= Missing-count EM for capture-recapture
{title2=$\mathbb E[n_{ij}\mid m,\phi,P]=N_iw_{ij}/C_i$}
Let $N_i=R_i-\sum_{s=i+1}^Jm_{is}$ count individuals never recaptured after release $i$. Partition them into death-interval counts $n_{ij}$ and terminal survivors $n_{iJ}$. Their unnormalized weights are $w_{ij}=(1-\phi_j)\prod_{\ell=i}^{j-1}\phi_\ell(1-P_{\ell+1})$ for $j<J$, and $w_{iJ}=\prod_{\ell=i}^{J-1}\phi_\ell(1-P_{\ell+1})$. Conditional counts are <multinomial> with normalizer $C_i=\sum_{j=i}^Jw_{ij}$. 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.
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