The method of the likelihood and the Fisher information in the construction of physical models

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 (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Information channel capacity in the field theory estimation

“…At every point, P Θ , the d × ddimensional observed Fisher information (FI) matrix can be defined [13,5,8]:…”

“…where i := P (Θ) N n=1 iFnn is the channel information density [5,8,9]. The channel information capacity I F n is invariant under the smooth invertible mapping Y → X, where X is the new variable [16].…”

“…The central quantity of EPI analysis is the information channel capacity I, which is the trace of the expectation value of the I F matrix. Fundamentally, it enters into the EPI formalism as the second order coefficient in the Taylor expansion of the log-likelihood function [5]. Originally, EPI was proposed by Frieden and Soffer [2].…”

“…In [7] two information principles were also postulated, but with a different interpretation of the structural information, which is denoted by Q. In [5] the derivations, firstly of the observed and secondly of the expected structural information principle from basic principles was given. It was based on the analyticity of the logarithm of the likelihood function, which allows for its Taylor expansion in the neighborhood of the true value of the vector parameter Θ, and on the metricity of the statistical space S of the system.…”

“…The fact that the formalism of the information principles is used for the derivation of the distribution generating equation signifies that 2 the microcanonical description of the thermodynamic properties of a compound system has to meet the analyticity and metricity assumptions as well. Thus, it is obvious from the former work that both the form of I and its density [5,8,9] play crucial roles in the construction of a particular physical model. Frieden, Soffer along with Plastino and Plastino [2] put into practice the solution of the differential information principles for various EPI models.…”