Maximum-Entropy Models in Science and Engineering
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1 Maximum-Entropy Models in Science and Engineering (Revised Edition) J. N. Kapur JOHN WILEY & SONS New York Chichester Brisbane Toronto Singapore
2 p Contents Preface iü 1. Maximum-Entropy Probability Distributions: Principles, Formalism and Techniques Measures of entropy Maximum-entropy principle Maximum-entropy formalism Proofs of some needed mathematical results Bibliographical and historical remarks Maximum-Entropy Discrete Univariate Probability Distributions When the discrete variate takes only a finite set of values When the discrete variate takes a countable infinite set of values When the prior probability distribution is given and mean is prescribed Lagrangian distributions Bibliographical and historical remarks Maximum-Entropy Continuous Univariate Probability Distributions When the ränge of the variate is ( oo, co) When the variate varies over the semi-infinite interval [0, oo) When the variate ranges over a finite interval [0, R] Bibliographical and historical Remarks Maximum-Entropy Discrete Multivariate Probability Distributions The Bose-Einstein, Fermi-Dirac and multinomial distributions A general class of discrete multivariate distributions Multivariate generalised geometrical, negative binomial, generalised negative binomial and Poisson distributions Multivariate Lagrangian distributions General discussion 110
3 viii Contents 4.6 Discrete multivariate distributions for queueing networks Bibliographical and historical remarks Maximum-Entropy Continuous Multivariate Probability Distributions Maximum-Entropy distributions obtained by specifying covariances Maximum-Entropy distributions when sumof variates is a constant Maximum-Entropy distributions obtained by also specifying expected values of functions of all variates Maximum-Entropy distribution obtained by finding distributions of function of given independent variates Some other multivariate distributions Bibliographical and historical remarks Maximum-Entropy Distributions in Statistical Mechanics Maxwell-Boltzmann distribution Bose-Einstein distribution Fermi-Dirac distribution Intermediate statistics distributions Bibliographical and historical remarks Minimum Discrepancy Measures Some general classes of measures of directed divergence or discrepancy Some special cases Minimum discriminationprobability distributions Properties of minimum discrimination functions Efficiency of minimum discrepancy estimators Generalised measures of generalised directed divergence Bibliographical and historical remarks Concavity (Convexity) of Maximum-Entropy (Minimum Information) Functions Concavity and convexity of functions Concavity and convexity of functions useful in Information theory Consequences of convexity or concavity of functions Concavity of S mix for Renyi's entropy of second order Extension of results to continuous variate distributions and density matrices Bibliographical and historical remarks 226
4 Contents ix 9. Eqaivalence of Maximum-Entropy Principle and Gauss's Principle of Density Estimation Minimum discrimination information principle Maximum likelihood estimators for the parameters of the minimum discrimination information distribution An alternative proof of equivalence and naturalness of measures Relations among six basic principles of Statistical inference Bibliographical and historical remarks Maximum-Entropy Principle and Contingency Tables Multidimensional contingency tables Contingency tables with elements as random variables The maximum-entropy principle and chi-square test Bibliographical and historical remarks Maximum-Entropy Principle and Statistics Statistical inference Jayne's entropy concentration theorem The role of MEP in statistics Non-parametric density estimation Bibliographical and historical remarks Entropy Maximization and Statistical Thermodynamics Thermodynamics of closed Systems Thermodynamics of open or diffusive systems Thermodynamic functions Bose-Einstein and Fermi-Dirac distributions Bibliographical and historical remarks Maximum-Entropy Models in Regional and Urban Planning Discrete population distribution modeis Transportation modeis Population distribution and transportation Systems (continuous modeis) Fermi-Dirac, Bose-Einstein and intermediate distributions for residential location and trip distribution Bibliographical and historical remarks Maximum-Entropy Models in Marketing and Elections Herniter's model for brand switching Generalisations of Herniter's model Maximum-Entropy brand switching and vote-switching modeis with different structures 378
5 x Contents 14.4 Vote-Switching among political parties Derivation of some purchase incidence stochastic modeis from maximum-entropy principle Bibliographical and historical remarks Maximum-Entropy Models in Economics, Finance, Insurance and Accountancy The effect of taxation on reduction of inequalities of income International trade modeis The maximum-entropy distribution of the future market price of a stock Maximum-entropy distributions in accounting and insurance Bibliographical and historical remarks Maximum-Entropy Spectral Analysis Maximum-Entropy probability distribution The spectral density function Relation of MESA with other methods of spectral analysis Multidimensional maximum-entropy spectral analysis Bibliographical and historical remarks Maximum-Entropy Image Reconstruction Some physical problems requiring image reconstruction The mathematical problems and discretisation Solutions of the problem of estimation Measures of entropy used in image reconstruction Entropiegray-level picture thresholding Bibliographical and historical remarks Maximum and Minimum-Entropy Models in Pattern Recognition Optimizing properties of Karhunen-Loeve expansion Use of measures of divergence in feature extraction Pattern recognition as a quest for minimumentropy Bibliographical and historical remarks Maximum-Entropy Principle in Operations Research Maximum-Entropy principle in search theory Maximum-Entropy principle in reliability theory 531' 19.3 Maximum-Entropy principle in queuing theory Maximum-Entropy principle in theory of games Optimal portfolio selection 544
6 Contents xi 19.6 Maximum-Entropy principle in risk sharing Maximum-Entropy principle and mathematical Programming Bibliographical and historical remarks Maximum-Entropy Models in Biology, Medicine and Agriculture Maximum-Entropy principle in pharmacokinetics (or in compartment analysis) Maximum-Entropy principle in stochastic prey-predator and epidemic modeis Maximum-Entropy principle in ecological modelling Maximum-Entropy principle in multispecies ecological modeis Maximum-Entropy estimation of missing data in design of experiments MEP derivation of logistic law of population growth Bibliographical and historical remarks 588 References Author Index Subject Index
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