Exact sample complexity in the weak detection regime under a uniform prior

Determine the exact minimal number of i.i.d. samples n required for Bayesian simple binary hypothesis testing under a uniform prior (alpha = 1/2) to achieve Bayes error delta = (1 − gamma)/2 for small gamma ∈ (0,1). Specifically, establish n_B(p, q, 1/2, (1 − gamma)/2) for arbitrary discrete distributions p and q, refining the current bounds gamma^2 / h^2(p, q) ≲ n_B(p, q, 1/2, (1 − gamma)/2) ≲ gamma / h^2(p, q), where h^2(p, q) denotes the Hellinger divergence.

Background

The paper tightly characterizes sample complexity for Bayesian simple binary hypothesis testing across several regimes of error probability, including linear, sublinear, and asymptotic regimes. However, in the weak detection regime—where the desired Bayes error is delta = alpha(1 − gamma) with gamma close to 0—the behavior is notably different.

For the uniform prior (alpha = 1/2), the authors highlight that existing results only provide bounds differing by a factor of gamma, indicating that the dependence on gamma is not precisely understood. The current best-known inequalities show that n_B scales between gamma^2 / h^2(p, q) and gamma / h^2(p, q), but the exact characterization remains elusive.