Sanov's theorem

In information theory, Sanov's theorem gives a bound on the probability of observing an atypical sequence of samples from a given probability distribution.

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 $x^n = x_1, x_2, \ldots, x_n$. Further, let us ask that the empirical distribution, $\hat{p}_{x^n}$, of the samples falls within the set A -- formally, we write $\{x^n : \hat{p}_{x^n} \in A\}$. Then,

$q^n(x^n) \le (n+1)^{|X|} 2^{-nD_{\mathrm{KL}}(p^*||q)}$,

where

• $q^n(x^n)$ is shorthand for $q(x_1)q(x_2) \cdots q(x_n)$, and
• $p^*$ is the information projection of q onto A.

In words, the probability of drawing an atypical distribution is proportional to the KL distance 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,

$\lim_{n\to\infty}\frac{1}{n}\log q^n(x^n) = -D_{\mathrm{KL}}(p^*||q).$

References

• Cover, Thomas M.; Thomas, Joy A. (2006). Elements of Information Theory (2 ed.). Hoboken, New Jersey: Wiley Interscience. p. 362. 

• Sanov, I. N. (1957) "On the probability of large deviations of random variables". Mat. Sbornik 42, 11–44.