📝 Abstract

The subjects of the paper are the likelihood method (LM) and the expected Fisher information (FI) considered from the point od view of the construction of the physical models which originate in the statistical description of phenomena. The master equation case and structural information principle are derived. Then, the phenomenological description of the information transfer is presented. The extreme physical information (EPI) method is reviewed. As if marginal, the statistical interpretation of the amplitude of the system is given. The formalism developed in this paper would be also applied in quantum information processing and quantum game theory.

💡 Analysis

The subjects of the paper are the likelihood method (LM) and the expected Fisher information (FI) considered from the point od view of the construction of the physical models which originate in the statistical description of phenomena. The master equation case and structural information principle are derived. Then, the phenomenological description of the information transfer is presented. The extreme physical information (EPI) method is reviewed. As if marginal, the statistical interpretation of the amplitude of the system is given. The formalism developed in this paper would be also applied in quantum information processing and quantum game theory.

📄 Content

arXiv:0811.3554v1 [physics.data-an] 21 Nov 2008 physica status solidi, 1 November 2018 The method of the likelihood and the Fisher information in the construction of physical models E.W. Piotrowski1, J. Sładkowski2, J. Syska2,*, S. Zaja¸c2 1 Institute of Mathematics, The University of Białystok, Pl-15424 Białystok, Poland 2 Institute of Physics, University of Silesia, Uniwersytecka 4, 40-007 Katowice, Poland Received XXXX, revised XXXX, accepted XXXX Published online XXXX PACS 03.65.Bz, 03.65.Ca, 03.65.Sq, 02.50.Rj ∗Corresponding author: e-mail jacek.syska@us.edu.pl, Phone +xx-xx-xxxxxxx, Fax +xx-xx-xxx The subjects of the paper are the likelihood method (LM) and the expected Fisher information (FI) considered from the point od view of the construction of the physical models which originate in the statistical description of phenomena. The master equation case and structural information principle are derived. Then, the phenomenological description of the information transfer is presented. The extreme physical information (EPI) method is reviewed. As if marginal, the statistical interpretation of the amplitude of the system is given. The formalism developed in this paper would be also applied in quantum information processing and quantum game theory. Copyright line will be provided by the publisher 1 Introduction. The subjects of the paper are the like- lihood method (LM) and the expected Fisher information (FI), I, considered from the point of view of the construc- tion of the physical models, which originate in the statisti- cal description of phenomena. The FI had been introduced by Fisher [1] as a part of the new technique of parameter estimation of models which undergo the statistical investi- gation in the language of the maximization of the value of the likelihood function L (signed below as P). The notion of the FI is also connected with the Cram´er-Rao inequal- ity, being the maximal inverse of the possible value of the mean-square error of the unbiased estimate from the true value of the parameter. The set Θ = (θn) of the parame- ters1 composes the components in the statistical space S. Under the regularity conditions the expected FI (here be- 1 In the most general case of the estimation procedure the di- mensions of Θ ≡(θi)k 1 and the sample y ≡(yn)N 1 are usually different, yet because (θi) is below the set of physical ”positions” of a physical nature (e.g. positions in the case of an equation of motion or energies in the case of a statistical physics generating equation [3]) we have E(yn) = θn and the dimensions of Θ and y are the same, i.e. k = N. cause of the summation over n, the channel capacity): I ≡ N X n=1 Z dNy−∂2lnP(y; Θ) ∂θ2n P(y)

N X n=1 Z dNy(∂lnP(y; Θ) ∂θn )2P(y) (1) is the variance of the score function S(Θ) ≡∂lnP(Θ)/∂Θ [2]. It describes the local properties of P(y; θ1, …, θN) which is the joint probability (density) of the N data values y ≡(y1, …, yN), but is understood as a function of the parameters θn. Below with xn ≡yn −θn we note the added fluctuations of the data yn from θn. The development of the statistical methods introduced by Fisher [1] has followed along two different routes. The first one, of the differential geometry origin, began when C.R. Rao [4] noticed that the FI matrix defines a Rieman- nian metric. For a short review of the following work, until the consistent definition of the α-connection on the statis- tical spaces see [5]. The second one is based on the con- struction of the information (entropical) principles and we will describe this approach in details. It was put forward by Frieden and his coauthors, especially Soffer [3], and is known under the notion of the extreme physical informa- Copyright line will be provided by the publisher 2 Piotrowski, Sładkowski, Syska, Zaja¸c: Likelihood method and Fisher information in construction of physical models tion (EPI). Frieden began the construction of the physical models by obtaining the kinetic term from the FI2. Then he postulated the variational (scalar) information prin- ciple (IP) and internal (structural) one. Each of the IP has the universal form, yet its realization depends on the partic- ular phenomenon.The variational IP leads to the dispersion relation proper for the system. The structural IP describes the inner characteristics of the system and enabled [3] the fixing of the relation between the FI (the channel capac- ity) I and the structural information (SI) Q [7]. The inter- esting point is that a lot of calculations is performed when the so called physical information K is partitioned equally into I and Q (or with the factor 1/2), having the total value equal to zero. The method is fruitful as Frieden derived 2 Let for a system the square distance between two of its states denoted by q(x) and q′(x) = (q +dq)(x) could be written in the Euclidean form ds2 = R X dx dq dq where X is the space of the positions x (in one measurement). Supposing that the states are described by the probability distrib

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