Value of information, Bayes risks, and rate-distortion theory

In the previous post we have seen that access to additional information is not always helpful in decision-making. On the other hand, extra information can never hurt, assuming one is precise about the quantitative meaning of “extra information.” In this post, I will show how Shannon’s information theory can be used to speak meaningfully about the value of information for decision-making. This particular approach was developed in the 1960s and 1970s by Ruslan Stratonovich (of the Stratonovich integral fame, among other things) and described in his book on information theory, which was published in Russian in 1975. As far as I know, it was never translated into English, which is a shame, since Stratonovich was an extremely original thinker, and the book contains a deep treatment of the three fundamental problems of information theory (lossless source coding, noisy channel coding, lossy source coding) from the viewpoint of statistical physics.

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