Definition:Conditional Entropy of Finite Partitions

Definition
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $\xi, \eta$ be finite partitions of $\Omega$.

The (conditional) entropy of $\xi$ given $\eta$ is defined as:
 * $\ds \map H {\xi \vert \eta} := \sum_{\substack {B \mathop \in \eta \\ \map \Pr B \mathop > 0}} \sum_{A \mathop \in \xi} \map \Pr B \map \phi {\dfrac {\map \Pr {A \cap B} } {\map \Pr B} }$

where:
 * $\map \phi x := \begin {cases}

-x \map \ln x & : x > 0 \\ 0 & : x = 0 \end {cases}$

Here $\ln$ denotes the natural logarithm.