TITLE: Sanov's Theorem and Chernoff bound 
QUESTION [8 upvotes]: Sanov's Theorem (p.292, Thomas/Cover "Elements of Information Theory" (1991)) says that probability of a hypothesis $E$ according to distribution $Q$ is bounded above by 
$$(n+1)^k \exp (-n D(P^* \|Q) )$$ where $D$ is KL-divergence, $k$ is the size of sample space and $P^*$ is an element of $E$ closest to $Q$ according $D(\cdot\|Q)$.
This bound is quite loose for small $n$ because when $k=2$ and $E$ is simply connected, Chernoff's bound tells us that the probability is at most the following
$$\exp (-n D(P^* \|Q) )$$
Is there a natural restriction of $E$ that makes the bound above generalize to $k>2$? Alternatively, is there a natural restriction of $E$ and a modification of Sanov's theorem that gives it a similar level of tightness to Chernoff's bound for small n?

REPLY [4 votes]: For convex sets E of probability measures, the bound you want (i.e., probability of empirical measure being in E is smaller than $e^{-nD(E\|Q)}$) is true in great generality, not just for finite alphabet settings. This is contained, for instance, in the spectacular 1984 paper of Csiszar titled "Sanov Property, Generalized $I$-Projection and a Conditional Limit Theorem":
http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.aop/1176993227 ,
which also contains (when you reduce it to its essence) a beautifully short and elegant proof.