Past exam of the communication Theory course of the University of Oxford / 2007 / 4 / a

Maximum a posteriori decoding chooses an input with the largest posterior probability given the received word. By Bayes' theorem this maximizes $P(X=x)p(y\mid x)$ over candidate codewords, minimizing the average probability of a decoding error.

Maximum-likelihood decoding chooses a codeword maximizing $p(y\mid x)$ for the received word $y$. It uses the likelihood alone, without weighting by the prior probabilities of the codewords.

Minimum-distance decoding with Hamming distance chooses a codeword minimizing the number of positions in which it differs from the received word. Ties may be resolved by a fixed rule.

With equally likely codewords, the prior factor in the ideal-observer criterion is constant, so it agrees with maximum likelihood. On the binary symmetric channel,For $0<p<1/2$ this is strictly decreasing in Hamming distance, so the likelihood maximizers are precisely the nearest codewords. With the same tie convention all three decoders therefore agree. At $p=0$ every possible received word equals its transmitted codeword, and the rules agree on all positive-probability outputs; outputs having zero probability need no posterior assignment.