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Dimensions of Uncertainty in Communication Engineering is a comprehensive and self-contained introduction to the problems of nonaleatory uncertainty and the mathematical tools needed to solve them. The book gathers together tools derived from statistics, information theory, moment theory, interval analysis and probability boxes, dependence bounds, nonadditive measures, and Dempster–Shafer theory. While the book is mainly devoted to communication engineering, the techniques described are also of interest to other application areas, and commonalities to these are often alluded to through a number of references to books and research papers. This is an ideal supplementary book for courses in wireless communications, providing techniques for addressing epistemic uncertainty, as well as an important resource for researchers and industry engineers. Students and researchers in other fields such as statistics, financial mathematics, and transport theory will gain an overview and understanding on these methods relevant to their field. Uniquely brings together a variety of tools derived from statistics, information theory, moment theory, interval analysis and probability boxes, dependence bounds, nonadditive measures, and Dempster—Shafer theory Focuses on the essentials of various, wide-ranging methods with references to journal articles where more detail can be found if required Includes MIMO-related results throughout Contents Notations and symbols Acronyms and abbreviations Bibliography Preface 1 Model selection 1.1 Parametric models 1.2 Wireless channel models 1.2.1 Rayleigh fading 1.2.2 Rice fading 1.2.3 Nakagami-m fading 1.3 Parameter estimation 1.3.1 A word of caution 1.4 Differential entropy and Kullback–Leibler divergence 1.4.1 Differential entropy 1.4.2 Kullback–Leibler divergence 1.5 KKT optimality conditions 1.6 Choosing the best model: the maximum-entropy principle 1.6.1 Maximizing entropy with order-statistics constraints 1.6.2 Spherically invariant processes 1.7 Choosing the best model in a set: Akaike Information Criterion 1.8 Choosing the best model in a set: minimum description length criterion Sources and parerga 2 Performance bounds from epistemic uncertainty 2.1 Model robustness 2.2 Performance optimization with divergence constraints 2.3 Scenarios of uncertainty 2.3.1 Defining the constraints 2.3.1.1 Constraints on P 2.3.1.2 Constraints on h 2.3.1.3 Presence of unknown unknowns 2.4 Concentration-of-measure inequalities 2.5 Some applications 2.5.1 P unknown, h known 2.5.2 A trivial bound 2.5.3 Some moments of X known, h known 2.5.4 McDiarmid constraint sets 2.5.5 Hoeffding constraint sets 2.6 The certification problem 2.6.1 Empirical measures 2.6.2 Certification under epistemic uncertainty Sources and parerga 3 Moment bounds 3.1 Some classical results 3.2 The general moment-bound problem 3.3 Calculation of moments 3.3.1 Using multinomial expansions 3.3.2 Using moment-generating functions 3.3.3 Using recursive computations 3.4 Moments of unimodal pdfs 3.4.1 Moment transfer 3.5 When is a sequence a valid moment sequence? 3.5.1 Probability measures represented by moments 3.5.1.1 The truncated moment problem 3.5.1.2 Probability measures determined by moments 3.6 Moments in a parallelepiped 3.7 Geometric bounds 3.7.1 Two-dimensional case 3.7.2 Application to spectrum sensing 3.7.2.1 Coherent sensing 3.7.2.2 Energy sensing 3.8 Quadrature-rule approximations and bounds 3.8.1 The algebraic moment problem 3.8.2 Čebyšev form of orthogonal polynomials 3.8.3 Recursive generation of orthogonal polynomials 3.8.4 Numerical generation of orthogonal polynomials 3.9 Čebyšev systems and principal