Proof: Relation of continuous mutual information to marginal and conditional differential entropy
Theorem: Let $X$ and $Y$ be continuous random variables with the joint probability $p(x,y)$ for $x \in \mathcal{X}$ and $y \in \mathcal{Y}$. Then, the mutual information of $X$ and $Y$ can be expressed as
\[\label{eq:cmi-mcde} \begin{split} \mathrm{I}(X,Y) &= \mathrm{h}(X) - \mathrm{h}(X|Y) \\ &= \mathrm{h}(Y) - \mathrm{h}(Y|X) \end{split}\]where $\mathrm{h}(X)$ and $\mathrm{h}(Y)$ are the marginal differential entropies of $X$ and $Y$ and $\mathrm{h}(X \vert Y)$ and $\mathrm{h}(Y \vert X)$ are the conditional differential entropies.
Proof: The mutual information of $X$ and $Y$ is defined as
\[\label{eq:MI} \mathrm{I}(X,Y) = \int_{\mathcal{X}} \int_{\mathcal{Y}} p(x,y) \log \frac{p(x,y)}{p(x)\,p(y)} \, \mathrm{d}y \, \mathrm{d}x \; .\]Separating the logarithm, we have:
\[\label{eq:MI-s1} \mathrm{I}(X,Y) = \int_{\mathcal{X}} \int_{\mathcal{Y}} p(x,y) \log \frac{p(x,y)}{p(y)} \, \mathrm{d}y \, \mathrm{d}x - \int_{\mathcal{X}} \int_{\mathcal{Y}} p(x,y) \log p(x) \, \mathrm{d}x \, \mathrm{d}y \; .\]Applying the law of conditional probability, i.e. $p(x,y) = p(x \vert y) \, p(y)$, we get:
\[\label{eq:MI-s2} \mathrm{I}(X,Y) = \int_{\mathcal{X}} \int_{\mathcal{Y}} p(x|y) \, p(y) \log p(x|y) \, \mathrm{d}y \, \mathrm{d}x - \int_{\mathcal{X}} \int_{\mathcal{Y}} p(x,y) \log p(x) \, \mathrm{d}y \, \mathrm{d}x \; .\]Regrouping the variables, we have:
\[\label{eq:MI-s3} \mathrm{I}(X,Y) = \int_{\mathcal{Y}} p(y) \int_{\mathcal{X}} p(x|y) \log p(x|y) \, \mathrm{d}x \, \mathrm{d}y - \int_{\mathcal{X}} \left( \int_{\mathcal{Y}} p(x,y) \, \mathrm{d}y \right) \log p(x)\, \mathrm{d}x \; .\]Applying the law of marginal probability, i.e. $p(x) = \int_{\mathcal{Y}} p(x,y) \, \mathrm{d}y$, we get:
\[\label{eq:MI-s4} \mathrm{I}(X,Y) = \int_{\mathcal{Y}} p(y) \int_{\mathcal{X}} p(x|y) \log p(x|y) \, \mathrm{d}x \, \mathrm{d}y - \int_{\mathcal{X}} p(x) \log p(x) \, \mathrm{d}x \; .\]Now considering the definitions of marginal and conditional differential entropy
\[\label{eq:MDE-CDE} \begin{split} \mathrm{h}(X) &= - \int_{\mathcal{X}} p(x) \log p(x) \, \mathrm{d}x \\ \mathrm{h}(X|Y) &= \int_{\mathcal{Y}} p(y) \, \mathrm{h}(X|Y=y) \, \mathrm{d}y \; , \end{split}\]we can finally show:
\[\label{eq:MI-qed} \mathrm{I}(X,Y) = - \mathrm{h}(X|Y) + \mathrm{h}(X) = \mathrm{h}(X) - \mathrm{h}(X|Y) \; .\]The conditioning of $X$ on $Y$ in this proof is without loss of generality. Thus, the proof for the expression using the reverse conditional differential entropy of $Y$ given $X$ is obtained by simply switching $x$ and $y$ in the derivation.
- Wikipedia (2020): "Mutual information"; in: Wikipedia, the free encyclopedia, retrieved on 2020-02-21; URL: https://en.wikipedia.org/wiki/Mutual_information#Relation_to_conditional_and_joint_entropy.
Metadata: ID: P58 | shortcut: cmi-mcde | author: JoramSoch | date: 2020-02-21, 16:53.