Rényi and Tsallis entropy monotonicity for Bernoulli sums

For independent Bernoulli variables, does increasing each success probability within [0,1/2] increase the Rényi and Tsallis entropies of their sum for every entropy order 0≤q≤2? The endpoints include support entropy at q=0 and Shannon entropy at q=1.

Why it matters and what counts as progress

Why it matters: The conjecture identifies the full entropy-order range of probability-monotonicity, including boundary probabilities; joint concavity asks something different.

What counts as progress: Prove probability-monotonicity throughout the order range, including zero and fair probabilities. Joint concavity is a separate question.

How the problem developed

2019 · Conjecture [1]
Aurélien Hillion and Oliver Johnson Posed a conjecture on Rényi and Tsallis entropy monotonicity as Bernoulli success parameters increase, extending known Shannon-entropy results.
17 Oct 2026 · Theorem [2]
JD Jones Proved the parameter-monotonicity claim for Rényi and Tsallis entropies of Bernoulli sums for orders 0≤q≤2 when all success probabilities stay in [0,1/2].

AI contribution

AI-led proof of Rényi and Tsallis entropy monotonicity for Bernoulli sums

For every finite family of independent Bernoulli variables, increasing each success probability from pᵢ to rᵢ with 0 ≤ pᵢ ≤ rᵢ ≤ 1/2 cannot decrease either the Rényi or Tsallis entropy of their sum, for any fixed entropy order 0 ≤ q ≤ 2. The statement includes zero and fair probabilities, support entropy at q=0 and Shannon entropy at q=1.

AI involvementAI-led

ImportanceNotable

Technical verificationFormal (Lean)

Human understandingFew