A diary on information theory by Alfred Renyi

By Alfred Renyi

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34 ON THE MATHEMATICAL NOTION OF INFORMATION Fourth lecture Today we analyzed mutual information further. One can see easily that (1) i{^,n)=^Hi^)+H(n)~Hi^,n). From (1) and from the fact that mutual information is a non-negative entity, it follows that if t, and t] are random variables, then (2) H(a,n))^H(0+mn), with equality if and only if £, and t] are independent. (1) can be written in the form: (10 Hi(^, rj)) = H(0 + H(^)-ia, n) = H(ri) + H,{0. (1') may be looked upon as the generalization of the law of additivity of information so that, if ^ and t] are two arbitrary random variables, then, observing the value of t, and ri, we can get the information contained in these two observations if we add the information contained in the observation of t] to the conditional information contained in an observation given a particular value of (^.

The unexpectedness of A^ is log2 — Pk and the probability of A^ is p^, therefore the entropy of ^ is (9) H(0 = Pl^Og^-~+PZ^Og2--+^^^+PN^Og2'—' Pl Pi PN We have arrived at the Shannon formula again, but in a different way. This way also brings us to the concept of relative information. Let A and B be arbitrary event relating to the same experiment. If we observe the outcome of event B, this will change the unexpectedness of event A. The unexpectedness of ^ was originally loga - . , when A and B are independent.

Most students are exam-centric. But I doubt that students alone are to blame. The root of the problem lies in the present educational system — the exam-centricity originates with the university, we are just influenced by it. Whenever THIRD LECTURE 31 an evaluation is made of the work of individual students, or of various groups of students, for instance, for the purpose of selecting those who will receive scholarships or grants, the only criterion used in the evaluation seems to be the exam grades of those involved.

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