  Capture-recapture (CRC) surveys are used to estimate the size of a population
whose members cannot be enumerated directly. CRC surveys have been used to
estimate the number of Covid-19 infections, people who use drugs, sex workers,
conflict casualties, and trafficking victims. When $k$ capture samples are
obtained, counts of unit captures in subsets of samples are represented
naturally by a $2^k$ contingency table in which one element -- the number of
individuals appearing in none of the samples -- remains unobserved. In the
absence of additional assumptions, the population size is not identifiable
(i.e. point-identified). Stringent assumptions about the dependence between
samples are often used to achieve point-identification. However, real-world CRC
surveys often use convenience samples in which the assumed dependence cannot be
guaranteed, and population size estimates under these assumptions may lack
empirical credibility. In this work, we apply the theory of partial
identification to show that weak assumptions or qualitative knowledge about the
nature of dependence between samples can be used to characterize a non-trivial
confidence set for the true population size. We construct confidence sets under
bounds on pairwise capture probabilities using two methods: test inversion
bootstrap confidence intervals, and profile likelihood confidence intervals.
Simulation results demonstrate well-calibrated confidence sets for each method.
In an extensive real-world study, we apply the new methodology to the problem
of using heterogeneous survey data to estimate the number of people who inject
drugs in Brussels, Belgium.