representations 3.9.1 Principal and canonical representations as quadrature rules 3.9.1.1 Gauss–Radau quadrature rules 3.9.1.2 Gauss–Lobatto quadrature rules 3.9.2 Moment bounds and quadrature rules 3.9.3 Bounds on CDFs and quadrature rules 3.9.4 Bounds on Laplace–Stieltjes transform 3.9.5 Extension to nondifferentiable functions: the method of contact polynomials 3.10 Moment problems as semidefinite programs 3.10.1 Moment problem in a parallelepiped 3.10.2 Generalized Čebyšev bounds 3.11 Multidimensional moment bounds and approximations 3.11.1 Multidimensional moment bounds 3.11.1.1 Multidimensional Čebyšev inequality 3.11.1.2 Method of contact polynomials 3.11.1.3 Method of contact polynomials: the bivariate case 3.11.1.4 The role of cubature rules Sources and parerga 4 Interval analysis 4.1 Some definitions 4.2 Set operations on intervals 4.3 Algebraic operations on interval numbers 4.3.1 Properties that may not be shared with ordinary real algebra 4.4 Interval vectors and matrices 4.5 Interval functions 4.5.1 The vertex method 4.6 The interval dependence problem 4.7 Integrals 4.8 Choosing a representative in an interval Sources and parerga 5 Probability boxes 5.1 Interval probabilities 5.2 Generating probability boxes 5.3 Aggregating probability boxes 5.4 Combining probability boxes of random variables 5.5 Using probability boxes in performance evaluation Sources and parerga 6 Dependence bounds 6.1 Copulas 6.1.1 Copulas and joint CDFs 6.1.2 Some explicit copula families 6.1.3 Fréchet–Hoeffding inequalities 6.1.4 A version of Fréchet–Hoeffding bounds involving probabilities 6.1.5 Dual of a copula and the survival copula 6.1.6 d-Dimensional copulas 6.2 Dependence p-boxes from copula bounds 6.2.1 Special operations 6.2.2 Using p-boxes of FX and FY 6.2.3 Operations on d RVs 6.2.4 Dependence bounds with order statistics 6.3 Some examples of application 6.4 Bivariate dependence bounds on expectations 6.5 Bounds with monotone marginal densities 6.5.1 Tail-monotone marginal densities 6.6 Deriving tighter dependence bounds 6.6.1 Degree of dependence 6.6.1.1 Using qualitative knowledge about dependence 6.6.1.2 Degree of dependence as an interval number 6.6.2 Linear correlation 6.6.3 Rank correlation: Kendall tau, Spearman rho, and Blomqvist beta 6.6.3.1 Kendall tau 6.6.3.2 Spearman rho 6.6.3.3 Blomqvist beta 6.6.4 Tail dependence 6.6.5 Quadrant/orthant dependence Sources and parerga 7 Beyond probability 7.1 Decisions under uncertainty 7.2 Epistemic vs. aleatory uncertainty 7.2.1 The Principle of Insufficient Reason 7.2.2 Objective vs. subjective probabilities 7.2.3 Using interval probabilities 7.3 Lotteries, prospects, and utility functions 7.3.1 Prospects 7.3.2 St. Petersburg paradox 7.3.3 Risk-averse and risk-seeking decisions 7.3.4 Reversing risk aversion: framing effect 7.4 Other paradoxes arising from EUT 7.4.1 Ellsberg paradox 7.4.2 Allais paradox 7.4.3 Entropy-modified expected utility of gambling 7.5 Upper and lower probabilities 7.5.1 Conditional upper and lower probabilities 7.5.2 Inference using upper and lower probabilities 7.5.3 The dilation phenomenon 7.6 Expected utility with interval probabilities 7.7 Some applications to digital communication 7.7.1 Optimum decisions: Bayes criterion 7.7.2 A case in which error probability does not tell the whole story 7.7.3 Neyman–Pearson criterion 7.7.3.1 Neyman–Pearson criterion with interval probabilities 7.8 Going beyond probability 7.9 Set functions and their properties 7.10 Infinite sets 7.11 Capacities and Choquet integral 7.12 Expected values and Choquet integral 7.13 Dempster–Shafer theory 7.13.1 Bayesian belief functions 7.13.2 Dempster rule of combination Sources and parerga Index