In mathematics and information theory, Sanov's theorem gives a bound on the probability of observing an atypical sequence of samples from a given probability distribution. In the language of large deviations theory, Sanov's theorem identifies the rate function for large deviations of the empirical measure of a sequence of i.i.d. random variables.

Let A be a set of probability distributions over an alphabet X, and let q be an arbitrary distribution over X (where q may or may not be in A). Suppose we draw n i.i.d. samples from q, represented by the vector ${\displaystyle x^{n}=(x_{1},x_{2},\ldots ,x_{n})}$. Then, we have the following bound on the probability that the empirical measure ${\displaystyle {\hat {p}}_{x^{n}}}$ of the samples falls within the set A:

where

In words, the probability of drawing an atypical distribution is bounded by a function of the KL divergence from the true distribution to the atypical one; in the case that we consider a set of possible atypical distributions, there is a dominant atypical distribution, given by the information projection.

Furthermore, if A is a closed set, then

Define:

$${\displaystyle \{(a_{1}/n,\dots ,a_{|\Sigma |}/n):\sum _{i}a_{i}=n,a_{i}\in \mathbb {N} \}}$$Then, Sanov's theorem states:

Here, ${\displaystyle int(S)}$ means the interior, and ${\displaystyle cl(S)}$ means the closure.