deletions | additions       diff --git a/However_the_previous_definitions_are__.tex b/However_the_previous_definitions_are__.tex  index b5b4533..874e397 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 k=joint distribution in case of independence between $X$ and $Y$^. So it is a measure of \textit{dependence} between two distribution (or random variables).  \subsubsection{Properties}  Mutual information as the following properties:  \[MI(X,Y) = MI(Y,X) (symmetry) \]  \[MI(X,X) = S(X) \]  The amount of information a random variable shared with itself is simply the entropy of $X$.  \[MI(X,Y) \leq S(X),  MI(X,Y) \leq S(Y) \]  The amount of information shared by two random variable cannot be greater than the information contained in one of those single one random variables.  \[