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\subsubsection{Mutual Information}  There are several equivalent ways to define Mutual Information.  The most intuitive definition is the following:  Let $X : P_1 P_{1}  \rightarrow E_1$ E_{1}$  and $Y: P_2 P_{2}  \rightarrow E_2$ E_{2}$  be two random variables, where $E_1$ $E_{1}$  and $E_2$ $E_{2}$  are two discrete probability spaces. We define the Mutual information of$ X$ of $X$  and $Y$, noted $I(X,Y)$ as:$ $I(X,Y)$, as:  \[ I(X,Y) = \sum{x \in E_1, y \in E_2}P_{(X,Y)}(x,y)*log(\frac{log(P_{(X,Y)}(x,y))}{P_{X}(x)*P_{Y}(y)}) \]