A New Interpretation of Information Rate 



By J. L. KELLY, JR. 



(Manuscript received March 21, 1956) 



7/ the input symbols to a communication channel represent the outcomes 

 of a chance event on which hets are available at odds consistent with their 

 probabilities (i.e., "fair'' odds), a gambler can use the knowledge given 

 him by the received symbols to cause his money to grow exponentially. The 

 maximum exponential rate of growth of the gambler's capital is equal to 

 the rate of transmission of information over the channel. This result is 

 generalized to include the case of arbitrary odds. 



Thus we find a situation in which the transmission rate is significant 

 even though no coding is contemplated. Previously this quantity was given 

 significance only by a theorem of Shannon's which asserted that, with suit- 

 able encoding, binary digits coidd be transmitted over the channel at this 

 rate with an arbitrarily small probability of error. 



INTRODUCTION 



Shannon defines the rate of transmission over a noisy communication 

 channel in terms of various probabilities. This definition is given sig- 

 nificance by a theorem which asserts that binary digits may be encoded 

 and transmitted over the channel at this rate with arbitrarily small 

 probability of error. Many workers in the field of communication theory 

 have felt a desire to attach significance to the rate of transmission in 

 cases where no coding was contemplated. Some have even proceeded 

 on the assumption that such a significance did, in fact, exist. For ex- 

 ample, in systems where no coding was desirable or even possible (such 

 as radar), detectors have been designed by the criterion of maximum 

 transmission rate or, what is the same thing, minimum equivocation. 

 Without further analysis such a procedure is unjustified. 



The problem then remains of attaching a value measure to a communi- 



^ C. E. Shannon, A Mathematical Theory of Communication, B.S.T.J., 27, 

 pp. 379-423, 623-656, Oct., 1948. 



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