# BENK: The Beran Estimator with Neural Kernels for Estimating the Heterogeneous Treatment Effect
## 3 CATE Estimation Problem Statement  
According to the CATE estimation problem, all patients are divided into two groups: control and treatment. Let the control group be the set \(\mathcal{C}=\{(\mathbf{x}_{1},f_{1}),\ldots,(\mathbf{x}_{c},f_{c})\}\) of \(c\) patients, such that the \(i\)-th patient is characterized by the feature vector \(\mathbf{x}_{i}=(x_{i1},\ldots,x_{id})\in\mathbb{R}^{d}\) and the \(i\)-th observed outcome \(f_{i}\in\mathbb{R}\) (time to event, temperature, the blood pressure, etc.). It is also supposed that the treatment group is the set \(\mathcal{T}=\{(\mathbf{y}_{1},h_{1}),\ldots,(\mathbf{y}_{t},h_{t})\}\) of \(t\) patients, such that the \(i\)-th patient is characterized by the feature vector \(\mathbf{y}_{i}=(y_{i1},\ldots,y_{id})\in\mathbb{R}^{d}\) and the \(i\)-th observed outcome \(h_{i}\in\mathbb{R}\). The indicator of a group for the \(i\)-th patient is denoted as \(T_{i}\in\{0,1\}\), where \(T_{i}=0\) (\(T_{i}=1\)) corresponds to the control (treatment) group.  
We use different notations \(\mathbf{x}_{i}\) and \(\mathbf{y}_{i}\) for controls and treatments in order to avoid additional indices. However, we use the vector \(\mathbf{z}\in\mathbb{R}^{d}\) instead of \(\mathbf{x}\) and \(\mathbf{y}\) when estimating the CATE.  
Suppose that the potential outcomes of patients from the control and treatment groups are \(F\) and \(H\), respectively. The treatment effect for a new patient with the feature vector \(\mathbf{z}\) is estimated by the individual treatment effect, defined as \(H-F\). The fundamental problem of computing the CATE is that only one of the outcomes \(f\) or \(h\) for each patient can be observed. An important assumption of unconfoundedness [54] is used to allow the untreated patients to be used to construct an unbiased counterfactual for the treatment group [55]. According to the assumption, potential outcomes are characteristics of a patient before the patient is assigned to a treatment condition, or, formally, the treatment assignment \(T\) is independent of the potential outcomes for \(F\) and \(H\) that conditional on the feature vector \(\mathbf{z}\), which can be written as  
\[T\perp\{F,H\}\mid\mathbf{z}. \tag{1}\]  
The second assumption, called the overlap assumption, regards the joint distribution of treatments and covariates. This assumption claims that a positive probability of being both treated and untreated for each value of \(\mathbf{z}\) exists. This implies that the following holds with probability 1:  
\[0<\Pr\{T=1\mid\mathbf{z}\}<1. \tag{2}\]  
Let \(\mathbf{Z}\) be the random feature vector from \(\mathbb{R}^{d}\). The treatment effect is estimated by means of CATE, which is defined as the expected difference between two potential outcomes, as follows [56]:  
\[\tau(\mathbf{z})=\mathbb{E}[H-F\mid\mathbf{Z}=\mathbf{z}]. \tag{3}\]  
By using the above assumptions, CATE can be rewritten as  
\[\tau(\mathbf{z})=\mathbb{E}[H\mid\mathbf{Z}=\mathbf{z}]-\mathbb{E}[F\mid \mathbf{Z}=\mathbf{z}]. \tag{4}\]  
The motivation behind unconfoundedness is that nearby observations in the feature space can be treated as having come from a randomized experiment [7].  
Suppose that functions \(g_{0}(\mathbf{z})\) and \(g_{1}(\mathbf{z})\) express outcomes of the control and treatment patients, respectively. Then, they can be written as follows:  
\[f=g_{0}(\mathbf{z})+\varepsilon,\;h=g_{1}(\mathbf{z})+\varepsilon, \tag{5}\]  
where \(\varepsilon\) is noise governed by the normal distribution with the zero expectation.  
The above imply that the CATE can be estimated as  
\[\tau(\mathbf{z})=g_{1}(\mathbf{z})-g_{0}(\mathbf{z}). \tag{6}\]  
An example illustrating the controls (circle markers), treatments (triangle markers) and corresponding unknown function \(g_{0}\) and \(g_{1}\) are shown in Figure 1.