By Alfred Renyi

Info conception

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**Extra info for A diary on information theory**

**Example text**

When A and B are independent. If A and B are independent, then the observation of 5 will not change the unexpectedness of A, while in the case where is dependence, the observation of B will decrease or increase the unexpectedness of A depending on which of P ( ^ 5 ) and P ( J ) P ( 5 ) is greater. ,M). We want to see by how much, on the average, the value of the unexpectedness of ^ will change with the observation of t]. In other words, we want to calculate the expected value of V{Ak, Bj). We obtain: (11) 2 2P(ABj)v(A,B^)=^ 2 2p(ABj)log, P;;7P7^ .

Denote as ^ the (a, ^) pair and as rj the (p, y) pair. Obviously, a, p and y contain 1 bit of information each, and ^ and t] contain two bits each, while the observation of the pair (^, rj) is equivalent to the observation of a, P, y (P twice but that is unimportant for the present) and so H{(^, ri))= 0. Finally, /(^, ri)=l since if we observe the outcome of ^, then we know what values a and p have assumed, of which a gives no information about the independent r], while p supplies 1 bit (knowing p from the t] = ip, y) pair, it is only y which remains unknown).

If they are not independent, then the observation of t] will contain some information about ^, too. The professor joked that from this we can conclude that whatever we learn at the University, we can only end up smarter and not stupider since in the worst case, it will only be a zero amount of information that we get out of our studies. e. , full information of ^ by observing ;; (which will dissolve the H(0 uncertainty about ^ completely). , I(^,0==H(0c) /(<^, »;)=/(»;, ^), meaning that the observation of tj gives as much information about ^ as the observation of ^ about rj.