Sanov's theorem

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:

${\displaystyle q^{n}({\hat {p}}_{x^{n}}\in A)\leq (n+1)^{|X|}2^{-nD_{\mathrm {KL} }(p^{*}||q)}}$,

where

• ${\displaystyle q^{n}}$ is the joint probability distribution on ${\displaystyle X^{n}}$, and
• ${\displaystyle p^{*}}$ is the information projection of q onto A.

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 the closure of its interior,

${\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\log q^{n}({\hat {p}}_{x^{n}}\in A)=-D_{\mathrm {KL} }(p^{*}||q).}$