Fix list intendation

docs/background.md

@@ -192,99 +192,99 @@ The X-Learner was introduced by [Kuenzel et al. (2019)](https://arxiv.org/pdf/17
 
 1. Estimate the conditional average outcomes for each variant:
 
-\[
-\begin{align*}
-\mu_0 (x) &:= \mathbb{E}[Y(0) | X = x] \\
-\mu_1 (x) &:= \mathbb{E}[Y(1) | X = x]
-\end{align*}
-\]
+   \[
+   \begin{align*}
+   \mu_0 (x) &:= \mathbb{E}[Y(0) | X = x] \\
+   \mu_1 (x) &:= \mathbb{E}[Y(1) | X = x]
+   \end{align*}
+   \]
 
-2. Impute the treatment effect for the observations in the treated group based on the control-outcome estimator as well as the treatment effect for the observations in the control group based on the treatment-outcome estimator:
+1. Impute the treatment effect for the observations in the treated group based on the control-outcome estimator as well as the treatment effect for the observations in the control group based on the treatment-outcome estimator:
 
-\[
-\begin{align*}
-\widetilde{D}\_1^i &:= Y^i_1 - \hat{\mu}\_0(X^i_1) \\
-\widetilde{D}\_0^i &:= \hat{\mu}\_1(X^i_0) - Y^i_0
-\end{align*}
-\]
+   \[
+   \begin{align*}
+   \widetilde{D}\_1^i &:= Y^i_1 - \hat{\mu}\_0(X^i_1) \\
+   \widetilde{D}\_0^i &:= \hat{\mu}\_1(X^i_0) - Y^i_0
+   \end{align*}
+   \]
 
-Then estimate $\tau_1(x) := \mathbb{E}[\widetilde{D}^i_1 | X=x]$ and $\tau_0(x) := \mathbb{E}[\widetilde{D}^i_0 | X=x]$ using the observations in the treatment group and the ones in the control group respectively.
+   Then estimate \(\tau_1(x) := \mathbb{E}[\widetilde{D}_1^i | X=x]\) and \(\tau_0(x) := \mathbb{E}[\widetilde{D}_0^i | X=x]\) using the observations in the treatment group and the ones in the control group respectively.
 
-3. Define the CATE estimate by a weighted average of the two estimates in stage 2:
+1. Define the CATE estimate by a weighted average of the two estimates in stage 2:
 
-$$
-\hat{\tau}^X(x) := g(x)\hat{\tau}_0(x) + (1-g(x))\hat{\tau}_1(x)
-$$
+   $$
+   \hat{\tau}^X(x) := g(x)\hat{\tau}_0(x) + (1-g(x))\hat{\tau}_1(x)
+   $$
 
-Where $g(x) \in [0,1]$. We take $g(x) := \mathbb{E}[W = 1 | X=x]$ to be the propensity score.
+   Where \(g(x) \in [0,1]\). We take \(g(x) := \mathbb{E}[W = 1 | X=x]\) to be the propensity score.
 
 #### More than binary treatment
 
 In the case of multiple discrete treatments, the stages are similar to the binary case:
 
 1. One outcome model is estimated for each variant (including the control), and one propensity model is trained as a multiclass classifier, $\forall k \in \{0,\dots, K-1\}$:
 
-\[
-\begin{align*}
-\mu_k (x) &:= \mathbb{E}[Y(k) | X = x]\\
-e(x, k) &:= \mathbb{E}[\mathbb{I}\{W = k\} | X=x] = \mathbb{P}[W = k | X=x]
-\end{align*}
-\]
+   \[
+   \begin{align*}
+   \mu_k (x) &:= \mathbb{E}[Y(k) | X = x]\\
+   e(x, k) &:= \mathbb{E}[\mathbb{I}\{W = k\} | X=x] = \mathbb{P}[W = k | X=x]
+   \end{align*}
+   \]
 
