Given $(X,Y)\sim p_{XY}$ on $\mathscr{X}\times\mathscr{Y}$, the mutual information between $X$ and $Y$, denoted by $I(X;Y)$, is given by $$\begin{align*} I(X;Y):&=D(p_{XY}\|p_Xp_Y)\\ &=E_{p_{XY}}\left[\log_2\left(\frac{p_{XY}(X,Y)}{p_X(X)p_Y(Y)}\right)\right]\\ &=\sum_{a\in\mathscr{X}}\sum_{b\in\mathscr{Y}}p_{XY}(a,b)\log_2\frac{p_{XY}(a,b)}{p_X(a)p_Y(b)} \end{align*}$$

Let $(X,Y,Z) \sim p_{XYZ}$ on $\mathscr{X}\times\mathscr{Y}\times\mathscr{Z}$, the conditional mutual information between $X$ and $Y$ given $Z$ is: $$\begin{align*} I(X;Y|Z):&=D(p_{XY|Z}\|p_{X|Z}p_{Y|Z}|p_Z)\\ &=\sum_{a\in\mathscr{X}}\sum_{b\in\mathscr{Y}}\sum_{c\in\mathscr{Z}}p_{XY|Z}(a,b|c)p_Z(c)\log_2\frac{p_{XY|Z}(a,b|c)}{p_{X|Z}(a|c) \ p_{Y|Z}(b|c)}\\ &=\sum_{c\in\mathscr{Z}}p_Z(c)\sum_{a\in\mathscr{X}}\sum_{b\in\mathscr{Y}}p_{XY|Z}(a,b|c)\log_2\frac{p_{XY|Z}(a,b|c)}{p_{X|Z}(a|c) \ p_{Y|Z}(b|c)}\\ &= \mathbb{E}_{z\sim P_{Z}}[D(p_{XY|z}\Vert p_{X|z}p_{Y|z})] \end{align*}$$

Let $(X,Y)\sim f_{XY}$ with $S_{X}\subset\mathbb{R}^{2}$. Then the mutual information between $X$ and $Y$ is $$\begin{align*} I(X;Y):&=D(f_{X}\|f_{y})\\ &=\int_{S_{X}}f_{XY}(x,y)\log_2\frac{f_{XY}(x,y)}{f_{X}(x)f_{Y}(y)}dxdy\\ &=h(X)+h(Y)-h(X,y)\\ &=h(X)-h(X|Y)\\ &=h(Y)-h(Y|X) \end{align*}$$