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Exploratory Data Analysis Using Fisher Information Reviews

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Summary: What can Fisher information tell us?
Comment: B.R. Frieden and R.A. Gatenby (Eds), Exploratory Data Analysis using Fisher Information (Springer, London 2007)
For some years now, Roy Frieden has been exploring the consequences of studying physical phenomena on the basis of Fisher information and extreme physical information (EPI). From the very beginning, the results were spectacular. From the slenderest beginnings, many of the fundamental equations of physics emerged from these EPI principles: the Klein-Gordon and Dirac equations of quantum mechanics as well as the Schrödinger equation; Newton's second law; Maxwell's equations; many of the equations of general relativity; and this does not exhaust the list. These ideas, gradually developed in a series of publications in very respectable and severely refereed scientific journals, were brought together in Physics from Fisher Information (1998) and its successor, Science from Fisher Information (2004).
It was clear from that work that the approach should not be limited to physics but the extent to which it has shown itself fruitful, charted in Frieden's latest book, is a revelation. This is not a monograph but a collection of essays, edited by Frieden and R.A. Gatenby, a life scientist, on a very wide range of topics, all of which are shown to benefit from the use of EPI. The book begins with an introduction by Frieden, in which the reader is told what Fisher information is and how to use it, employing the EPI approach. Eight chapters follow, contributed by the editors and 11 other authors, on financial economics (Frieden, R.J. Hawkins and and J.L. d'Anna); tissue growth and cancer (by the editors); statistical mechanics and `thermal physics' - not very different from what I was taught to call thermodynamics (A. and A.R. Plastino); astrobiology (by Frieden and B.H. Soffer), which is described as a unification of biology and astrophysics; encryption (R.C. Venkatesan); the management of sustainable environmental systems (A.L. Mayer, C.W. Pawlowski, B.D. Fath and H. Cabezas); ecology (by the editors); and to conclude, `Sociohistory: an information theory of social change' (M.I. Yolles and Frieden).
This makes for a very adventurous book, all of which makes fascinating reading though some chapters are more readable than others and occasionally, the authors seem unnecessarily on the defensive, as though they expect readers to have a red pencil at the ready. The list of chapters already gives a good idea of the diversity of the contents and even within individual chapters, the coverage is often surprising; thus Chapter 7 (Environmental systems) ends with a section on `Sociopolitical data', in which "state failure", the risk of a "catatastrophic collapse of a nation's governing body" is examined and illustrated with a histogram showing the stability of five countries, Sweden, France, Argentina, Sierra Leone and Haiti. The Fisher index based on eight criteria is very high (indicating great stability) for Sweden, low for Argentina, Sierra Leone and Haiti and only marginally better for France (in the years between 1961 and 1995)! The concluding chapter (Sociohistory) is the most difficult for readers from the exact sciences, unaccustomed to Kant's notion of the noumenon, the Hegelian doctrine of the dialectic and the autonomous holon, though the authors have tried hard to render the vocabulary of the sociologist palatable.
The very different nature of the topics examined makes it less easy to appreciate the remarkable role of EPI than in the earlier books, addressed to physicists in language with which they were familiar, however revolutionary the theory presented. I imagine that readers of this latest offering will peruse only the chapter that deals with their own particular interest. I therefore wish to emphasize that the truly original feature of this book - apart from EPI itself - is precisely its broad coverage; its demonstration that such a simple principle, easily grasped, is capable of yielding valuable results in such a wide range of fields of enquiry. I found Frieden's earlier books immensely original and intellectually thrilling and this one adds yet more weight to that opinion.
P.W. Hawkes
(M.A., Ph.D., Sc.D., Cambridge; emeritus Director of Research, CNRS)

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Summary: Fisher information forever
Comment: In a simple-minded way Fisher information can be expressed in one dimension
as an integral of (P') squared over P where P is a suitable probability density.
For example P could be the square modulus of a quantum wave function.
Many action principles for physical systems in quantum mechanics or
relativity involve extremizing a Lagrangian which contains such Fisher information
(FI) terms. The book applies such ideas to a huge variety of physical, biological,
economic, ecological, social, game theoretical, and informational systems.
One uses FI in a unified approach to statistically based science called
EPI (extreme physical information). This leads to a program (EPA) of
exploratory data analysis whose inputs are real or Gedanken data and whose
outputs are the natural laws governing a system. The results often appear in the form of
differential equations. Here one thinks of the
universe as information-dominated and "participatory", of Harrison type,
allowing maximum
information gain at each observation and "favoring" the intelligent
observation of information.

One speaks of three levels of solution for EPI,
depending on the three levels of prior knowledge categorized by the
19th century philosopher C. Pierce. These are
(A) The highest level
or "abduction", giving exact (quantum) solutions; (B) The next highest level
or "deduction", giving accurate but inexact (non-quantum) solutions of
classical physics; and
(C) The lowest level or "induction" using merely empirical data giving approximate
but smooth solutions.
The exact type (A) solutions of EPI require a measurement space connected
via unitary transformations with some other space having a physical reality.
In this event one arrives via EPI at the correct dynamical equations for the
measurement space. There are two earlier books in these directions:
(1) Physics from Fisher information, Cambridge Univ. Press, 1998 and
(2) Science from Fisher information, Cambridge Univ. Press, 2004, both
by B.R. Frieden.

The present book is is a collaboration by H. Cabezas, J.L. D'Anna, B.D. Fath,
B.R. Frieden, R.A. Gatenby, R.J. Hawkins, A.L. Mayer, C. Pawlowski, A. Plastino,
A.R. Plastino, B.H. Soffer, R.C. Venkatesan, and M. Yolles. The topics include (1)
A tutorial on FI and background mathematics plus chapters on
(2) Financial economics from FI, (3) Growth characteristics of organisms,
(4) Information theory and thermal physics, (5) Parallel information phenomena
in biology and astrophysics, (6) Encryption of covert information through a
Fisher game, (7) Applications of FI to the management of sustainable
environmental systems, (8) FI in ecological systems, and (9) Sociohistory:
An information theory of social change.

There is much to reflect on here and strong evidence that this is indeed the
way to go. I personally have used and exploited the Fisher information theme
in numerous papers related to quantum mechanics and relativity, in particular
via relations of FI to the quantum potential. I would even say that this theme
seems to have "cosmic significance" and in the present arena of information
technology, processing, retrieval, and distortion the book should be considered as
must reading.

Robert Carroll, Emeritus Professor, University of Illinois

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