  About forty years ago, in a now--seminal contribution, Rosenbaum & Rubin
(1983) introduced a critical characterization of the propensity score as a
central quantity for drawing causal inferences in observational study settings.
In the decades since, much progress has been made across several research
fronts in causal inference, notably including the re-weighting and matching
paradigms. Focusing on the former and specifically on its intersection with
machine learning and semiparametric efficiency theory, we re-examine the role
of the propensity score in modern methodological developments. As Rosenbaum &
Rubin (1983)'s contribution spurred a focus on the balancing property of the
propensity score, we re-examine the degree to which and how this property plays
a role in the development of asymptotically efficient estimators of causal
effects; moreover, we discuss a connection between the balancing property and
efficient estimation in the form of score equations and propose a score test
for evaluating whether an estimator achieves balance.
