Information Measures

• Abstract.- Structure and Structuring.- 1 Introduction.- Science and information.- Man as control loop.- Information, complexity and typical sequences.- Concepts of information.- Information, its technical dimension and the meaning of a message.- Information as a central concept.- 2 Basic considerations.- 2.1 Formal derivation of information.- 2.2 Application of the information measure (Shannon’s information).- 2.3 The law of Weber and Fechner.- 2.4 Information of discrete random variables.- 3 Historic development of information theory.- 3.1 Development of information transmission.- 3.2 Development of information functions.- 4 The concept of entropy in physics.- The laws of thermodynamics:.- 4.1 Macroscopic entropy.- 4.2 Statistical entropy.- 4.3 Dynamic entropy.- 5 Extension of Shannon’s information.- 5.1 Rényi’s Information 1960.- 5.2 Another generalized entropy (logical expansion).- 5.3 Gain of information via conditional probabilities.- 5.4 Other entropy or information measures.- 6 Generalized entropy measures.- 6.1 The corresponding measures of divergence.- 6.2 Weighted entropies and expectation values of entropies.- 7 Information functions and gaussian distributions.- 7.1 Rényi’s information of a gaussian distributed random variable.- 7.2 Shannon’s information.- 8 Shannon’s information of discrete probability distributions.- 8.1 Continuous and discrete random variables.- 8.2 Shannon’s information of a gaussian distribution.- 8.3 Shannon’s information as the possible gain of information in an observation.- 8.4 Limits of the information, limitations of the resolution.- 8.5 Maximization of the entropy of a continuous random variable.- 9 Information functions for gaussian distributions part II.- 9.1 Kullback’s information.- 9.2 Kullback’sdivergence.- 9.3 Kolmogorov’s information.- 9.4 Transformation of the coordinate system and the effects on the information.- 9.5 Transformation, discrete and continuous measures of entropy.- 9.6 Summary of the information functions.- 10 Bounds of the variance.- 10.1 Cramér-Rao bound.- 10.2 Chapman-Robbins bound.- 10.3 Bhattacharrya bound.- 10.4 Barankin bound.- 10.5 Other bounds.- 10.6 Summary.- 10.7 Biased estimator.- 11 Ambiguity function.- 11.1 The ambiguity function and Kullback’s information.- 11.2 Connection between ambiguity function and Fisher’s information.- 11.3 Maximum likelihood estimation and the ambiguity function.- 11.4 The ML estimation is asymptotically efficient.- 11.5 Transition to the Akaike information criterion.- 12 Akaike’s information criterion.- 12.1 Akaike’s information criterion and regression.- 12.2 BIC, SC or HQ.- 13 Channel information.- 13.1 Redundancy.- 13.2 Rate of transmission and equivocation.- 13.3 Hadamard’s inequality and Gibbs’s second theorem.- 13.4 Kolmogorov’s information.- 13.5 Kullbacks divergence.- 13.6 An example of a transmission.- 13.7 Communication channel and information processing.- 13.8 Shannon’s bound.- 13.9 Example of the channel capacity.- 14 ‘Deterministic’ and stochastic information.- 14.1 Information in state space models.- 14.2 The observation equation.- 14.3 Transmission faster than light.- 14.4 Information about state space variables.- 15 Maximum entropy estimation.- 15.1 The difference between maximum entropy and minimum variance.- 15.2 The difference from bootstrap or resampling methods.- 15.3 A maximum entropy example.- 15.4 Maximum entropy: The method.- 15.5 Maximum entropy and minimum discrimination information.- 15.6 Generation of generalized entropy measures.- 16 Concludingremarks.- 16.1 Information, entropy and self-organization.- 16.2 Complexity theory.- 16.3 Data reduction.- 16.4 Cryptology.- 16.5 Concluding considerations.- 16.6 Information.- A.1 Inequality for Kullback’s information.- A.2 The log-sum inequality.- A.3 Generalized entropy, divergence and distance measures.- A.3.1 Entropy measures.- A.3.2 Generalized measures of distance.- A.3.3 Generalized measures of the directed divergence.- A.3.4 Generalized measures of divergence.- A.3.4.1 Information radius and the J-divergence.- A.3.4.2 Generalization of the R-divergence.- A.3.4.3 Generalization of the J-divergence.- A.4 A short introduction to probability theory.- A.4.1 Axiomatic definition of probability.- A.4.1.1 Events, elementary events, sample space.- A.4.1.2 Classes of subsets, fields.- A.4.1.3 Axiomatic definition of probability according to Kolmogorov.- Probability space.- A.4.1.4 Random variables.- A.4.1.5 Probability distribution.- A.4.1.6 Probability space, sample space, realization space.- A.4.1.7 Probability distribution and distribution density function.- A.4.1.8 Probability distribution density function (PDF).- A.5 The regularity conditions.- A.6 State space description.

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