IPW逆概率加权

IPW是个非常优雅的纠偏方法。下面介绍如何利用它来实现纠偏：

uplift定义如下：

$$\begin{aligned} \tau(x) &= \mathbb{E}_{\mathbb{P}}(Y(1)-Y(0) \mid x) \\ \mu_1(x) &= \mathbb{E}_{\mathbb{P}}(Y \mid W=1, X=x) \\ \mu_0(x) &= \mathbb{E}_{\mathbb{P}}(Y \mid W=0, X=x) \\ \tau(x) &= \mu_1(x) - \mu_0(x) \end{aligned}$$

ESN纠偏：

$$\begin{aligned} \underbrace{P(Y, W=1 \mid X)}_{E S T R} & =\underbrace{P(Y \mid W=1, X)}_{T R} \cdot \underbrace{P(W=1 \mid X)}_\pi \\ & =\mu_1 \cdot \pi \\ \underbrace{P(Y, W=0 \mid X)}_{E S C R} & =\underbrace{P(Y \mid W=0, X)}_{C R} \cdot \underbrace{P(W=0 \mid X)}_{1-\pi} \\ & =\mu_0 \cdot(1-\pi) \\ \mu_1 &= \frac{ESTR}{\pi} \\ \mu_0 &= \frac{ESCR}{1-\pi} \\ \tau &= \mu_1 - \mu_0 \end{aligned}$$

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参考

• DESCN: Deep Entire Space Cross Networks | 多任务、端到端、IPW的共舞