# 因果推断｜干预影响（三）

Posted by Derek on June 7, 2020

# 1. 条件干预（Conditional Intervention）与具体的协变量效应（Covariate-Specific Effect）

$Z$的具体效应可以通过类似后门调整的过程来确定。当我们的目标是估计$P(Y=y|do(X=x))$时，如果$S$阻塞了从$X$到$Y$的所有后门路径，那么对$S$的调整是合理的。现在，我们希望确定$P(Y=y|do(X=x), Z=z),$ 我们需要确保当我们添加新变量$Z$到条件集中时，这些路径仍然被阻塞。

\begin{aligned} &\ \ \ \ \ \ P(Y=y|do(X=g(Z))) \\&=\sum_z P(Y=y|do(X=g(Z)), Z=z)P(Z=z|do(X=g(Z))) \\&=\sum_z P(Y=y|do(X=g(z)), Z=z)P(Z=z) \end{aligned}

# 2. 逆概率加权（Inverse Probability Weighing）

Table 2.1 Joint probability distribution $P(X, Y, Z)$ for the drug-gender-recovery1
 $X$ $Y$ $Z$ % of population Yes Yes Male 0.116 Yes Yes Female 0.274 Yes No Male 0.009 Yes No Female 0.101 No Yes Male 0.334 No Yes Female 0.079 No No Male 0.051 No No Female 0.036

Table 2.2 Conditional probability distribution $P(Y, Z|X)$ for drug users in the population
 $X$ $Y$ $Z$ % of population Yes Yes Male 0.232 Yes Yes Female 0.548 Yes No Male 0.018 Yes No Female 0.202

\begin{aligned} P(X=\text{Yes}|Z=\text{Male})&=\frac{0.116+0.009}{0.116+0.009+0.334+0.051} \approx 0.245 \\ P(X=\text{Yes}|Z=\text{Female})&=\frac{0.274+0.101}{0.274+0.101+0.079+0.036} \approx 0.765 \end{aligned}

Table 2.3 Probability distribution for the population of Table 2.1 under the
intervention $do(X=\text{Yes}),$ determined via the inverse probability method
 $X$ $Y$ $Z$ % of population Yes Yes Male 0.473 Yes Yes Female 0.358 Yes No Male 0.037 Yes No Female 0.132

# Reference

1. Pearl, J., Glymour, M., & Jewell, N. P. (2016). Causal inference in statistics: A primer. John Wiley & Sons, 73.