DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Differential entropy for real densities

Definition

If $X$ has Lebesgue density $f$ on $R^{d}$ and $\int f \ln f$ is a finite real number, its differential entropy in nats is $h (X) : = - \int_{R^{d}} f (x) \ln f (x) d x$ . If $(X, S)$ has conditional densities $f_{X ∣ S = s}$ for almost every $s$ and the indicated integral is finite, set $h (X ∣ S) : = \int h (f_{X ∣ S = s}) d P_{S} (s)$ . We use only these finite-integral cases; differential entropy is not defined here by a formal subtraction of possibly infinite quantities.

Facts & Assumptions

Given: A Euclidean random vector with a Lebesgue density and an absolutely finite entropy integral; for the conditional version, conditional densities and an absolutely finite averaged entropy integral.

Proof

1.1givenalgebra

The integrability hypothesis makes $\int f \ln f$ a finite real number, so the displayed negative integral is a defined real number.

2.1givenalgebraalgebra∎

In the conditional definition, almost-everywhere existence of the conditional densities and integrability of their entropy values make the outer expectation a finite real number; changing a conditional density on a $P_{S}$ -null set does not change it.

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Thomas M. Cover and Joy A. Thomas, Elements of Information Theory, 2nd ed., Chapter 8, §8.6 (standard reference, not scraped)
Abbas El Gamal and Young-Han Kim, Network Information Theory, Lecture Notes 4, Gaussian MAC converse (standard reference, not scraped)