Missing-count EM for capture-recapture ($\mathbb{E}[n_{ij}\mid m,\varphi ,P]=N_i{w}_{ij}/C_i$)

Let $N_i=R_i-{\sum }_{s=i+1}^J{m}_{is}$ count individuals never recaptured after release $i$. Partition them into death-interval counts $n_{ij}$ and terminal survivors $n_{iJ}$. Their unnormalized weights are $w_{ij}=(1-{\varphi }_j){\prod }_{\ell =i}^{j-1}{\varphi }_{\ell }(1-P_{\ell +1})$ for $j<J$, and $w_{iJ}={\prod }_{\ell =i}^{J-1}{\varphi }_{\ell }(1-P_{\ell +1})$. Conditional counts are multinomial with normalizer $C_i={\sum }_{j=i}^J{w}_{ij}$. The EM algorithm replaces all missing counts by the displayed conditional expectations, then maximizes the resulting complete-data log likelihood. Survival and detection updates are expected successes divided by expected opportunities. A flat observed-likelihood ridge remains nonidentifiable even if each conditional complete-data maximization is explicit.