Dimensions of Uncertainty in Communication Engineering

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
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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

2.4 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
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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

3.6 Moments in a parallelepiped

3.7 Geometric bounds

3.7.1 Two-dimensional case

3.7.2 Application to spectrum sensing

3.8 Quadrature-rule approximations and bounds

3.8.1 The algebraic moment problem

3.8.2 Cebyšev form of orthogonal polynomials

3.8.3 Recursive generation of orthogonal polynomials

3.8.4 Numerical generation of orthogonal polynomials

3.9 Cebyšev systems and principal representations

3.9.1 Principal and canonical representations as 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 Cebyšev bounds

3.11 Multidimensional moment bounds and approximations

3.11.1 Multidimensional moment bounds
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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
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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
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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 probabilites

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

6.6.2 Linear correlation

6.6.3 Rank correlation: Kendall tau, Spearman rho, and Blomqvist beta

6.6.4 Tail dependence

6.6.5 Quadrant/orthant dependence
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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

7.8 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
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