In causal inference studies, interest often lies in understanding the mechanisms through which a treatment affects an outcome. One approach is principal stratification (PS), which introduces well-defined causal effects in the presence of confounded post-treatment variables, or mediators, and clearly defines the assumptions for identification and estimation of those effects. The goal of this paper is to extend the PS framework to studies with continuous treatments and continuous post-treatment variables, which introduces a number of unique challenges both in terms of defining causal effects and performing inference. This manuscript provides three key methodological contributions: 1) we introduce novel principal estimands for continuous treatments that provide valuable insights into different causal mechanisms, 2) we utilize Bayesian nonparametric approaches to model the joint distribution of the potential mediating variables based on both Gaussian processes and Dirichlet process mixtures to ensure our approach is robust to model misspecification, and 3) we provide theoretical and numerical justification for utilizing a model for the potential outcomes to identify the joint distribution of the potential mediating variables. Lastly, we apply our methodology to a novel study of the relationship between the economy and arrest rates, and how this is potentially mediated by police capacity.

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