Binary hypothesis testing in information theory

For two probability mass functions $P,Q$, a decision region $B_n$ accepts $P^{\otimes n}$. Its two error probabilities are $P^{\otimes n}(B_n^c)$ and $Q^{\otimes n}(B_n)$.

For distinct probability distributions $P,Q$, the best exponential decay rate of the type-II error $Q^{\otimes n}(B_n)$ among tests whose type-I error $P^{\otimes n}(B_n^c)$ stays below any fixed number in $(0,1)$ is $D(P\Vert Q)$.

The Neyman-Pearson lemma says that an optimal region accepting $P$ has the likelihood-ratio formwith possible randomization on the boundary.