The Sample Complexity of Simple Binary Hypothesis Testing

Understand

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 $\alpha$ and type-II error at most $\beta$; or (ii) the Bayesian setting, with Bayes error at most $\delta$ and prior distribution $(\pi, 1-\pi)$.

• This problem has only been studied when $\alpha = \beta$ (prior-free) or $\pi = 1/2$ (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 $0 \le \alpha, \beta \le 1/8$ in the prior-free setting; and (ii) all $\delta \le \pi/4$ in the Bayesian setting.
• In particular, the formula admits equivalent expressions in terms of certain divergences from the Jensen--Shannon and Hellinger families.