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当研究人员不能进行控制实验，但有观测数据进行建模时，逆概率加权估计量可用于证明因果关系。因为假设治疗不是随机分配的，如果总体中的所有受试者被分配了任何一种治疗，则目标是估计反事实或潜在结果。

当研究人员不能进行控制实验，但有观测数据进行建模时，逆概率加权估计量可用于证明因果关系。因为假设治疗不是随机分配的，如果总体中的所有受试者被分配了任何一种治疗，则目标是估计反事实或潜在结果。

Suppose observed data are <math>\{\bigl(X_i,A_i,Y_i\bigr)\}^{n}_{i=1}</math> drawn [[Independent and identically distributed random variables|i.i.d (independent and identically distributed)]] from unknown distribution P, where

Suppose observed data are <math>\{\bigl(X_i,A_i,Y_i\bigr)\}^{n}_{i=1}</math> drawn [[Independent and identically distributed random variables|i.i.d ()]] from unknown distribution P, where

* <math>X \in \mathbb{R}^{p}</math> covariates

* <math>X \in \mathbb{R}^{p}</math> covariates

* <math>A \in \{0, 1\}</math> are the two possible treatments.

* <math>A \in \{0, 1\}</math> are the two possible treatments.

The goal is to estimate the potential outcome, Y^{*}\bigl(a\bigr), that would be observed if the subject were assigned treatment a. Then compare the mean outcome if all patients in the population were assigned either treatment: \mu_{a} = \mathbb{E}Y^{*}(a). We want to estimate \mu_a using observed data \{\bigl(X_i,A_i,Y_i\bigr)\}^{n}_{i=1}.

The goal is to estimate the potential outcome, Y^{*}\bigl(a\bigr), that would be observed if the subject were assigned treatment a. Then compare the mean outcome if all patients in the population were assigned either treatment: \mu_{a} = \mathbb{E}Y^{*}(a). We want to estimate \mu_a using observed data \{\bigl(X_i,A_i,Y_i\bigr)\}^{n}_{i=1}.

假设观测数据是<math>\{\bigl(X_i,A_i,Y_i\bigr)\}^{n}_{i=1}</math>，这些数据是从未知的分布中抽取出来的独立同分布数据，其中

假设观测数据是<math>\{\bigl(X_i,A_i,Y_i\bigr)\}^{n}_{i=1}</math>，这些数据是从未知的分布中抽取出来的独立同分布（[[Independent and identically distributed random variables|independent and identically distributed, i.i.d]]）数据，其中

* <math>X \in \mathbb{R}^{p}</math> 为协变量；  

* <math>X \in \mathbb{R}^{p}</math> 为协变量；  

* <math>A \in \{0, 1\}</math> 是两个可能的处理；

* <math>A \in \{0, 1\}</math> 是两个可能的处理；