Editing Channel capacity (section)

=== Main results on feedback capacity ===
Let <math>X</math> and <math>Y</math> be modeled as random variables. The [[causal conditioning]] <math>P(y^n||x^n) \triangleq \prod_{i=1}^n P(y_i|y^{i-1},x^{i})</math> describes the given channel. The choice of the [[causally conditional distribution]]  <math>P(x^n||y^{n-1}) \triangleq \prod_{i=1}^n P(x_i|x^{i-1},y^{i-1})</math> determines the [[Joint probability distribution|joint distribution]] <math>p_{X^n,Y^n}(x^n,y^n)</math> due to the chain rule for causal conditioning<ref name="2008.2009849">{{cite journal |last1=Permuter |first1=Haim Henry |last2=Weissman |first2=Tsachy |last3=Goldsmith |first3=Andrea J. |date=February 2009 |title=Finite State Channels With Time-Invariant Deterministic Feedback |journal=IEEE Transactions on Information Theory |volume=55 |issue=2 |pages=644–662 |arxiv=cs/0608070 |doi=10.1109/TIT.2008.2009849 |s2cid=13178}}</ref> <math>P(y^n, x^n) = P(y^n||x^n) P(x^n||y^{n-1})</math> which, in turn, induces a [[directed information]] <math>I(X^N \rightarrow Y^N)=\mathbf
E\left[ \log \frac{P(Y^N||X^N)}{P(Y^N)} \right]</math>.

The '''feedback capacity''' is given by

: <math>\ C_{\text{feedback}} = \lim_{n \to \infty} \frac{1}{n} \sup_{P_{X^n||Y^{n-1}}} I(X^n \to Y^n)\, </math>,

where the [[Infimum and supremum|supremum]] is taken over all possible choices of <math>P_{X^n||Y^{n-1}}(x^n||y^{n-1})</math>.