= Discrete memoryless channel
{title2=$W^n(y^n\mid x^n)=\prod_{j=1}^nW(y_j\mid x_j)$}
{wiki}
A <discrete memoryless channel> models transmission through finite input and output <alphabets> using fixed <conditional probabilities> $W(y\mid x)$. For each input $x$, these nonnegative numbers sum to one over the output symbols. Discrete refers to the symbols, and memoryless means that the noise of one use does not remember earlier uses.
For a fixed input word $x^n=(x_1,\ldots,x_n)$ sent without feedback, the output law is
$$
W^n(y^n\mid x^n)=\prod_{j=1}^nW(y_j\mid x_j).
$$
This is conditional independence of the outputs given the prescribed input word. It does not require an <encoder> to choose <independent> input symbols: the <codewords> of a <block code> deliberately impose relationships among them. Those relationships let a <decoder> distinguish possible messages after noise.
The <binary symmetric channel> and <binary erasure channel> make two different memoryless noise models with a binary input. One hides flipped <bits>; the other marks missing <bits>. Their different <channel capacities> show why the whole <transition matrix> matters, as the <flips versus erasures in a binary channel> demonstrate.
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