  We introduce a novel Bayesian estimator for the class proportion in an
unlabeled dataset, based on the targeted learning framework. Our procedure
requires the specification of a prior (and outputs a posterior) only for the
target of inference, instead of the prior (and posterior) on the full-data
distribution employed by classical non-parametric Bayesian methods .When the
scientific question can be characterized by a low-dimensional parameter
functional, focus on such a prior and posterior distributions is more aligned
with Bayesian subjectivism, compared to focus on entire data distributions. We
prove a Bernstein-von Mises-type result for our proposed Bayesian procedure,
which guarantees that the posterior distribution converges to the distribution
of an efficient, asymptotically linear estimator. In particular, the posterior
is Gaussian, doubly robust, and efficient in the limit, under the only
assumption that certain nuisance parameters are estimated at slow rates. We
perform numerical studies illustrating the frequentist properties of the
method. We also illustrate their use in a motivating application to estimate
the proportion of embolic strokes of undetermined source arising from occult
cardiac sources or large-artery atherosclerotic lesions. Though we focus on the
motivating example of the proportion of cases in an unlabeled dataset, the
procedure is general and can be adapted to estimate any pathwise differentiable
parameter in a non-parametric model.
