The Sample Complexity of Simple Binary Hypothesis Testing

The sample complexity of simple binary hypothesis testing is the smallest number of i.i.d. samples required to distinguish between two distributions p and q in either: (i) the prior-free setting, with type-I error at most α and type-II error at most β; or (ii) the Bayesian setting, with Bayes error at most δ and prior distribution (α,1−α). This problem has only been studied when α=β (prior-free) or α=1/2 (Bayesian), and the sample complexity is known to be characterized by the Hellinger divergence between p and q, up to multiplicative constants. In this paper, we derive a formula that characterizes the sample complexity (up to multiplicative constants that are independent of p, q, and all error parameters) for: (i) all 0≤α,β≤1/8 in the prior-free setting; and (ii) all δ≤α/4 in the Bayesian setting. In particular, the formula admits equivalent expressions in terms of certain divergences from the Jensen–Shannon and Hellinger families. The main technical result concerns an f-divergence inequality between members of the Jensen–Shannon and Hellinger families, which is proved by a combination of information-theoretic tools and case-by-case analyses. We explore applications of our results to robust and distributed (locally-private and communication-constrained) hypothesis testing.