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

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 Ln*=exp{−nDCw(P,Q)+o(n)},n→∞, where DCw is the weighted Chernoff information. The single-letter form of the exponent relies on a structural assumption that the weight factorises across observations, φ(x1n)=∏i=1nφ(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.