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Background information

According to Cover's text book on information theory Mutual Information is calculated as:

$$(1) I(W;C) = \Sigma_k\Sigma_i P(C_k,W_i)log(\frac{P(C_k,W_i)}{P(C_k)P(W_i)}) $$

Now if we were interested in a single Word W_i we would just calculate this with one sum:

$$(2) I(W_i;C) = \Sigma_k P(C_k,W_i)log(\frac{P(C_k,W_i)}{P(C_k)P(W_i)}) $$

OR we can rewrite this as

$$ (3) I(W_i;C) = \Sigma_k P(C_k,W_i)log(\frac{P(W_i|C_k)}{P(W_i)}) $$

because $$ (4) P(W_i|C_k) = \frac{P(W_i,C_k)}{P(C_k)}$$ (unless i am incorrect)

Question While reading a paper on feature selection using information theory measurements I came across A segment of the text:

enter image description here

Should the equation be more like (3), and I should contact the Author, or have I mis-understood/interpreted mutual information?

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  • $\begingroup$ I don't see any difference between Prof. Cover's equation $(3)$ and the Author's equation $(2)$ from the paper. $\endgroup$ – Sudarsan Sep 12 '13 at 23:46
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No there is nothing wrong with the formulas in the paper.

  • MI(S) is the average mutual information of set S over all classes c in C. It is NOT a reformulation of the formula for MI(W,C).

  • MI(w) is identical to your formula for I(Wi,C) (except it uses slightly different notation).

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