Causal - Propensity Score

focus on the treatment assignment

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• Observation: { X , D , Y ( 1 ) , Y ( 0 ) } \{X, D, Y(1), Y(0) \} { X,D,Y(1),Y(0)}

$$P(X,D,Y(1),Y(0))=P(X)P(Y(1),Y(0)|X)P(D|X,Y(1),Y(0))$$

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Definition:

Define the propensity score
$$e(X,Y(1),Y(0))=P(D=1|X,Y(1),Y(0))$$

Under strong ignorability
$$e(X,Y(1),Y(0))=P(D=1|X):=e(X)$$
where $e(X)$ is the conditional probability of receiving the treatment given the observed covariates.

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Useful theorem

Theorem: If D ⊥ { Y ( 1 ) , Y ( 0 ) } ∣ X D\bot\{Y(1), Y(0)\}|X D⊥{ Y(1),Y(0)}∣X, then D ⊥ { Y ( 1 ) , Y ( 0 ) } ∣ e ( X ) D\bot\{Y(1), Y(0)\}|e(X) D⊥{ Y(1),Y(0)}∣e(X).

Thus, $e(X)$ can be used as a dimensional reduction tool.
【$X\to e(X)$】

把问题从$X$转换到$e(X)$上的话，$e(X)$一定是一维的，这样如果把$e(X)$离散化