(i) Let X have probability density p(x) with one of the values 1,...,n, and consider the problem of determining the correct value of ,

(i) Let X have probability density pθ(x) with θ one of the values

θ1,...,θn, and consider the problem of determining the correct value of θ, so that the choice lies between the n decisions d1 = θ1,...,dn = θn with gain a(θi) if di = θi and 0 otherwise. Then the Bayes solution (which maximizes the average gain) when θ is a random variable taking on each of the n values with probability 1/n coincides with the maximum-likelihood procedure. (ii) Let X have probability density pθ(x) with 0 ≤ θ ≤ 1. Then the maximum-likelihood estimate is the mode (maximum value) of the a posteriori density of Θ given x when Θ is uniformly distributed over (0, 1).