Download Relative Information: Theories and Applications by Professor Guy Jumarie (auth.) PDF

By Professor Guy Jumarie (auth.)

For 4 many years, info concept has been considered virtually completely as a idea established upon the Shannon degree of uncertainty and data, frequently known as Shannon entropy. because the book of Shannon's seminal paper in 1948, the speculation has grown super quickly and has been utilized with diversified good fortune in just about all components of human recreation. at present, the Shannon details thought is a good demonstrated and built physique of data. between its most important fresh contributions were using the complementary ideas of minimal and greatest entropy in facing quite a few basic platforms difficulties reminiscent of predic­ tive structures modelling, trend acceptance, snapshot reconstruction, etc. due to the fact that its inception in 1948, the Shannon concept has been seen as a constrained details conception. It has usually been argued that the idea is in a position to dealing in basic terms with syntactic points of data, yet now not with its semantic and pragmatic features. This limit was once thought of a v~rtue through a few specialists and a vice through others. extra lately, even though, numerous arguments were made that the idea could be correctly transformed to account for semantic features of in­ formation to boot. the most convincing arguments during this regard are in­ cluded in Fred Dretske's Know/edge & circulation of data (The M.LT. Press, Cambridge, Mass., 1981) and during this booklet by way of man lumarie.

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1) could turn out to be very interesting. 1) is that the value H(X) = - 00 does not characterize a deterministic event only, but refers to any discrete probability distribution. If we consider a random variable that is continuous everywhere except at a given point Xo where one has Pr{X = x o} =f. 0, then H(X) = - 00. In our opinion, the main defect of this measure of uncertainty is not the fact that it may take on the value - 00, but rather that it cannot absolutely characterize a deterministic event.

1. Let &' := (Pl,PZ' ... E1 := (ql, qz, ... , qn) denote two complete sets of probabilities Pi = qi = 1). E1) of fJ with respect to oc is defined by the expression (I n \' I q. E1):= L. qi ln --'- . 1) 0 Motivation. The following example provides the heuristics of this definition. i) Consider a set E := {e 1 , e z , .. , , eN} of N elements ei' each characterized by its position in E. If we randomly choose one element of E, the index of this element is a random variable X whose the Shannonian entropy is H(X) = In N.

Iv) Let f3 be another uniform random experiment f3 := (B 1 , Bz , ... , Bn) and consider the new random experiment rxf3:= (AiBj, 1 ~ i ~ m, 1 ~ j ~ n) which has mn outcomes occurring with the same probability limn. Assume that rx and f3 are independent from a probabilistic standpoint, that is to say, loosely speaking, that they do not interact with one another. 1) . 2) klnn . 2 Non Uniform Random Experiments Our task now is to guess the explicit form of H(rx) for any rx:= (PI,PZ' ... , Pm); and, to this end, we shall determine the contribution of each outcome Ai to the total uncertainty.

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