  A powerful tool for the analysis of nonrandomized observational studies has
been the potential outcomes model. Utilization of this framework allows
analysts to estimate average treatment effects. This article considers the
situation in which high-dimensional covariates are present and revisits the
standard assumptions made in causal inference. We show that by employing a
flexible Gaussian process framework, the assumption of strict overlap leads to
very restrictive assumptions about the distribution of covariates, results for
which can be characterized using classical results from Gaussian random
measures as well as reproducing kernel Hilbert space theory. In addition, we
propose a strategy for data-adaptive causal effect estimation that does not
rely on the strict overlap assumption. These findings reveal the stringency
that accompanies the use of the treatment positivity assumption in
high-dimensional settings.
