diff --git a/docs/background.rst b/docs/background.rst index 457cc9ef..969eeaf2 100644 --- a/docs/background.rst +++ b/docs/background.rst @@ -421,13 +421,13 @@ It consists of two stages: \DeclareMathOperator*{\argmin}{arg\,min} \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[{\tilde{W}^i}^2\bigg(\frac{\tilde{Y}^i}{\tilde{W}^i} - \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\} Where .. math:: - \tilde{W}^i &= W^i - \hat{e}(X^i) \\ - \tilde{Y}^i &= Y^i - \hat{m}(X^i) + \widetilde{W}^i &= W^i - \hat{e}(X^i) \\ + \widetilde{Y}^i &= Y^i - \hat{m}(X^i) And therefore any ML model which supports weighting each observation differently can be used for the final model.