## 2 Background and Relevant Research
### Machine Learning Models for Treatment Effect Estimation  
Alongside SCMs, purely data-driven machine learning models have been proposed in the bibliography to directly estimate the various types of treatment effects, introduced in Section 2.1, without the use of DAGs. Following a Bayesian non-parametric approach, [PERSON] et al. [28] developed a Bayesian \"sum-of-trees\" model, named Bayesian Additive Regression Trees (BART). The BART approach fits a parameter-rich model by using a strongly influential prior distribution. Specifically, the model uses the sum of trees to approximate the average value of the outcomes given a set of covariates, \(\mathbb{E}[Y\,|\,\mathbf{x}]\). The main idea behind BART is to impose a prior, which regularizes the fit by keeping the individual tree effects small in order to elaborate the sum-of-trees model. Additionally, BART uses a tailored version of Bayesian back-fitting Markov Chain Monte Carlo [29] for fitting the sum-of-trees model. [PERSON] et al. [30] proposed two methodologies, named S-learner and T-learner, for the accurate estimation of treatment effects. S-learner estimates the outcomes by using all the features of the dataset, along with the treatment as an additional feature. Notice that the treatment indicator is handled by the base learner like any other feature; hence, it does not play any special role in the estimation of the effects. In contrast, T-learner utilizes two independent models: one trained on the treated data to predict outcomes under treatment and the other one on the control data to predict outcomes without treatment. Next, the treatment effect is then estimated by subtracting the predicted outcome for the control model from the predicted outcome for the treatment model. In the literature, a variety of causal inference models were proposed based on T-learner and S-learner methodologies using a variety of ML models as base learners, providing some interesting results [30; 31]. [PERSON] at al. [32] proposed a new framework for estimating individual treatment effects, named Counterfactual Regression (CFR). This framework uses a prediction model, focusing on learning a balanced representation of the control and treatment groups. Under this balanced representation, the distributions of the two groups are considered to be similar. Specifically, the authors used two different integral probability metrics, Wasserstein (Wass) distance [33] and Maximum Mean Discrepancy (MMD) [34], for calculating the distances between the treatment and control distributions, as well as proposed a generalization bound for estimating the individual treatment effect. In addition, the authors demonstrated the performance of the proposed models, CFR (MMD) and CFR (Wass), which stand for MMD and Wass distances, respectively, and compared their performance with state-of-the-art models. Finally, they proposed a variant without balance regularization, the neural network-based model TARNet. [PERSON] et al. [35] proposed a neural network-based model for the estimation of treatment effects, named Dragonnet. The authors mainly focused on the use of observational data for treatment effect estimation, giving special attention to the average and individual level. The proposed model aimed to improve these estimations by exploiting the propensity score's sufficiency. Additionally, the authors proposed targeted regularization, which is based on non-parametric estimation theory and focuses on reducing the bias of the estimator, therefore further improving the estimation of treatment effects. The presented experiments provided evidence about the superiority of Dragonnet over TarNet, CFR (MMD) and CFR (Wass) on a variety of challenging causal inference benchmark datasets. In more recent works, [PERSON] and [PERSON] [36; 37] proposed NN-Dragonnet for estimating treatment effects, which consists of a modification of the neural network-based model Dragonnet.