Economics of disagreement -- financial intuition for the Rényi divergence
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Lack of accurate intuition is often cited as a scientific challenge, especially when interpreting probabilistic and statistical research. A popular technique for developing statistical intuition involves imagining a game of chance with well-defined financial outcomes. In 1956 Kelly used this technique to propose an intuitive interpretation of relative entropy [1]. He considered a growth-optimizing investor in a game with mutually exclusive outcomes (a "horse race") and showed that the rate of return expected by such an investor is equal to the relative entropy which measures disagreement between the investor's believed probabilities and the official odds. Effectively, Kelly showed that a growth-optimizing investor would expect on average to convert 1 bit of additional information into a 100 mathematics, this interpretation has been vigorously criticized by Samuelson as dangerous on the grounds that most people are not growth-optimizing [2,3]. Here we show that a deeper connection between information and expected returns is in fact true. Relative to the Kelly information benchmark, variation in people's risk aversion does indeed cause a drop in expected returns. However, the amount of the drop is also information-driven: it is proportional to the absolute difference between the relative entropy and its celebrated generalization -- the Rényi divergence [4]. Given the widespread use of the Rényi divergence in science and engineering, and because financial returns are much easier to imagine, we expect our intuition to be useful in a broad range of fields. Financial intuition might also be useful in its own right as an instrument for raising financial support and accelerating the adoption of sound scientific developments.
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