12. (i) Let X have probability density plJ(x) with 8 one of the values 81" " , 8n , and consider the problem of determining the correct value of 8, so that the choice lies between the n decisions dl = 81 " , . , d; = 8n with gain a(8;) if

d, = 8; and 0 otherwise. Then the Bayes solution (which maximizes the average gain) when 8 is a random variable taking on each of the n values with probability l /n coincides with the maximum-likelihood procedure. (ii) Let X have probability density plJ(x) with 0 s 8 s 1. Then the maximum-likelihood estimate is the mode (maximum value) of the a posteriori density of e given x when e is uniformly distributed over (0,1) .