The likelihood-ratio event depends only on the type $R=\widehat P_n$ and is
$$
\sum_xR(x)\log\frac{P(x)}{Q(x)}\geq0,
$$
equivalently $D(R\Vert Q)\geq D(R\Vert P)$. This constraint defines a closed subset of the finite probability simplex, so Sanov's upper bound gives the exponent
$$
C(P,Q)=\inf_{R:\,D(R\Vert Q)\geq D(R\Vert P)}D_e(R\Vert Q).
$$
It is strictly positive: the only distribution with zero divergence from $Q$ is $R=Q$, but $Q$ violates the constraint because $0=D(Q\Vert Q)<D(Q\Vert P)$. Compactness and continuity under full support keep the infimum away from zero. This exponent is the <Chernoff information> between $P$ and $Q$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 224 / 2 / b / Solution

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The Chernoff information between $P$ and $Q$ is
$$
C(P,Q)=-\min_{0\leq s\leq1}\log\sum_xP(x)^sQ(x)^{1-s}.
$$
It is the optimal exponential error rate for Bayesian testing of two independent identically distributed simple hypotheses.


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