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：
$$\begin{array}{rl}\tau (x)& ={\mathbb{E}}_{\mathbb{P}}(Y(1)-Y(0)\mid x)\\ {\mu }_1(x)& ={\mathbb{E}}_{\mathbb{P}}(Y\mid W=1,X=x)\\ {\mu }_0(x)& ={\mathbb{E}}_{\mathbb{P}}(Y\mid W=0,X=x)\\ \tau (x)& ={\mu }_1(x)-{\mu }_0(x)\end{array}$$

ESN纠偏：
$$\begin{array}{rl}\underset{ESTR}{P(Y,W=1\mid X)}& =\underset{TR}{P(Y\mid W=1,X)}\cdot \underset{\pi }{P(W=1\mid X)}\\ & ={\mu }_1\cdot \pi \\ \underset{ESCR}{P(Y,W=0\mid X)}& =\underset{CR}{P(Y\mid W=0,X)}\cdot \underset{1-\pi }{P(W=0\mid X)}\\ & ={\mu }_0\cdot (1-\pi )\\ {\mu }_1& =\frac{ESTR}{\pi }\\ {\mu }_0& =\frac{ESCR}{1-\pi }\\ \tau & ={\mu }_1-{\mu }_0\end{array}$$

---

参考

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