  Three steps in the development of the maximum likelihood (ML) method are
presented. At first, the application of the ML method and Fisher information
notion in the model selection analysis is described (Chapter 1). The
fundamentals of differential geometry in the construction of the statistical
space are introduced, illustrated also by examples of the estimation of the
exponential models. At second, the notions of the relative entropy and the
information channel capacity are introduced (Chapter 2). The observed and
expected structural information principle (IP) and the variational IP of the
modified extremal physical information (EPI) method of Frieden and Soffer are
presented and discussed (Chapter 3). The derivation of the structural IP based
on the analyticity of the logarithm of the likelihood function and on the
metricity of the statistical space of the system is given. At third, the use of
the EPI method is developed (Chapters 4-5). The information channel capacity is
used for the field theory models classification. Next, the modified Frieden and
Soffer EPI method, which is a nonparametric estimation that enables the
statistical selection of the equation of motions of various field theory models
(Chapter 4) or the distribution generating equations of statistical physics
models (Chapter 5) is discussed. The connection between entanglement of the
momentum degrees of freedom and the mass of a particle is analyzed. The
connection between the Rao-Cramer inequality, the causality property of the
processes in the Minkowski space-time and the nonexistence of tachions is
shown. The generalization of the Aoki-Yoshikawa sectoral productivity
econophysical model is also presented (Chapter 5). Finally, the Frieden EPI
method of the analysis of the EPR-Bhom experiment is presented. It differs from
the Frieden approach by the use of the information geometry methods.
