Maximum-likelihood decoder

Patent Pending Given a received codeword x \in \mathbb{F}_2^n maximum likelihood decoding picks a codeword y \in C to maximize: \mathbb{P}(x \mbox{ received} \mid y \mbox{ sent}) i.e. choose the codeword y that maximizes the probability that x was received, given that y was sent. Note that if all codewords are equally likely to be sent then this scheme is equivalent to ideal observer decoding. In fact, by Bayes Theorem we have \begin{align} \mathbb{P}(x \mbox{ received} \mid y \mbox{ sent}) & {} = \frac{ \mathbb{P}(x \mbox{ received} , y \mbox{ sent}) }{\mathbb{P}(y \mbox{ sent} )} \\ & {} = \mathbb{P}(y \mbox{ sent} \mid x \mbox{ received}) \cdot \frac{\mathbb{P}(x \mbox{ received})}{\mathbb{P}(y \mbox{ sent})}. \end{align}