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How to calculate the entropy in the ID3 decision tree algorithm?

Suppose you have data: ...
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What does the product of probabilities raised to own powers used for entropy calculation quantify?

I don't know if $N(X)$ has a name or has any applicability in AI, but I can comment on how this function varies as the $H(X)$ based on your equation $$N(X) = \dfrac{1}{2^{H(X)}}$$ which looks correct ...
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