Weighted Chernoff information and optimal loss exponent in context-sensitive hypothesis testing

Abstract

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 $$ L_n^* = \exp\{-n D_C^{\mathrm{w}}(\mathbb{P}, \mathbb{Q}) + o(n)\}, \quad n \to \infty, $$ where $D_C^{\mathrm{w}}$ is the weighted Chernoff information. The single-letter form of the exponent relies on a structural assumption that the weight factorises across observations, $\varphi(x_1^n) = \prod_{i=1}^n \varphi(x_i)$; 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 $\varphi p^\alpha q^{1-\alpha}$ 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.