Conditional dependence of sets of discrete random variables \(\mathbf{X}\) and \(\mathbf{Y}\), given the set \(\mathbf{Z}\), is measured via the conditional Shannon mutual information [Cover and Thomas, 1991]

\[

I(\mathbf{X}: \mathbf{Y} \mid \mathbf{Z}):=\sum_{\mathbf{x}, \mathbf{y}, \mathbf{Z}} p(\mathbf{x}, \mathbf{y}, \mathbf{z}) \log \frac{p(\mathbf{x}, \mathbf{y} \mid \mathbf{z})}{p(\mathbf{x} \mid \mathbf{z}) p(\mathbf{y} \mid \mathbf{z})}

\]