T-Learner

\begin{aligned} \hat\tau &= E [Y(1) - Y(0)|X = x] \\ &= E [Y(1)|X=x] - E[Y(0)|X = x] \\ &= \hat\mu_{1}(x) - \hat\mu_{0}(x) \end{aligned}

where \hat\mu_{0}(x) = M_{0}(Y^0 \sim X^0), \hat\mu_{1}(x) = M_{1}(Y^1 \sim X^1)

S-Learner

\begin{aligned} \hat\tau(x) &= E [Y|X=x, T = 1] - E[Y)|X = x, T = 0] \\ &= \hat\mu_{1}(x, 1) - \hat\mu_{0}(x, 0) \end{aligned}

where \hat\mu = M(Y \sim (X, T))

X-Learner

\begin{aligned} \hat\mu_{0} &= M_{1}(Y^0 \sim X^0) \\ \hat\mu_{1} &= M_{2}(Y^1 \sim X^1) \\ \hat{D}^1 &= Y^1 - \hat\mu_{0}(X^1) \\ \hat{D}^0 &= \hat\mu_{1}(X^0) - Y^0 \\ \hat\tau_{0} &= M_{3}(\hat{D}^0 \sim X^0) \\ \hat\tau_{1} &= M_{4}(\hat{D}^1 \sim X^1) \\ \hat\tau &= g(x)\hat\tau_{0}(x) + (1 - g(x))\hat\tau_{1}(x) \end{aligned}

ATE

群全体における処置の効果

\begin{aligned} ATE &= E[Y^1-Y^0] \end{aligned}

ITE

個体iにおける因果効果

\begin{aligned} D_i &= E[Y_i^1-Y_i^0] \end{aligned}

CATE

\begin{aligned} \tau(x) &= E[D|X=x_i = E[Y^1 - Y^0|X = x_i]] \end{aligned}

we note that the best estimator for the CATE is also
the best estimator for the ITE in terms of the MSE.

let \hat\tau_i be an estimator for D_i and decompose the MSE at x_i

\begin{aligned} E[(D_i − \hat\tau_i)^2|X_i = x_i] =E[(D_i − \tau(x_i))^2|Xi = xi] + E[(\tau (x_i) − \hat\tau)^2] \end{aligned}