Suggest edit — mutual information

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---
title: "mutual information"
type: concept
related: [Information Theory, Kl Divergence, Collocation Extractio, Random Variables, Mutual Information]
source: https://www.jemoka.com/posts/kbhmutual_information/
confidence: high
status: active
---

mutual information a measure of the dependence of two random variables in information theory. Applications include collocation extraction, which would require finding how two words co-occur (which means one would contribute much less entropy than the other.)
constituents \(X, Y\) random variables \(D_{KL}\) KL Divergence function \(P_{(X,Y)}\) the joint distribution of \(X,Y\) \(P_{X}, P_{Y}\) the marginal distributions of \(X,Y\) requirements mutual information is defined as
\begin{equation} I(X ; Y) = D_{KL}(P_{ (X, Y) } | P_{X} \otimes P_{Y}) \end{equation}
which can also be written as:
\begin{equation} I(X:Y) = H(X) + H(Y) - H(X,Y) \end{equation}
where \(H\) is entropy (recall that \(H(X,Y) \leq H(X)+H(Y)\) always.
&ldquo;mutual information between \(X\) and \(Y\) is the additional information contributed by the &quot;
additional information mutual information and independence if \(X \perp Y\), then \(H(X)+H(Y) = H(X,Y)\), so \(I(X:Y) = 0\) otherwise, we have \(I(X:Y) &gt; 0\) This is important because