Han’s conjecture on distributed testing against dependence

Let $Q_{XY}$ be a joint distribution on finite alphabets, and consider distributed hypothesis testing with null hypothesis $\mathcal{H}_0: P_{XY}=Q_XQ_Y$ and alternative hypothesis $\mathcal{H}_1: P_{XY}=Q_{XY}$ . An encoder observing $X^n$ sends a message of rate at most $R$ to a decoder, which observes $Y^n$ and decides between the hypotheses. For every fixed type-I error constraint, let $\theta(R)$ denote the optimal type-II error exponent. Han's conjecture asserts that

\theta(R)=\max_{\substack{U-X-Y\\ I_Q(U;X)\le R}} L_Q(U;Y),

where $Q_{UXY}=Q_{XY}Q_{U|X}$ and $L_Q(U;Y)=D(Q_UQ_Y\,\|\,Q_{UY})$ is the lautum information. The August 2026 paper cited in the source claims that this conjectured characterization is false and gives a different exact single-letter characterization.

References

Primary source

arXiv

Additional references

Distributed Hypothesis Testing Against Dependence — arXiv

distributed-hypothesis-testing information-theory

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A new August 2026 paper claims to settle the exact error-decay rate in Han’s problem and shows that the conjectured information formula is wrong, but the claim has not been independently verified.

Han’s conjecture concerns the exact exponent for distributed testing against dependence, where the null model asserts independence. The conjecture proposes a characterization using lautum information; the latest paper claims this characterization is false and replaces it with an exact single-letter formula.

August 2026 claimed resolution

A paper titled Distributed Hypothesis Testing Against Dependence claims to prove the exact exponent by single-letterizing Han’s exponent and establishing matching upper and lower bounds. It also treats product and conditional-dependence variants, but the conditional-dependence result is only a converse bound and is tight in some cases.

Current status (as of August 2026): The central exponent is claimed to be settled and Han’s lautum-information conjecture claimed to be disproved, but the paper’s result remains unverified; the conditional-dependence extension is not fully settled.

Sources

No solutions have been posted yet.