Definition:Conditional Entropy of Finite Sub-Sigma-Algebra

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

Let $\AA, \BB \subseteq \Sigma$ be finite sub-$\sigma$-algebras.

The (conditional) entropy of $\AA$ given $\BB$ is defined as:
 * $\ds \map H {\AA \mid \BB} := \map H {\map \xi \AA \mid \map \xi \BB}$

where:
 * $\map H {\cdot \mid \cdot}$ on the denotes the conditional entropy of finite partitions
 * $\map \xi \cdot$ denotes the generated finite partition

Also see

 * Definition:Entropy of Finite Sub-Sigma-Algebra
 * Conditional Entropy Given Trivial Sigma-Algebra is Entropy
 * Conditional Entropy Decreases if More Given
 * Conditional Entropy of Join as Sum