Abstract

Recently authors pointed to Bayesian causal inference as important pathway for future development of causal methodologies (Li, 2022; Imbens, 2022). Bayesian additive regression trees (BART) perspective has been developed by Chipman et al. (2010) and popularized in recent years in its usage in regression and causal inference problems (for example Tan and Roy, 2019; Hahn, Murray and Carvalho, 2020). Commonly, BART uses a specific regularization prior, sometimes combined with Gaussian, Dirichlet, Dirichlet Process Mixture (DPM) and semiparametric perspectives (Tan and Roy, 2019). Despite the success of BART, there has been a growing number of papers that point out its limitations. Hahn, Murray and Carvalho have developed Bayesian Causal Forests (BCF - Hahn, Murray and Carvalho, 2020) as a novel regularization approach for nonlinear models geared specifically towards situations with small effect sizes, heterogeneous effects, and strong confounding, to improve on the earlier BART perspective. We develop a novel nonparametric regularization prior for BART based on Pitman-Yor Mixture (PYM) partition-based process which has to date to our knowledge rarely been used in causal inference (but is suggested for classification and mixture modelling). Pitman–Yor process mixtures (Ishwaran and James, 2001; 2003) are a generalization of DPMs based on the Pitman–Yor process, also known as the two-parameter Poisson–Dirichlet process. Our novel BART perspective is studied in more detail for several different causal perspectives: regression discontinuity design; causal maximally partially directed acyclic graph (MPDAG); direct causal clause; and causal mediation. Our results on simulated and real data confirm improved properties as compared to earlier BART priors and the performance is similar to the Hahn et al. BCF model, but improved in the presence of strong confounding. Performance of the prior is different for causal mediation and we provide suggestions for future work in this perspective. We address computational issues by using importance sampling with the integrated nested Laplace approximation (Outzen Berild et al., 2021). In conclusion we discuss extensions to endogeneity corrections in the line of BCF-IV approach of Bargagli Stoffi et al. (2022) and extensions to Single World Intervention Graph perspective (Richardson and Robins, 2013; 2014).