  We contribute a general and flexible framework to estimate the size of a
closed population in the presence of $K$ capture-recapture lists and
heterogeneous capture probabilities. Our novel identifying strategy leverages
the fact that it is sufficient for identification that a subset of the $K$
lists are not arbitrarily dependent \textit{within the subset of the population
unobserved by the remaining lists}, conditional on covariates. This
identification approach is interpretable and actionable, interpolating between
the two predominant approaches in the literature as special cases:
(conditional) independence across lists and log-linear models with no
highest-order interaction. We derive nonparametric doubly-robust estimators for
the resulting identification expression that are nearly optimal and
approximately normal for any finite sample size, even when the heterogeneous
capture probabilities are estimated nonparametrically using machine learning
methods. Additionally, we devise a sensitivity analysis to show how deviations
from the identification assumptions affect the resulting population size
estimates, allowing for the integration of domain-specific knowledge into the
identification and estimation processes more transparently. We empirically
demonstrate the advantages of our method using both synthetic data and real
data from the Peruvian internal armed conflict to estimate the number of
casualties. The proposed methodology addresses recent critiques of
capture-recapture models by allowing for a weaker and more interpretable
identifying assumption and accommodating complex heterogeneous capture
probabilities depending on high-dimensional or continuous covariates.
