Definition (Conditional mutual information). Let $X$ , $Y$ and $Z$ be random variables. The conditional mutual information of $X$ and $Y$ given $Z$ , denoted by $I [X : Y | Z]$ is

\begin{array}{l} \sum_{z} ℙ [Z = z] I [X | Z = z : Y | Z = z] \\ = \sum_{z} ℙ [Z = z] (H [X | Z = z] + H [Y | Z = z] - H [X, Y | Z = z]) \\ = H [X | Z] + H [Y | Z] - H [X, Y | Z] \\ = H [X, Z] + H [Y, Z] - H [X, Y, Z] - H [Z] \end{array}