A discrete communication channel (with memory) is a triplet $$(\mathcal{X}, \mathcal{Y}, \{P_{Y^{n}|X^{n}}\}_{n=1}^\infty)$$with - a finite input alphabet $\mathcal{X}$ - a finite output alphabet $\mathcal{Y}$ - a sequence of $n$-dimensional transition distributions $$P_{Y^{n}|X^{n}}(b^{n}|a^{n})=P(Y^{n}=b^{n}|X^{n}=a^{n}), \ n\ge1, \ a^{n}\in\mathcal{X}^{n}, \ b^{n}\in\mathcal{Y}^{n}$$ ## Assumption We assume that the channel’s $n$-dimensional distribution is consistent (i.e. we can obtain the $i$-dimensional conditional distribution by marginalizing the $i+1$th conditional distribution): $$\begin{align*} P_{Y^{n}|X^{n}}(b^{n}|a^{n})= \frac{p_{X^{i}Y^{i}}(a^{i},b^{i})}{p_{X^{i}}(a^{i})}&=\frac{\sum\limits_{a_{i+1}\in\mathcal{X}}\sum\limits_{b_{i+1}\in\mathcal{Y}}p_{X^{i+1}Y^{i+1}}(a^{i+1},b^{i+1})}{p_{X^{i}}(a^{i})}\\\\ &=\sum\limits_{a_{i+1}\in\mathcal{X}}\sum\limits_{b_{i+1}\in\mathcal{Y}}p_{X_{i+1}|X^{i}}(a_{i+1}|a^{i})p_{Y^{i+1}|X^{i+1}}(b^{i+1}|a^{i+1}) \end{align*}$$$\forall i\ge1, \ a^i\in\mathcal{X}^i, \ b^i\in\mathcal{Y}^i, \ p_{X_{i+1}|X^{i}}$.