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By Solomon Kullback

Hugely important textual content experiences the logarithmic measures of knowledge and their program to trying out statistical hypotheses. issues comprise creation and definition of measures of knowledge, their courting to Fisher's info degree and sufficiency, basic inequalities of knowledge concept, even more. a variety of labored examples and difficulties. References. thesaurus. Appendix.

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Note that the coarser grouping of a sufficient partitioning is as informative for discrimination as the finer grouping of the space %. In terms of the concept that a statistic is a partitioning of % into sets of equivalent x's [Lehmann (1950b, pp. 2 is satisfied. This is consistent with the original criterion of sufficiency introduced by R. A. Fisher (1922b, p. 316): "the statistic chosen should summarise the whole of the relevant information supplied by the sample," and further developments, for example, by Fisher (1925a, b), Neyman (1935), DuguC (1936a, b), Koopman (1936), Pitman (1936), Darmois (1945), Halmos and Savage (1949), Lehmann and Scheffi (1950), Savage (1954), Blackwell and Girshick (1954, pp.

Cf. 1 will play an important part in subsequent applications to testing statistical hypotheses. 1 (and its consequences) and the classical information inequality of the theory of estimation in sections 5 and 6. 2. MINIMUM DISCRIMINATION INFORMATION Suppose thatf,(x) and f,(x) are generalized densities of a dominated set of probability measures on the measurable space ( X , 9') so,that (see sections 2, 4, and 7 of chapter 2) a ( ~=)LL(x) c/i(x), E E 9, i = I, 2. 1 of chapter 2), it is clear that we must impose some additional restriction on f,(x) if the desired "nearest" probability measure is to be some other than the probability measure p, itself.

4. ( x ) - ) ( ) d l ( ) f*(x) ( f-*',,q~~))2dl(x)=l. 5. I f ~ (:T(x) x = 8) # 1 , then O(T) ir a strict@ increasing fwrction of T and log M2(r) is strictly convex. 6. If O(0) = ~T(x)fe(x) dl(%) = J y g 2 Q dy(y), then O(0) = M,'(O), M2(0) = 1, Of(0)= E((y - 8(0))21~ = 0 ) = var ( y l ~= 0). 7, I f 8 = M2'(7(8)) and ~ M2(7(8)) (:q x ) = 8) # 1, then and 7(8) ir a strictly increasing function of 0. 8. 1(* :2) = Or(@- log M2(7(8))2 0, with equality i f and only i/ ~ ( 8=) 0, that Q 8 = 8(0) = Jyg2(y)d y O .

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