Main

centropy1_uniform_fiber

File dumas2017dual/entropy_fiber/entropy_fiber.v

Section fiber_entropy

Type Lemma

Proof Lines 6

Signature

Lemma centropy1_uniform_fiber (c : CodomainT) : `Pr[Y = c] != 0 -> let fiber_c := fiber f c in (forall x, x \in fiber_c -> `Pr[X = x | Y = c] = #|fiber_c|%:R ^-1) -> (forall x, x \notin fiber_c -> `Pr[X = x | Y = c] = 0) -> (0 < #|fiber_c|)%N -> `H[X | Y = c] = log (#|fiber_c|%:R : R).

Description

If X is uniform over fiber(c) given Y=c, then H(X|Y=c) = log|fiber(c)|. Key lemma for computing conditional entropy via fiber cardinality.

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