Some Statistical Observations on a Cooperative Study of Human Pulmonary Pathology. Ii.

the place of s8 is taken by an arbitrarily fixed number s' which is put for so in equation (7). For the channel of the present section Theorems 8 and 9 then hold verbatim with C4 replaced by C(. For a channel slightly more general than that of this section a noncomputable definition of C5 was given by Blackwell, Breiman, and Thomasian,8 who proved a coding theorem and weak converse. (Brief proofs of these are now available, for the coding theorem is a special case of Theorem 2, and the weak converse is a special case of Theorem 4.) The new results of this section are the definition of C5, the result corresponding to equation (6), and the strong converse. By the same methods by which the present results follow, one could treat the channel where equation (7) is replaced by an expression analogous to (4) in all the previously received letters. Moreover, one can combine the results of the present and immediately preceding sections to obtain the corresponding results when the probability which corresponds to equations (4) and (7) involves all the previously received and transmitted letters in the analogous manner. The formal proofs of these are entirely routine.