-2. The treatment effects are imputed using the corresponding outcome estimator, $\forall k \in \{1,\dots, K-1\}$:
+1. The treatment effects are imputed using the corresponding outcome estimator, $\forall k \in \{1,\dots, K-1\}$:
 
-\[
-\begin{align*}
-\widetilde{D}*k^i &:= Y^i*k - \hat{\mu}\_0(X^i_k) \\
-\widetilde{D}*{0,k}^i &:= \hat{\mu}\_k(X^i*0) - Y^i_0
-\end{align*}
-\]
+   \[
+   \begin{align*}
+   \widetilde{D}*k^i &:= Y^i*k - \hat{\mu}\_0(X^i_k) \\
+   \widetilde{D}*{0,k}^i &:= \hat{\mu}\_k(X^i*0) - Y^i_0
+   \end{align*}
+   \]
 
-Then $\tau_k(x) := \mathbb{E}[\widetilde{D}^i_k | X=x]$ is estimated using the observations which received treatment $k$ and $\tau_{0,k}(x) := \mathbb{E}[\widetilde{D}^i_{0,k} | X=x]$ using the observations in the control group.
+   Then $\tau_k(x) := \mathbb{E}[\widetilde{D}^i_k | X=x]$ is estimated using the observations which received treatment $k$ and $\tau_{0,k}(x) := \mathbb{E}[\widetilde{D}^i_{0,k} | X=x]$ using the observations in the control group.
 
-3. Finally, the CATE for each variant is estimated as a weighted average:
+1. Finally, the CATE for each variant is estimated as a weighted average:
 
-$$
-\hat{\tau}_k^X(x) := g(x, k)\hat{\tau}_{0,k}(x) + (1-g(x,k))\hat{\tau}_k(x)
-$$
+   $$
+   \hat{\tau}_k^X(x) := g(x, k)\hat{\tau}_{0,k}(x) + (1-g(x,k))\hat{\tau}_k(x)
+   $$
 
-Where
+   Where
 
-$$
-g(x,k) := \frac{\hat{e}(x,k)}{\hat{e}(x,k) + \hat{e}(x,0)}
-$$
+   $$
+   g(x,k) := \frac{\hat{e}(x,k)}{\hat{e}(x,k) + \hat{e}(x,0)}
+   $$
 
 ### R-Learner
 
 The R-Learner was introduced by [Nie et al. (2017)](https://arxiv.org/pdf/1712.04912). It consists of two stages:
 
 1. Estimate a general outcome model and a propensity model:
 
-\[
-\begin{align*}
-m(x) &:= \mathbb{E}[Y | X=x] \\
-e(x) &:= \mathbb{P}[W = 1 | X=x]
-\end{align*}
-\]
+   \[
+   \begin{align*}
+   m(x) &:= \mathbb{E}[Y | X=x] \\
+   e(x) &:= \mathbb{P}[W = 1 | X=x]
+   \end{align*}
+   \]
 
-2. Estimate the treatment effect by minimizing the R-Loss:
+1. Estimate the treatment effect by minimizing the R-Loss:
 
