Mutual Information second order expansion is the Pearson's chi-square statistic

We show that MI connects subtly and elegantly the two best-known state-of-the-art independence test statistics: the $G^2$ and the Pearson's chi-square statistic $χ^2$. Furthermore, we show that the MI connects directly those statistics by an elegant formula arising from a stochastic Taylor expansion of MI ($δ$-method). MI second order term is precisely $χ^2$ up to a scale factor. As a consequence, by this connection, the difference between $G^2$ and $χ^2$ can be explicitly quantified.