  Motivated by environmental policy questions, we address the challenges of
estimation, change point detection, and uncertainty quantification of a causal
exposure-response function (CERF). Under a potential outcome framework, the
CERF describes the relationship between a continuously varying exposure (or
treatment) and its causal effect on an outcome. We propose a new Bayesian
approach that relies on a Gaussian process (GP) model to estimate the CERF
nonparametrically. To achieve the desired separation of design and analysis
phases, we parametrize the covariance (kernel) function of the GP to mimic
matching via a Generalized Propensity Score (GPS). The hyper-parameters as well
as the form of the kernel function of the GP are chosen to optimize covariate
balance. Our approach achieves automatic uncertainty evaluation of the CERF
with high computational efficiency, and enables change point detection through
inference on derivatives of the CERF. We provide theoretical results showing
the correspondence between our Bayesian GP framework and traditional approaches
in causal inference for estimating causal effects of a continuous exposure. We
apply the methods to 520,711 ZIP-code-level observations to estimate the causal
effect of long-term exposures to PM2.5, ozone, and NO2 on all-cause mortality
among Medicare enrollees in the US. A computationally efficient implementation
of the proposed GP models is provided in the GPCERF R package, which is
available on CRAN.
