  Accounting for exposure measurement errors has been recognized as a crucial
problem in environmental epidemiology for over two decades. Bayesian
hierarchical models offer a coherent probabilistic framework for evaluating
associations between environmental exposures and health effects, which take
into account exposure measurement errors introduced by uncertainty in the
estimated exposure as well as spatial misalignment between the exposure and
health outcome data. While two-stage Bayesian analyses are often regarded as a
good alternative to fully Bayesian analyses when joint estimation is not
feasible, there has been minimal research on how to properly propagate
uncertainty from the first-stage exposure model to the second-stage health
model, especially in the case of a large number of participant locations along
with spatially correlated exposures. We propose a scalable two-stage Bayesian
approach, called a sparse multivariate normal (sparse MVN) prior approach,
based on the Vecchia approximation for assessing associations between exposure
and health outcomes in environmental epidemiology. We compare its performance
with existing approaches through simulation. Our sparse MVN prior approach
shows comparable performance with the fully Bayesian approach, which is a gold
standard but is impossible to implement in some cases. We investigate the
association between source-specific exposures and pollutant (nitrogen dioxide
(NO$_2$))-specific exposures and birth outcomes for 2012 in Harris County,
Texas, using several approaches, including the newly developed method.
