  Principal stratification analysis evaluates how causal effects of a treatment
on a primary outcome vary across strata of units defined by their treatment
effect on some intermediate quantity. This endeavor is substantially challenged
when the intermediate variable is continuously scaled and there are infinitely
many basic principal strata. We employ a Bayesian nonparametric approach to
flexibly evaluate treatment effects across flexibly-modeled principal strata.
The approach uses Bayesian Causal Forests (BCF) to simultaneously specify two
Bayesian Additive Regression Tree models; one for the principal stratum
membership and one for the outcome, conditional on principal strata. We show
how the capability of BCF for capturing treatment effect heterogeneity is
particularly relevant for assessing how treatment effects vary across the
surface defined by continuously-scaled principal strata, in addition to other
benefits relating to targeted selection and regularization-induced confounding.
The capabilities of the proposed approach are illustrated with a simulation
study, and the methodology is deployed to investigate how causal effects of
power plant emissions control technologies on ambient particulate pollution
vary as a function of the technologies' impact on sulfur dioxide emissions.
