Sanov's theorem

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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 . Further, let us ask that the empirical distribution, , of the samples falls within the set A—formally, we write . Then,



  • is shorthand for , and
  • 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 the closure of its interior,


  • 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(84), No. 1, 11–44.
  • Санов, И. Н. (1957) "О вероятности больших отклонений случайных величин". МАТЕМАТИЧЕСКИЙ СБОРНИК' 42(84), No. 1, 11–44.