Editing Typical set (section)

==Applications of typicality==
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===Typical set encoding===
{{see|Shannon's source coding theorem}}
In [[information theory]], typical set encoding encodes only the sequences in the typical set of a stochastic source with fixed length block codes. Since the size of the typical set is about ''2''<sup>nH(X)</sup>, only ''nH(X)'' bits are required for the coding, while at the same time ensuring that the chances of encoding error is limited to ε. Asymptotically, it is, by the AEP, lossless and achieves the minimum rate equal to the entropy rate of the source.

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===Typical set decoding===
In [[information theory]], typical set decoding is used in conjunction with [[random coding]] to estimate the transmitted message as the one with a codeword that is jointly ε-typical with the observation. i.e.
:<math>\hat{w}=w \iff (\exists w)( (x_1^n(w),y_1^n)\in A_{\varepsilon}^n(X,Y)) </math>
where <math>\hat{w},x_1^n(w),y_1^n</math> are the message estimate, codeword of message <math>w</math> and the observation respectively. <math>A_{\varepsilon}^n(X,Y)</math> is defined with respect to the joint distribution <math>p(x_1^n)p(y_1^n|x_1^n)</math> where <math>p(y_1^n|x_1^n)</math> is the transition probability that characterizes the channel statistics, and <math>p(x_1^n)</math> is some input distribution used to generate the codewords in the random codebook.

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===Universal null-hypothesis testing===
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===Universal channel code===
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{{See also|algorithmic complexity theory}}