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Lucas Fidon edited subsection_MI_based_metric_for__.tex
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\subsection{MI-based metric for trajectories}
According to the previous properties of mutual information we can
now define a mutual information-based metric between two players' trajectories as:
\[d_{MI}(X,Y) = 1 - \frac{2MI(X,Y)}{S(X) + S(Y)} \]
Where $X$ and $Y$ are the positions of the two players.
Indeed it comes from the previous section that:
\begin{enumerate}
\item $d_{MI}$ is symmetric.
\item $d_{MI}(X,Y) \in [0,1]$.
\item $d_{MI}(X,Y) = 0$ iff $X$ and $Y$ have the same probability law.
\item in particular $d_{MI}(X,X) = 0$ for any trajectory.
\end{enumerate}
\subsection{Empirical MI-based metric for trajectories}