Characterize sample complexity in the weak detection regime for general priors

Characterize the sample complexity n_B(p, q, alpha, alpha(1 − gamma)) for Bayesian simple binary hypothesis testing in the weak detection regime (delta = alpha(1 − gamma)) for arbitrary discrete distributions p and q and priors alpha ∈ (0, 1/2]. Establish tight, constant-factor expressions that capture the dependence on alpha, gamma, and the divergences between p and q.

Background

Beyond the uniform prior, the weak detection regime presents additional subtleties: for non-uniform priors, the sample complexity may exhibit behavior that is independent of gamma for small gamma, and the known upper and lower bounds differ by a factor of gamma. The paper provides bounds and illustrative examples but emphasizes that a complete characterization remains unresolved.

The authors’ main results address regimes where delta is at most alpha/4 and reduce certain sublinear error regimes to linear ones, but they explicitly leave the weak detection regime (delta approaching alpha from below) open for future resolution.