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 right hand side 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

Sources

• 2013: Peter Walters: An Introduction to Ergodic Theory (4th ed.) $4.3$: Conditional Entropy