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By Guiseppe Longo

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Extra info for Quantitative — Qualitative Measure of Information: Course Held at the Department of Automation and Information June–July 1972

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Pi. , + ... ~), n 1 n 1 whence immediately (3. 9, 1J,) + n (n- f) H(S') ~ - niA. tog n U, nt u. og :. 31) u.. tog m u, is upper and lower bounded as follows: H(5l) + J.. _ B:t L':2 < n n . 32) One additional element for studying the ptotic behaviour of Lemma 3. t defined by ( 3. 30) and ( 3. 34) where the equality signs hold iff the utilities u,~ are all e- qual. 34) is quite similar, Since, apart from the positive factor 1 togm ' 81 k E. 1. £. P·~ log..!... ,* s. ~ ' The Main Theorem 42 one gets B = ~.

D. n -block code, its average useful length L<:> can be rendered small er than H'"' ( 1', U)- u, 'log u.. +~"'log~"'' --------------~------~--- (3. og m Proof It is sufficient to choose vi"(1t) as to satisfy (trEA") which is always possible, and then one proceeds as usual. =1 p~ u,~ +... + t. e. 25) n n 11 + ... + u, (n)/ 39 Asympt otic Behavi our of L u. A" -E E ... ~+1 ... ~ )- k "1=1 "n= 1 P· - ... oo a ~1 ~1 . =1 •1=1 i-,_•1 · ( p~1.. · P~ .. og ( P~ 1 P~:t.. lnap. , ~" -a ~" . k k - ... ~t1-f p~ n- 1 p~nLog P•n = H(5>, 'U,) + ...

E. 1 " (3. 6) pj, 1 where the q~•S lkj, f:. u,. d i ~· " = k r::. l,. " ~ d" defined by qi. = (3. :~-=-- ~~pdlki form an auxiliary probability distribution Q . d. Q and not to the original p. d. = { p1 , p,_, •.. , p K} . This could give the im- ~ pression that the classical theory applies also when utilities are involved, and that nothing new comes out. This is only partially true, and now we shall see the details. 1 , inequality ( 1. ,_, ... 4) satisfies the following inequality (3. 8) where H(9, 'U,) iA.

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