 4580. A.Plastino, A.R. Plastino, H. G. Miller, F. C. Khanna, A lower bound for Fisher's information measure, Physics Letters A 221 (1996) 29. A.R. Plastino, A.Plastino, Symmetries of the Fokker-Planck equation and the Fisher)Frieden arrow time, The Physical Review E 54 (1996) 4423. L. Diambra, A.Plastino, Perturbative treatments and learning techniques, The Physical Review E 53 (1996) 3970. 24. A.R. Plastino, A.Plastino, Statistical treatment of autonomous systems with divergenceless flows, Physica A 232 (1996) 458. 25. F. Pennini, A.Plastino, A.R. Plastino, Nonextensive thermostatistics, Pauli principle and the structure of the Fermi surface, Physica A 234 (1966) 471. 26. A.Plastino, A.R. Plastino, H. G. Miller, Nonextensive thermostatistics and Fisher's information measure, Physica A 235 (1996) 577. 27. A.Kowalski, A.Plastino, A.N. Proto, A semi classical statistical model for quantum dissipation, Physica A 236 (1997) 429. 28. M. Casas, A.Plastino, Gibbs-like ensembles and the inference of pure states, Physica A 241 (1997) 704. 29. L. G. Gamero, A.Plastino, M. E. Torres, Wavelet analysis and nonlinear dynamics in a nonextensive setting, Physica A 246 (1997) 487. 30. F. Pennini, A.Plastino, Fisher's information measure a Tsallis nonextensive setting and its application to diffusive processes, Physica A 247 (1997) 559. 31. A.Plastino, A.R. Plastino, H. G. Miller, Continuity equations, H-theorems, and maximum entropy, Physics Letters A 232 (1997) 349. 32. C. M. Giordano, A.R. Plastino, A.Plastino, Robe's restricted three-body problem with drag, Celestial Mechanics and Dynamical Astronomy 66 (1997) 299. 33. A.R. Plastino, A.Plastino, H. G. Miller, Thermodynamics paths to Jensen's inequality, American Journal of Physics 65 (1997) 1102. 34. M. Casas, F. Pennini, A.Plastino, Soffe-Frieden principle and wave functions that maximize Shannon's measure, Physics Letters A 235 (1997) 457. 35. L. Rebollo, J. Fernandez-Rubio, A.Plastino, Frames: A maximum entropy statistical estimate of the inverse problem, Journal of Mathematical Physics 38 (1997) 4863. 36. D. Torres, H. Vucetich, A.Plastino, Early universe test of nonextensive statistics, Physical Review Letters 79 (1997) 1588. 38. C. M. Giordano, A.R. Plastino, A.Plastino, Comment on a simple, conservative understanding of many time-driven systems, American Journal of Physics, 65 (1997) 1183. 39. A.Plastino, A.R. Plastino, H. G. Miller, On the relationship between the Fisher-Frieden-Soffer arrow of time, and the behaviour of the Boltzmann and Kullback entropies, Physics Letters A 235 (1997) 129. 40. J. Fernandez, A.Plastino, Dynamical mechanisms for biological evolution, Physical Review E 56 (1997) 841. 41. A.R. Plastino, H. G. Miller, A.Plastino Minimum Kullback entropy approach to the Fokker-Planck equation, Physical Review E 56 (1997) 3927. 42. A.R. Plastino, H. G. Miller, A.Plastino, G. D. Yen, Entropic measures and the maximum entropy-minimum norm solution to the generalized inverse problem, Journal of Mathematical Physics 38 (1997) 6675. 10.A. Plastino, A.R. Plastino & H.G. Miller: "Tsallis nonextensive thermostatistics and Fisher's information measure." Physica A 235 (1997) 577. 11.A. Plastino, A.R. Plastino & H.G. Miller: "On the relationship between the Fisher- Frieden-Soffer arrow of time, and the behaviour of the Boltzmann and Kullback entropies." Physics Letters A 235 (1997) 129. 12.A. Plastino, A.R. Plastino & H.G. Miller: "Continuity equations, H-theorems, and maximum entropy." Physica A 232 (1997) 349. Reginatto, M. "Derivation of the Pauli Equation Using the Principle of Minimum Fisher Information" Physics Letters A, Vol. 249, pp. 355-357, December, 1998 Reginatto, M. "Derivation of the Equations of Non-Relativistic Quantum Mechanics Using the Principle of Minimum Fisher Information" Physical Review A, Vol. 58, pp. 1775-1778, September, 1998 ** http://www.mth.kcl.ac.uk/publications/98prep5.html KCL-MTH-98-22 H Nencka, RF Streater Information Geometry for Lie Algebras KCL-MTH-98-28 Dorje C. Brody, Lane P. Hughston Thermalisation of Quantum States KCL-MTH-98-29 Dorje C. Brody, Lane P. Hughston Geometry of Thermodynamic States KCL-MTH-98-32 RF Streater The Soret and Dufour effects in statistical dynamics KCL-MTH-98-71 R. F. Streater The Information Manifold for Relativity Bounded Potentials