Weighted Chernoff Information and Optimal Loss Exponent in Context-Sensitive Hypothesis Testing

Entropy. 2026. Vol. 28. Article 536.

Kelbert M., Kalimulina E. Y.

We study binary hypothesis testing for i.i.d. observations under a multiplicative context
weight. For the optimal weighted total loss, defined as the sum of weighted type-I and type-
II losses, we prove the logarithmic asymptotic L∗n = exp{−nDwC (P,Q) + o(n)}, n →∞, where Dw
C is the weighted Chernoff information. The single-letter form of the exponent
relies on a structural assumption that the weight factorises across observations,
φ(xn1 ) =nΠi=1φ(xi); this restriction is essential for the single-letter representation and should
be distinguished from the weaker qualitative description “multiplicative context weight”.
The proof embeds the weighted geometric mixtures φpαq1−α into a likelihood-ratio exponential
family and identifies the rate through its log-normaliser. We also derive concentration
bounds for the tilted weighted log-likelihood, obtain closed forms for Gaussian,
Poisson, and exponential models, and extend the exponent characterisation to finitely many
hypotheses

Keywords: information geometry, Context-sensitive hypothesis testing, weighted Chernoff information, weighted Bhattacharyya coefficient,