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18). Hei a) is positive or zero, and satisfies the condition lim He)a) = H(a) as s ~ 1. 20) even when a and f3 are mutually independent. More explicitly, the joint structural entropy can be written in the form m He... e. 22) with p( Bj IA; ) denoting the conditional probability of the issue Bj of f3 given Aj. 24) in such a manner that one has the inequalities He ,(a,fJ) ~ He,,,(a) + He, , (/3). : 15 INFORMATION TIIEORY which qualify the structural entropy as a valuable candidate to deal with information in the presence of sUbjectivity.

A). How can we measure this amount of information? 8. Shannon mutual information or transinformation. Following Shannon, the amount of mutual information I( a,[3) provided by a about [3 is defined by the expression I(a,[3) = H(/3) - H{[3la) . 32) H([3)+H(al[3) . /3) = 1(/3, a), therefore the expression of mutual information which is used instead of information (alone). Important remark. Many authors refer to H( [3) as a measure of information, but this is a bit confusing. 31), it is also the maximum value of the mutual information I( a,[3), therefore one is used to saying that H( [3) is the maximum value of the amount of information involved in [3.

I PN , and one can consider the N numbers {~, ~, ... , PN } as being an original alphabet which is in a one-to-one correspondence with the sequence of the initial alphabet {xi!. We then have to encode the P-alphabet, that is to say, we have to associate a sequence of bits with each number Pi, in such a manner that no mistake be possible in the corresponding identification procedure. In other words, encoding the sequence {xiJ amounts to encode the sequence {PiJ and the amount of information so involved should be the same, at least in absolute value.

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A density lemma


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