David Ellerman - New Foundations for Information Theory: Logical Entropy and Shannon Entropy
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- Book:New Foundations for Information Theory: Logical Entropy and Shannon Entropy
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Information is based on distinctions, differences, distinguishability, and diversity. Information sets are defined that express the distinctions made by a partition, e.g., the inverse-image of a random variable so they represent the pre-probability notion of information. Then logical entropy is a probability measure on the information sets, the probability that on two independent trials, a distinction or dit of the partition will be obtained.
The formula for logical entropy is a new derivation of an old formula that goes back to the early twentieth century and has been re-derived many times in different contexts. As a probability measure, all the compound notions of joint, conditional, and mutual logical entropy are immediate. The Shannon entropy (which is not defined as a measure in the sense of measure theory) and its compound notions are then derived from a non-linear dit-to-bit transform that re-quantifies the distinctions of a random variable in terms of bitsso the Shannon entropy is the average number of binary distinctions or bits necessary to make all the distinctions of the random variable. And, using a linearization method, all the set concepts in this logical information theory naturally extend to vector spaces in generaland to Hilbert spaces in particularfor quantum logical information theory which provides the natural measure of the distinctions made in quantum measurement.
Relatively short but dense in content, this work can be a reference to researchers and graduate students doing investigations in information theory, maximum entropy methods in physics, engineering, and statistics, and to all those with a special interest in a new approach to quantum information theory.
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To the memory of Gian-Carlo Rotamathematician, philosopher, mentor, and friend.
This book presents a new foundation for information theory where the notion of information is defined in terms of distinctions, differences, distinguishability, and diversity. The direct measure is logical entropy which is the quantitative measure of the distinctions made by a partition. Shannon entropy is a transform or re-quantification of logical entropy that is especially suited for Claude Shannons mathematical theory of communications. The interpretation of the logical entropy of a partition is the two-draw probability of getting a distinction of the partition (a pair of elements distinguished by the partition) so it realizes a dictum of Gian-Carlo Rota: 
. Andrei Kolmogorov suggested that information should be defined independently of probability, so logical entropy is first defined in terms of the set of distinctions of a partition and then a probability measure on the set defines the quantitative version of logical entropy. We give a history of the logical entropy formula that goes back to Corrado Ginis 1912 index of mutability and has been rediscovered many times. In addition to being defined as a (probability) measure in the sense of measure theory (unlike Shannon entropy), logical entropy is always non-negative (unlike three-way Shannon mutual information) and finitely-valued for countable distributions. One perhaps surprising result is that in spite of decades of MaxEntropy efforts based maximizing Shannon entropy, the maximization of logical entropy gives a solution that is closer to the uniform distribution in terms of the usual (Euclidean) notion of distance. When generalized to metrical differences, the metrical logical entropy is just twice the variance, so it connects to the standard measure of variation in statistics. Finally, there is a semi-algorithmic procedure to generalize set-concepts to vector-space concepts. In this manner, logical entropy defined for set partitions is naturally extended to the quantum logical entropy of direct-sum decompositions in quantum mechanics. This provides a new approach to quantum information theory that directly measures the results of quantum measurement.
After Gian-Carlo Rotas untimely death in 1999, many of us who had worked or coauthored with him set about to develop parts of his unfinished work. My own work focused on developing his initial efforts on the logic of equivalence relations or partitions. Since the notion of a partition on a set is the category-theoretic dual to notion of a subset of a set, the first product was the logic of partitions (see the Appendix) that is dual to the Boolean logic of subsets (usually presented as propositional logic). Rota had additionally emphasized that the quantitative version of subset logic was logical probability theory and that probability is to subsets as information is to partitions. Hence the next thing to do in this Rota-suggested program was to develop the quantitative version of partition logic as a new approach to information theory. The topic of this book is this logical theory of information based on the notion of the logical entropy of a partitionthat then transforms in a uniform way to the well-known entropy formulas of Shannon. The book is dedicated to Rotas memory to acknowledge his influence on this whole project.
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