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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 . Then, we have the following bound on the probability that the empirical measure of the samples falls within the set A:
- ,
where
- is the joint probability distribution on , and
- 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 a closed set, then
Technical statement
editDefine:
- is a finite set with size . Understood as “alphabet”.
- is the simplex spanned by the alphabet. It is a subset of .
- is a random variable taking values in . Take samples from the distribution , then is the frequency probability vector for the sample.
- is the space of values that can take. In other words, it is
Then, Sanov's theorem states:[1]
- For every measurable subset ,
- For every open subset ,
References
edit- ^ Dembo, Amir; Zeitouni, Ofer (2010). "Large Deviations Techniques and Applications". Stochastic Modelling and Applied Probability. 38: 16–17. doi:10.1007/978-3-642-03311-7. ISBN 978-3-642-03310-0. ISSN 0172-4568.
- Cover, Thomas M.; Thomas, Joy A. (2006). Elements of Information Theory (2 ed.). Hoboken, New Jersey: Wiley Interscience. pp. 362. ISBN 9780471241959.
- 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.