From Wikipedia we can read:
In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables.
in fact, $I(X;Y)=I(Y;X)$. But we know that MI can measure also non-linear dependence. To clarify that concept, I made this Venn diagram to describe what I know about dependence, linear correlation and causality in probability and statistics.
$Y = X^2$ is an example of dependence between two RVs, that is not contained in the set of correlation, and cannot be detected by Pearson's coefficient. In that case, mutual information will be greater than zero suggesting a mutual dependence. But that's not true! I mean: Y depends on X but not viceversa.
If Mutual Information measures dependence, why is it symmetric, while dependence is not?