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The inverse probability weighting estimator can be used to demonstrate causality when the researcher cannot conduct a controlled experiment but has observed data to model. Because it is assumed that the treatment is not randomly assigned, the goal is to estimate the counterfactual or potential outcome if all subjects in population were assigned either treatment.

The inverse probability weighting estimator can be used to demonstrate causality when the researcher cannot conduct a controlled experiment but has observed data to model. Because it is assumed that the treatment is not randomly assigned, the goal is to estimate the counterfactual or potential outcome if all subjects in population were assigned either treatment.

Suppose observed data are [[文件:Wiki-IPWE-Figure1.png]]  <nowiki><math>\{\bigl(X_i,A_i,Y_i\bigr)\}^{n}_{i=1}</math></nowiki> drawn i.i.d (independent and identically distributed) from unknown distribution P, where

Suppose observed data are [[文件:Wiki-IPWE-Figure1.png]]  drawn i.i.d (independent and identically distributed) from unknown distribution P, where<syntaxhighlight lang="mathematica">

 

<math>\begin{align}

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

 

        =  \mathbb{E}_{(X,Y)}\left[ Y(X,a)\right]  =  \mathbb{E}_{(X,A,Y)}\left[ \frac{  Y \mathbf{1}(A=a) }{ P(A=a|X)} \right]. \qquad \cdots \cdots (*)

\end{align}</math>

</syntaxhighlight>

* [[文件:Wiki-IPWE-Figure2.png]] covariates

* [[文件:Wiki-IPWE-Figure2.png]] covariates

* [[文件:Wiki-IPWE-Figure3.png]] are the two possible treatments.

* [[文件:Wiki-IPWE-Figure3.png]] are the two possible treatments.