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An error exponent is the limiting exponential decay rate of an error probability $p_n$ as sample size $n$ grows. Different constraints on the Type I error and Type II error give different optimal rates. The Chernoff information describes the optimal average-error rate for simple Bayesian hypotheses; Stein's lemma gives the optimal type-II rate when type-I error is held below a fixed nontrivial bound.

Stein's lemma states that for testing $P^{\otimes n}$ against $Q^{\otimes n}$, the smallest Type II error ${\beta }_n$ among tests with Type I error at most any fixed $0<\epsilon <1$ satisfies

Stein's lemma states that for distinct probability mass functions $P,Q$ on a finite alphabet, the best exponential decay rate of the Type II error ${\beta }_n=Q^{\otimes n}(B_n)$ among tests whose Type I error ${\alpha }_n=P^{\otimes n}(B_n^c)$ is eventually at most any fixed $ϵ\in (0,1)$ is

For the direct part, define the information-density typical regionThe weak law of large numbers under $P$ gives ${\alpha }_n\to 0$, while

For the converse, letAgain $P^{\otimes n}(C_n)\to 1$. Any test with $P^{\otimes n}(B_n)\ge 1-ϵ$ satisfiesOn $C_n$, $Q^{\otimes n}\ge 2^{-n(D+\delta )}{P}^{\otimes n}$, henceThus ${\mathrm{limsup}}_n-n^{-1}{\log }_2{\beta }_n\le D+\delta$. Letting $\delta \downarrow 0$ proves optimality.

Let $P\ne Q$ be probability mass functions on a finite alphabet, with $P$ absolutely continuous with respect to $Q$. A decision region $B_n$ accepts the null law $P^{\otimes n}$. Writefor its type-I and type-II errors. Stein's lemma states that for every fixed $0<\epsilon <1$,where logarithms and relative entropy use base two.

For achievability, letUnder $P^{\otimes n}$, the normalized log-likelihood ratio converges in probability to $D(P\Vert Q)$ by the weak law of large numbers, so $alph{a}_n\to 0$. On $B_n$, $Q^{\otimes n}\le 2^{-n(D(P\Vert Q)-\eta )}{P}^{\otimes n}$, whence ${\beta }_n\le 2^{-n(D(P\Vert Q)-\eta )}$.

For the converse, letAgain $P^{\otimes n}(A_n)\to 1$. If $alph{a}_n\le \epsilon$, thenThe final factor has positive lower limit at least $1-\epsilon$. Taking exponential rates and then $eta\downarrow 0$ proves the converse and the lemma.