Estimating multinomial probabilities

Classical maximum likelihood (ML) as well as the uniformly minimum variance unbiased (UMVU) estimators of multinomial cell probabilities are given by the observed relative frequencies, Bayes estimators corresponding to symmetric Dirichlet prior distribution for p are the inflated observed relative cell frequencies of the type (n(i) + k)(M + kt)(-1). These estimators, which are more reasonable when the observed n(i)'s are 0 or very small, are justified classically by Johnson and are also reported without proof in Good. We give here a proof of Johnson's results that perhaps is easier to understand.