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A capture-recapture model is a statistical model for repeated observations of identifiable individuals, including those missed on some occasions. A closed-population capture-recapture model estimates a fixed population cardinality; models allowing individuals to enter or leave can instead describe survival and recruitment. The sampling assumptions determine which likelihood function is appropriate.

For the two-sample closed-population capture-recapture model, condition on the sample sizes $M$ and $C$. If the second sample selects $C$ distinct individuals from $N$ by simple random sampling, then the overlap $R$ has the hypergeometric distribution:Here $r$ is an integer and the probability mass function is zero outside these bounds. The binomial coefficients count choices of $r$ marked and $C-r$ unmarked individuals, divided by all equally likely size-$C$ subsets. The hypergeometric distribution has expected valueEquating the observed overlap with this expected value gives the Lincoln-Petersen estimator. The underlying sampling without replacement, probability mass function, and expected value are developed in Raymond P. Guiteras, MIT 14.30 hypergeometric distribution handout, §§1–4 and 8.

For the toy counts $M=20$, $C=25$, and $R=5$, hold the observations fixed and vary the unknown integer $N$. The likelihood function isValues below forty cannot contain the observed set union. Exact comparison of consecutive likelihoods gives two tied maximum-likelihood estimates, $N=99$ and $N=100$. The Lincoln-Petersen estimator returns $100$; the Chapman estimator returns $90$ and is a different rule for estimation.

The vertical scale is $L(N)/{\max }_{n\ge 40}L(n)$. This relative likelihood compares how well candidate statistical parameters explain the same observations; it is not a posterior probability. The displayed range $40\le N\le 300$ is only a plotting window. The likelihood function remains positive for every integer $N\ge 40$ and tends to zero as $N$ increases, so the curve has a continuing right tail. The data do not justify truncating statistical inference at three hundred. The conditional hypergeometric distribution formulation also appears in Louis-Paul Rivest and Mamadou Yauck, Small Sample Estimators for Two-way Capture Recapture Experiments, §3, equation (5).

Zero recaptures require care. With $R=0$ and positive $M,C$, the Lincoln-Petersen estimator has no finite ratio, but the capture-recapture model still gives a valid likelihood function:This likelihood function increases toward one as $N$ grows, so there is no finite maximum-likelihood estimate. For the same sample sizes twenty and twenty-five with zero overlap, forty-five distinct individuals would be observed. Lack of overlap supplies a lower bound but very weak information about large population cardinalities.

In more elaborate capture-recapture models, a first-recapture array $m_{ij}$ counts individuals released at occasion $i$ and next observed at $j$. A probability of survival ${\varphi }_j$ and a probability of detection $P_{j+1}$ contribute a no-observation continuation factor ${\varphi }_j(1-P_{j+1})$. The resulting likelihood function depends on both survival and detection; these probabilities need not be separately identifiable at the last observation interval. Missing-count EM for capture-recapture describes one way to fit such a model, and identifiability remains a separate issue from computing an estimator output.

The consistent missing-data interpretation is that each never-recaptured individual either dies in an interval $j<J$ or survives unobserved through the final occasion. Define the known never-recaptured total and its allocation byThe terminal cell $n_{iJ}$ is the source's separately named $n_i$. The printed identity equating that cell to the total observed recaptures is inconsistent with the supplied multinomial law. In particular, with no observed recaptures its printed value would be zero even though the terminal-cell probability is positive at interior parameters. The displayed accounting identity repairs that notation and makes all sums through $J$ well-defined.

For an individual released at $i$, define the unnormalized probabilitiesThe empty product is one. Each intermediate factor means surviving an interval and avoiding its subsequent detection; the death weight additionally includes failure to survive interval $j$. Let $K_i={\sum }_{j=i}^J{w}_{ij}$. This is the probability of never being recaptured. It is the source's normalizing constant $C_i$, renamed to avoid confusion with the detection-success counts above. Conditional on no recapture, division by $K_i$ gives precisely the supplied multinomial distribution cell probabilities.

Initialize an interior parameter vector $({\varphi }^{(0)},P^{(0)})$. At iteration $r$, the E step of missing-count EM for capture-recapture computes, for every cohort and missing cell,If $N_i=0$, set every missing count to zero without dividing, even if $K_i=0$. If $N_i>0$ and $K_i=0$, that parameter vector gives zero probability to the observations and cannot be used in the E step. These expected counts sum to $N_i$, preserving each cohort's observed accounting. The E step is conditional expectation, not replacement by the modal missing counts or a single random imputation.

The expected complete-data log likelihood is linear in the missing counts, apart from count-only terms irrelevant to the new parameter values. In the M step substitute these expectations into the success/failure ratios from part (a):andIf an exposure denominator vanishes, retain any admissible value of the unidentifiable parameter. Repeat until the observed likelihood and parameter updates stabilize. The conditional normalizers in the E step are evaluated at the old parameters; maximizing a newly normalized conditional missing-data distribution instead of the expected complete-data log likelihood would be the wrong M step.

One can monitor the observed likelihood directly. The probability of a first recapture at $s>i$ isand successive death, detection and continuation alternatives partition the possibilities, giving $K_i+{\sum }_{s=i+1}^J{p}_{is}=1$. Thus, up to count-only factors,The usual conditional Jensen argument for the EM algorithm makes this likelihood nondecreasing at each exact M step. It does not guarantee a unique global maximum. In this observation scheme the final survival and detection probabilities enter through the product ${\varphi }_{J-1}{P}_J$ only, so these two probabilities cannot generally be estimated separately without a further constraint. The algorithm remains valid, but may converge to different points on the same likelihood ridge.