-\[
-\begin{align*}
-\hat{\tau}^R (\cdot) &:= \argmin*{\tau}\Bigg\{\mathbb{E}\Bigg[\bigg(\left\{Y^i - \hat{m}(X^i)\right\} - \left\{W^i - \hat{e}(X^i)\right\}\tau(X^i)\bigg)^2\Bigg]\Bigg\} \\
-&=\argmin*{\tau}\left\{\mathbb{E}\left[\left\{W^i - \hat{e}(X^i)\right\}^2\bigg(\frac{\left\{Y^i - \hat{m}(X^i)\right\}}{\left\{W^i - \hat{e}(X^i)\right\}} - \tau(X^i)\bigg)^2\right]\right\} \\
-&= \argmin\_{\tau}\left\{\mathbb{E}\left[{\widetilde{W}^i}^2\bigg(\frac{\widetilde{Y}^i}{\widetilde{W}^i} - \tau(X^i)\bigg)^2\right]\right\}
-\end{align*}
-\]
+   \[
+   \begin{align*}
+   \hat{\tau}^R (\cdot) &:= \argmin*{\tau}\Bigg\{\mathbb{E}\Bigg[\bigg(\left\{Y^i - \hat{m}(X^i)\right\} - \left\{W^i - \hat{e}(X^i)\right\}\tau(X^i)\bigg)^2\Bigg]\Bigg\} \\
+   &=\argmin*{\tau}\left\{\mathbb{E}\left[\left\{W^i - \hat{e}(X^i)\right\}^2\bigg(\frac{\left\{Y^i - \hat{m}(X^i)\right\}}{\left\{W^i - \hat{e}(X^i)\right\}} - \tau(X^i)\bigg)^2\right]\right\} \\
+   &= \argmin\_{\tau}\left\{\mathbb{E}\left[{\widetilde{W}^i}^2\bigg(\frac{\widetilde{Y}^i}{\widetilde{W}^i} - \tau(X^i)\bigg)^2\right]\right\}
+   \end{align*}
+   \]
 
-Where
+   Where
 
-\[
-\begin{align*}
-\widetilde{W}^i &= W^i - \hat{e}(X^i) \\
-\widetilde{Y}^i &= Y^i - \hat{m}(X^i)
-\end{align*}
-\]
+   \[
+   \begin{align*}
+   \widetilde{W}^i &= W^i - \hat{e}(X^i) \\
+   \widetilde{Y}^i &= Y^i - \hat{m}(X^i)
+   \end{align*}
+   \]
 
 And therefore any ML model which supports weighting each observation differently can be used for the final model.
 
@@ -294,14 +294,14 @@ In the case of multiple discrete treatments, the stages are similar to the binar
 
 1. Estimate a general outcome model and a propensity model:
 
-\[
-\begin{align*}
-m(x) &:= \mathbb{E}[Y | X=x] \\
-e(x) &:= \mathbb{P}[W = k | X=x]
-\end{align*}
-\]
+   \[
+   \begin{align*}
+   m(x) &:= \mathbb{E}[Y | X=x] \\
+   e(x) &:= \mathbb{P}[W = k | X=x]
+   \end{align*}
+   \]
 
-2. For each $k \neq 0$, estimate the pairwise treatment effect $\hat{\tau}_{0,k}^R$ between 0 and $k$ by minimizing the R-Loss from above. In order to fit these models, we fit the pseudo outcomes only on observations of either the control group or the treatment variant group $k$.
+1. For each $k \neq 0$, estimate the pairwise treatment effect $\hat{\tau}_{0,k}^R$ between 0 and $k$ by minimizing the R-Loss from above. In order to fit these models, we fit the pseudo outcomes only on observations of either the control group or the treatment variant group $k$.
 
 Note that:
 
@@ -317,52 +317,52 @@ The DR-Learner was introduced by [Kennedy (2020)](https://arxiv.org/pdf/2004.144
 
 1. Estimate the conditional average outcomes for each variant and a propensity model:
 
-\[
-\begin{align*}
-\mu_0 (x, w) &:= \mathbb{E}[Y(0) | X = x] \\
-\mu_1 (x, w) &:= \mathbb{E}[Y(1) | X = x] \\
-e(x) &:= \mathbb{E}[W = 1 | X=x]
-\end{align*}
-\]
+   \[
+   \begin{align*}
+   \mu_0 (x, w) &:= \mathbb{E}[Y(0) | X = x] \\
+   \mu_1 (x, w) &:= \mathbb{E}[Y(1) | X = x] \\
+   e(x) &:= \mathbb{E}[W = 1 | X=x]
+   \end{align*}
+   \]
 
-And construct the pseudo-outcomes:
+   And construct the pseudo-outcomes:
 
