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Sanov's theorem

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Title: Sanov's theorem  
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Subject: Stochastic processes, Information theory, Martingale difference sequence, Bühlmann model, Continuous-time stochastic process
Collection: Information Theory, Probabilistic Inequalities
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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

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.


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