Heterogeneous Treatment Effects with Continuous Treatment (e.g. using BART) Overview:
Most of the causal inference literature (both theoretical and applied), I have seen on heterogeneous treatment effects, only considers the case with a binary treatment $T\in\{0,1\}$. However, I would like to estimate heterogeneous treatment effects in a design with a continuous treatment $D\in \mathbb{R}$. I would like to my model to be nonparametric, such as the regression tree based model BART (Bayesian Additive Regression Trees). I have three questions:

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*Do you know any good references for basic theory on heterogeneous treatment effects, when the treatment is continuous?

*Do you know if BART can handle continuous treatments? If so, do you know of any references to theoretical or applied research, where I can learn more about this?

*Do you know of other nonparametric models that could be useful for estimating heterogeneous treatment effects, when the treatment is continuous?

Thank you very much for your help.
Technical details:
Let $D\in \mathbb{R}$ denote a real-valued treatment variable (sometimes also called a dose). Let $Y$ be some real-valued or binary outcome variable, and let $Y^d_i$ denote the potential outcome of a unit $i$, if $i$ were treated with dose level $d$. Let $X_i\in \mathbb{R}^p$ be a vector of covariates for unit $i$. Let $\mu^d(x) = E(Y^d_i | X_i=x)$ denote the conditional average dose response function. That is, $\mu^d(x)$ denotes the average potential outcome for individuals with covariates equal to $x$, if they were treated with dose level $d$.
I consider $\mu^d(x)$ to the fundamental quantity of interest, since we can use it to calculate e.g. the conditional average treatment effect of changing the dose level from $d_0$ to $d_1$ for any $d_0,d_1 \in \mathbb{R}$: $\mu^{d_1}(x) - \mu^{d_0}(x)$. Likewise, we can use it to calculate the conditional marginal treatment effect of an infinitesimal change in dose level for any dose level $d_0$: $\frac{\partial}{\partial d} \mu^{d_0}(x)$.
All in all, I am therefore interested in methods for estimating nonparametric models of $\mu^d(x)$. I am especially interested if BART can be used for this task.
 A: *

*The first part is to identify what is the exact estimand you are interested in. In the context of $T \in \{0, 1\}$, things are fairly simple. However, you can picture different variations for a continuous $T$. For example, you can imagine 'shifting' all the observed $T$ values by some constant $\alpha$. See Dia-Munoz & van der Laan 2011 for a detailed discussion of this problem (and an example using several different estimators). You can also look at 'setting' everyone to $T=t$. This option is usually referred to as estimating the dose-response curve. See Kennedy et al. 2017 for discussion of this approach. Based on your description, it sounds like you are more interested in the later.

*I can't speak to BART, but the previous procedures allow for general functions to estimate the counterfactual quantity. The Kennedy et al. paper in particular describes a non-parametric approach.

*See above for alternative approaches. The proposed estimators in the above linked papers are general. You can also use super-learner with a 'binning' approach, which allows for any classification algorithm you would want. This procedure is described in Chapter 14.

A: With respect to BART's ability to handle continuous treatments, I have found two useful references:

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*(Hill, 2011): Section 6 of the paper (named "Estimating Dosage Effects") uses BART to estimate the causal effect of getting a dosage level $d\in \mathbb{R}$ compared to getting a dosage of 0.

*(Woody et al, 2020): The paper develops a BART model with a linear causal effect of a continuous variable, which is non-linearly moderated by a set of moderator variables.

References:

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*Jennifer L. Hill (2011) Bayesian Nonparametric Modeling for Causal Inference, Journal of Computational and Graphical Statistics, 20:1, 217-240, DOI: 10.1198/jcgs.2010.08162

*Spencer Woody, Carlos M. Carvalho, P. Richard Hahn, & Jared S. Murray. (2020). Estimating heterogeneous effects of continuous exposures using Bayesian tree ensembles: revisiting the impact of abortion rates on crime. https://arxiv.org/abs/2007.09845
