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Entropy lies between zero and the logarithm of the order

Lax263307.EntropyBounds · concepts/Lax263307/EntropyBounds.lean · lax-263307

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    Natural Language Statement

    Theorem

    For every probability vector on nn vertices,

    0≤H(π)≤log⁡2n.0\le H(\pi)\le\log_2 n.
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    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.
    Evidence

    Each proof establishes this claim relative to its assumptions.

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    1import Lax263307.Entropy
    2
    3/-!
    4---
    5title: Entropy lies between zero and the logarithm of the order
    6type: theorem
    7---
    8For every probability vector on nn vertices,
    90≤H(π)≤log⁡2n.0\le H(\pi)\le\log_2 n.
    10
    11# Attribution
    12
    13Bonnet–Chang, proof of Lemma 12. The upper bound follows from concavity
    14of −xlog⁡x-x\log x, with equality at the uniform distribution.
    15-/
    16
    17namespace Lax263307.EntropyBounds
    18
    19open Entropy
    20
    21/-- The entropy bounds used to limit successive conditioning steps. -/
    22axiom entropy_bounds {n : ℕ} (π : Fin n → ℝ) (probability : IsProbability π) :
    23 0 ≤ entropy π ∧ entropy π ≤ Real.logb 2 n
    24
    25end Lax263307.EntropyBounds
    26
    Show Proof

    Attribution

    Bonnet–Chang, proof of Lemma 12. The upper bound follows from concavity of −xlog⁡x-x\log x, with equality at the uniform distribution.

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