  We consider inference from non-random samples in data-rich settings where
high-dimensional auxiliary information is available both in the sample and the
target population, with survey inference being a special case. We propose a
regularized prediction approach that predicts the outcomes in the population
using a large number of auxiliary variables such that the ignorability
assumption is reasonable while the Bayesian framework is straightforward for
quantification of uncertainty. Besides the auxiliary variables, inspired by
Little & An (2004), we also extend the approach by estimating the propensity
score for a unit to be included in the sample and also including it as a
predictor in the machine learning models. We show through simulation studies
that the regularized predictions using soft Bayesian additive regression trees
yield valid inference for the population means and coverage rates close to the
nominal levels. We demonstrate the application of the proposed methods using
two different real data applications, one in a survey and one in an
epidemiology study.
