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 $\phi : \closedint 0 1 \to \R _{\ge 0}$ is defined by:
 * $\map \phi x := \begin {cases}

0 & : x = 0 \\ -x \map \ln x & : x \in \hointl 0 1 \end {cases}$

Here $\ln$ denotes the natural logarithm.

Also see

 * Definition:Entropy of Finite Partition
 * Definition:Conditional Entropy of Finite Sub-Sigma-Algebra