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Quantitative Finance · Begrippenlijst

Wat is Entropy, mutual information?

Ook bekend als: entropy · mutual information

Definition 29.10 Quantitative Methods · Hoofdstuk 29 — Games, Auctions and Information

The entropy of a discrete random variable XX is H(X)=−∑xp(x)log⁡2p(x)H(X) = -\sum_xp(x)\log_2p(x) bits (Shannon, 1948). The mutual information of XX and YY is I(X;Y)=∑x,yp(x,y)log⁡2p(x,y)p(x)p(y)=H(X)−H(X∣Y)I(X; Y) = \sum_{x,y}p(x, y)\log_2\frac{p(x, y)}{p(x)p(y)} = H(X) - H(X \mid Y), the Kullback–Leibler divergence of the joint law from the product of its marginals.

Gain in the Kelly bettor’s doubling rate from a noisy tip, against the tip’s accuracy, in a four-horse race: the mutual information and the gain measured over 200 000 simulated races for each accuracy. Data: the chapter’s tutorial, seeded.
Figure 29.4. Gain in the Kelly bettor’s doubling rate from a noisy tip, against the tip’s accuracy, in a four-horse race: the mutual information and the gain measured over 200 000 simulated races for each accuracy. Data: the chapter’s tutorial, seeded.
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