= 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, gives
$$
\boxed{\widehat P_j=\frac{(C_j+1)-1}{(C_j+1)+(U_j+1)-2}
=\frac{C_j}{C_j+U_j}.}
$$
This 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.
Back to article page
