# 因果推断｜干预影响（一）

Posted by Derek on May 24, 2020

# 1. 干预（Intervention）

考虑以下模型：

Figure 1.1 A graphical model1

Figure 1.2 A graphical model1

Figure 2.1 A graphical model2

Figure 2.2 A graphical model3

\begin{aligned} P(Y=y|do(X=x))&=P_m(Y=y|X=x) \\&=\sum_zP_m(Y=y|X=x, Z=z)P_m(Z=z|X=x) \\&=\sum_zP_m(Y=y|X=x, Z=z)P_m(Z=z) \\&=\sum_zP(Y=y|X=x, Z=z)P(Z=z) \end{aligned}

 Drug No drug Men $81$ out of $87$ recovered ($93\%$) $234$ out of $270$ recovered ($87\%$) Women $192$ out of $263$ recovered ($73\%$) $55$ out of $80$ recovered ($69\%$) Combined data $273$ out of $350$ recovered ($78\%$) $289$ out of $350$ recovered ($83\%$)

\begin{aligned}P(Y=1|do(X=1))&=P(Y=1|X=1, Z=1)P(Z=1)+P(Y=1|X=1, Z=0)P(Z=0) \\&=0.93 \times \frac{87+270}{700}+0.73 \times \frac{263+80}{700}=0.832 \end{aligned}

Figure 2.3 A graphical model3

## 2.1 调整还是不调整，这是一个问题

\begin{aligned} P(Y=y|do(X=x))=P(y|do(x))&=\sum_zP(Y=y|X=x, PA=z)P(PA=z) \\&=\sum_z\frac{P(X=x, Y=y, PA=z)}{P(X=x|PA=z)} \end{aligned}

## 2.2 多重干预和截断乘积法则

Figure 2.4 A graphical model4

# Reference

1. Pearl, J., Glymour, M., & Jewell, N. P. (2016). Causal inference in statistics: A primer. John Wiley & Sons, 54.

2. Pearl, J., Glymour, M., & Jewell, N. P. (2016). Causal inference in statistics: A primer. John Wiley & Sons, 55.

3. Pearl, J., Glymour, M., & Jewell, N. P. (2016). Causal inference in statistics: A primer. John Wiley & Sons, 58.

4. Pearl, J., Glymour, M., & Jewell, N. P. (2016). Causal inference in statistics: A primer. John Wiley & Sons, 49.