In the last post I gave an introduction of Kelly’s paper where he describes optimal gambling strategies based on information received over a potentially noisy channel. Here I’ll talk about the general case, where the channel has several inputs symbols, each with a given probability of being transmitted, and which represent the outcome of some chance event. First, we need to set up some notation:

– probability that the transmitted symbol is s.

– the conditional probability that the received symbols is r given that the transmitted symbol is s.

– the joint probability that the received symbol is r and the transmitted symbol is s.

– probability that the received symbol is r.

– the conditional probability that the transmitted symbol is s given that the received symbols is r.

– the odds paid on the occurrence of s, i.e. the number of dollars returned on a one-dollar bet on s.

– the fraction of capital that the gambler decides to bet on the occurrence of s after receiving r.

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