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Conditional KL for supplied probability kernels

Definition

Let $(Z, Z)$ and $(E, E)$ be standard-Borel spaces, let $μ$ be a probability measure on $Z$ , and let $K, L : Z ⇝ E$ be supplied probability kernels. Define their conditional KL divergence with base $μ$ by $D_{μ} (K ∥ L) : = \int_{Z} D (K (z, \cdot) ∥ L (z, \cdot)) μ (d z) \in [0, + \infty] .$ The integrand is measurable by Conditional KL is measurable for supplied kernels, so this is a well-defined nonnegative extended integral. No absolute-continuity assumption is imposed on individual fibers; a singular fiber has divergence $+ \infty$ . Changing supplied kernel versions on a $μ$ -null set does not change the integral.

Depends on

Used by

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Sources

Robert M. Gray, Entropy and Information Theory, corrected author-hosted first edition, Chapter 5 §5.3 (standard reference, not scraped)
Yury Polyanskiy and Yihong Wu, MIT 6.441 Information Theory, Chapter 2 §§2.2 and 2.6 (standard reference, not scraped)