 Solution

Within the merged third observed cell, the probability of belonging to the original fourth cell isThe allocation therefore has the conditional binomial distributionIts conditional expectation and the filled-in missing counts arewith $y_1^{(t)}=x_1$ and $y_2^{(t)}=x_2$.

Taking the conditional expectation of the complete log-likelihood gives the EM for merged multinomial cells E-step:This is the parameter-dependent part of $\log L(y^{(t)}\mid \theta )$. More precisely, $Q(\theta \mid {\theta }^{(t)})=\log L(y^{(t)}\mid \theta )+C_t^{\prime }$, where the constant is independent of the candidate $\theta$, interpreting factorials of noninteger filled counts through the gamma function. The expected log-factorials are not generally the log-factorials of expected counts, so literal equality including the normalization would be false. Equality up to this constant is exactly what the M-step needs. Filling in conditional means works here because the candidate-parameter terms are linear in the missing counts; it is not a universal rule for the expectation-maximization algorithm.