Lax Lean Archive

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Proof of `Entropy lies between zero and the logarithm of the order`

groundedproofs/Lax263307Proofs/EntropyBounds.lean · lax-263307

What this proof establishes

no assumptions

thm✓Lax263307.EntropyBounds

Assuming the claims on the left, the claim on the right holds — checked by the archive's pipeline. Proof code is not displayed here.

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Description

The entropy of a probability vector lies between zero and the logarithm of the number of vertices.

Proof strategy

Each entropy summand is nonnegative. Jensen's inequality for the concave function negMulLognegMulLog bounds their sum by the entropy of the uniform vector. Divide by the positive logarithm of two to obtain the bound in bits.

Attribution

Bonnet–Chang, proof of Lemma 12.