tion vanishes, since A(1) = 0. We can then continue this partitioning by splitting the second subgroup into two new subgroups with N2 and n3 elements (n2 +n3=1 N - n1) and respective probabilities p2 = n2/N, P3 = n3/N, yielding the information decomposition: A(N) = H(P1,P2,P3) + p1A(n1) + p2A(n2) + p3A(n3). (B18) 2 It is interesting to note that the Appendix in C.E Shannon, A mathematical theory of communication. Bell Syst. Tech. J., 27 (1948), 79-423, 623-56 proceeds differently: Eqs. (B8) and (B11) are written in the alternative forms |log T/ log S - m/n < E and |A(T)/A(S)-m/n| < E, respectively. From there, it is directly concluded that |A(T)/A(S) - log T/log S < 28, which is far from obvious (as the reader might easily check). As a matter of fact, this last inequality summarizes at once the results in Eqs. (B15) and (B16) without providing any of the details. This apparent omission could be explained by the author's concern to save room in a very mathematically intensive paper or, as another possibility, to challenge the reader with the proof. Appendix B Iteration of the above partitioning into m subgroups under the condition - = N finally yields: (B19) A general definition of the function H(p1, P2, pm) is, thus, obtained from Eq. (B19): H(p1, P2, (B20) For an m-symbol source, the only function that satisfies the requirements (1) to (3) is, therefore: (B21) Appendix C (Chapter 4) Maximum entropy of discrete sources In this appendix, I show first that for a discrete source of k independent events, X = {x1 x2, , xk}, the source entropy is maximized when all events are equiprobable, corresponding to the uniform distribution. Second, I derive the discrete distribution for which entropy is maximized when there is a constraint on the mean N = (x). With this constraint, I show that in the case where the events take integer values (x1 = 0, X2 = with k 80) the entropy is maximized for the discrete exponential distribution of mean N (Bose-Einstein or thermal distribution). In the case of a discrete source of finite siz