-\[
-\begin{align*}
-\varphi(X^i, W^i, Y^i) := \frac{W^i - \hat{e}(X^i)}{\hat{e}(X^i)(1-\hat{e}(X^i))}\big\{Y^i - \hat{\mu}*{W^i}(X^i)\big\} + \hat{\mu}_{1}(X^i) - \hat{\mu}\_{0}(X^i)
-\end{align_}
-\]
+   \[
+   \begin{align*}
+   \varphi(X^i, W^i, Y^i) := \frac{W^i - \hat{e}(X^i)}{\hat{e}(X^i)(1-\hat{e}(X^i))}\big\{Y^i - \hat{\mu}*{W^i}(X^i)\big\} + \hat{\mu}\_{1}(X^i) - \hat{\mu}\_{0}(X^i)
+   \end{align\*}
+   \]
 
-2. Estimate the CATE by regressing $\varphi$ on $X$:
+1. Estimate the CATE by regressing $\varphi$ on $X$:
 
-$$
-\hat{\tau}^{DR}(x) := \mathbb{E}[\varphi(X^i, W^i, Y^i) | X^i=x]
-$$
+   $$
+   \hat{\tau}^{DR}(x) := \mathbb{E}[\varphi(X^i, W^i, Y^i) | X^i=x]
+   $$
 
 #### More than binary treatment
 
 In the case of multiple discrete treatments, the stages are similar to the binary case:
 
 1. One outcome model is estimated for each variant (including the control), and one propensity model is trained as a multiclass classifier, $\forall k \in \{0,\dots, K-1\}$:
 
-\[
-\begin{align*}
-\mu_k (x) &:= \mathbb{E}[Y(k) | X = x]\\
-e(x, k) &:= \mathbb{E}[\mathbb{I}\{W = k\} | X=x] = \mathbb{P}[W = k | X=x]
-\end{align*}
-\]
+   \[
+   \begin{align*}
+   \mu_k (x) &:= \mathbb{E}[Y(k) | X = x]\\
+   e(x, k) &:= \mathbb{E}[\mathbb{I}\{W = k\} | X=x] = \mathbb{P}[W = k | X=x]
+   \end{align*}
+   \]
 
-The pseudo-outcomes are constructed for each treatment variant, $\forall k \in \{1,\dots, K-1\}$:
+   The pseudo-outcomes are constructed for each treatment variant, $\forall k \in \{1,\dots, K-1\}$:
 
-\[
-\begin{align*}
-\varphi*k(X^i, W^i, Y^i) := &\frac{Y^i - \hat{\mu}*{k}(X^i)}{\hat{e}(k, X^i)}\mathbb{I}\{W^i = k\} + \hat{\mu}*k(X^i) \\
-&- \frac{Y^i - \hat{\mu}_{0}(X^i)}{\hat{e}(0, X^i)}\mathbb{I}\{W^i = 0\} - \hat{\mu}\_0(X^i)
-\end{align_}
-\]
+   \[
+   \begin{align*}
+   \varphi*k(X^i, W^i, Y^i) := &\frac{Y^i - \hat{\mu}*{k}(X^i)}{\hat{e}(k, X^i)}\mathbb{I}\{W^i = k\} + \hat{\mu}*k(X^i) \\
+   &- \frac{Y^i - \hat{\mu}\_{0}(X^i)}{\hat{e}(0, X^i)}\mathbb{I}\{W^i = 0\} - \hat{\mu}\_0(X^i)
+   \end{align\*}
+   \]
 
 1. Finally, the CATE is estimated by regressing $\varphi_k$ on $X$ for each treatment variant, $\forall k \in \{1,\dots, K-1\}$:
 
-$$
-\hat{\tau}_k^{DR}(x) := \mathbb{E}[\varphi_k(X^i, W^i, Y^i) | X^i=x]
-$$
+   $$
+   \hat{\tau}_k^{DR}(x) := \mathbb{E}[\varphi_k(X^i, W^i, Y^i) | X^i=x]
+   $$