Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 47 / 6 / c / Solution

With complete counts fixed, the likelihood factor for $P_j$ is $P_j^{C_j}(1-P_j{)}^{U_j}$. As a function of $P_j\in (0,1)$, normalize it to the density of a beta distribution with parameters $C_j+1,U_j+1$. Normalization does not change its maximizing argument. The standard beta-mode formula, when both counts are positive, givesThis uses the standard distribution result without differentiating the likelihood, and introducing the normalized beta kernel does not require adopting a Bayesian prior. If only $C_j$ is zero the maximum is at zero; if only $U_j$ is zero it is at one; if both vanish every probability maximizes the constant factor. For the EM M step the same beta-mode argument applies to the expected, possibly noninteger counts, since positive real beta parameters are allowed.