deletions | additions       diff --git a/However_the_previous_definitions_are__.tex b/However_the_previous_definitions_are__.tex  index 3699191..86ea259 100644  --- a/However_the_previous_definitions_are__.tex  +++ b/However_the_previous_definitions_are__.tex 
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However the previous definitions are well nigh impossible to use in practice. Hopefully it can be proved that the Mutual Information between $X$ and $Y$ can be expressed as:  \[ MI(X,Y) = \sum_{(x,y) \in E_1\times E_2}P_{(X,Y)}(x,y)log\Bigg(\frac{P_{(X,Y)}(x,y)}{P_{X}(x)P_{Y}(y)}\Bigg) \]  The interpretation of this form is that it measures the distance between the joint distribution and the joint distribution in case of independence between $X$ and $Y$. So it is a measure of \textit{dependence} between two distribution distributions  (or random variables). \subsubsection{Properties}  Mutual information has the following properties: 
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$MI(X,Y) \leq S(Y) $  The amount of information shared by two random variable variables  cannot be greater than the information contained in one of thosesingle one  random variables. variable.  \item $MI(X,Y) \geq 0$  The uncertainty about $X$ cannot be increased by learning about $Y$.  \item $MI(X,Y) = 0$ iff $X$ and $Y$ are independent.  When $X$ and $Y$ are not in any way related, no information is gained about one of the random variables variable  when the other is known. \end{enumerate}  For more information about mutual information the reader can refer to \cite{Pluim_2003}. And for extra information about multivariate mutual information he can refer to \cite{srinivasa2005review}.