 Solution

For a received word $y$, the ideal-observer rule chooses a message $i$ maximizing the posterior probabilityMaximum-likelihood decoding chooses $i$ maximizingand minimum-distance decoding chooses a codeword minimizing its Hamming distance from $y$.

Bayes' formula givesEqual message priors therefore make ideal-observer and maximum-likelihood decoding identical. On a binary symmetric channel, if $d=d_H(y,c_i)$, thenFor $p<1/2$, this strictly decreases with $d$, so maximum likelihood and minimum distance agree.