Stochastic Trees and Heterogeneous Effects

Stochastic Trees and Heterogeneous Effects

This section delves into a powerful family of non-parametric Bayesian models grounded in Bayesian Additive Regression Trees (BART) and its modern derivatives. By representing unknown response surfaces as sums of shallow decision trees endowed with regularizing priors, these models capture complex non-linearities and high-order interactions automatically, without requiring hand-crafted functional forms or polynomial expansions.

Before applying these methods to causal questions, it is crucial to keep their identifying assumptions clearly in view. When estimating causal effects, tree-based estimators rely on the standard foundations of causal inference: the Stable Unit Treatment Value Assumption (SUTVA) and strong ignorability.

Stable Unit Treatment Value Assumption (SUTVA): SUTVA comprises two parts:
1. No interference: The treatment applied to one unit does not influence the potential outcomes of other units. In a marketplace or network setting, interference between units must be addressed by study design or specialized models.
2. No hidden variations of treatment: Each treatment level corresponds to a well-defined, uniform intervention across all recipient units.
Strong Ignorability: This assumption consists of two components:
1. Unconfoundedness: Given the observed covariates $X$, treatment assignment is conditionally independent of potential outcomes. No unmeasured confounding may distort the comparison.
2. Positivity (or overlap): Every unit in the target population has a non-zero probability of receiving each treatment condition ($0 < P(Z=1 \mid X) < 1$). Forests can extrapolate across sparse regions, but causal identification requires overlap.

Roadmap for this part

The chapters in this part explore the tree-based toolkit across diverse modeling roles. These methods do not share a single monolithic sampler or engine; different packages implement distinct sampling algorithms tailored to their mathematical structure:

BART: The classical Bayesian Additive Regression Tree framework, providing flexible surface estimation and coherent posterior uncertainty.
Bayesian Causal Forests (BCF): Tailored causal trees that separate prognostic baseline estimation from treatment effect moderation, incorporating the estimated propensity score to prevent regularization-induced confounding.
Causal Propensity: Modern machine learning and tree-based methods for modeling the assignment mechanism itself.
PhoBART: Ordinal tree models designed for discrete rating scales, survey scores, and bounded business metrics, illustrating that tree ensembles extend well beyond Gaussian regression.
Bayesian Adaptive Designs (BAD): Dynamic experimentation and multi-armed bandit allocation, connecting posterior effect estimates directly to sequential sample allocation and operational decisions.

When causal questions involve time—such as staggered feature rollouts, market-level interventions, or longitudinal panels—we take these ideas further. The subsequent part, Longitudinal and Panel Causal Inference, brings stochastic trees to panel data with LongBet.

Learn more

Krantsevich et al. (2023) Stochastic Tree Ensembles for Estimating Heterogeneous Effects.
Herren et al. (2024) Stochastic tree ensembles (XBART and BART) for supervised learning and causal inference.
Hahn et al. (2020) Bayesian Regression Tree Models for Causal Inference: Regularization, Confounding, and Heterogeneous Effects.

--- title: "Stochastic Trees and Heterogeneous Effects" description: "BART and its causal derivatives: flexible non-parametric models that capture non-linearities and interactions without hand-specified functional forms, ordinal outcomes, and adaptive designs." share: permalink: "https://book.martinez.fyi/stochastictrees.html" description: "Stochastic trees and heterogeneous effects: BART, BCF, and tree-based causal inference" linkedin: true email: true mastodon: true --- This section delves into a powerful family of non-parametric Bayesian models grounded in Bayesian Additive Regression Trees (BART) and its modern derivatives. By representing unknown response surfaces as sums of shallow decision trees endowed with regularizing priors, these models capture complex non-linearities and high-order interactions automatically, without requiring hand-crafted functional forms or polynomial expansions. Before applying these methods to causal questions, it is crucial to keep their identifying assumptions clearly in view. When estimating causal effects, tree-based estimators rely on the standard foundations of causal inference: the Stable Unit Treatment Value Assumption (SUTVA) and strong ignorability. 1. **Stable Unit Treatment Value Assumption (SUTVA):** SUTVA comprises two parts: a. **No interference:** The treatment applied to one unit does not influence the potential outcomes of other units. In a marketplace or network setting, interference between units must be addressed by study design or specialized models. b. **No hidden variations of treatment:** Each treatment level corresponds to a well-defined, uniform intervention across all recipient units. 2. **Strong Ignorability:** This assumption consists of two components: a. **Unconfoundedness:** Given the observed covariates $X$, treatment assignment is conditionally independent of potential outcomes. No unmeasured confounding may distort the comparison. b. **Positivity (or overlap):** Every unit in the target population has a non-zero probability of receiving each treatment condition ($0 < P(Z=1 \mid X) < 1$). Forests can extrapolate across sparse regions, but causal identification requires overlap. ### Roadmap for this part The chapters in this part explore the tree-based toolkit across diverse modeling roles. These methods do not share a single monolithic sampler or engine; different packages implement distinct sampling algorithms tailored to their mathematical structure: - **[BART](bart.qmd):** The classical Bayesian Additive Regression Tree framework, providing flexible surface estimation and coherent posterior uncertainty. - **[Bayesian Causal Forests (BCF)](bcf.qmd):** Tailored causal trees that separate prognostic baseline estimation from treatment effect moderation, incorporating the estimated propensity score to prevent regularization-induced confounding. - **[Causal Propensity](causalPropensity.qmd):** Modern machine learning and tree-based methods for modeling the assignment mechanism itself. - **[PhoBART](phobart.qmd):** Ordinal tree models designed for discrete rating scales, survey scores, and bounded business metrics, illustrating that tree ensembles extend well beyond Gaussian regression. - **[Bayesian Adaptive Designs (BAD)](bad.qmd):** Dynamic experimentation and multi-armed bandit allocation, connecting posterior effect estimates directly to sequential sample allocation and operational decisions. When causal questions involve time---such as staggered feature rollouts, market-level interventions, or longitudinal panels---we take these ideas further. The subsequent part, **[Longitudinal and Panel Causal Inference](longitudinal.qmd)**, brings stochastic trees to panel data with **[LongBet](longbet.qmd)**. ::: {.callout-tip} ## Learn more - @krantsevich23a Stochastic Tree Ensembles for Estimating Heterogeneous Effects. - @stochastictree Stochastic tree ensembles (XBART and BART) for supervised learning and causal inference. - @hahn2020bayesian Bayesian Regression Tree Models for Causal Inference: Regularization, Confounding, and Heterogeneous Effects. :::