逆確率重み付け
μ ^ a , n I P W E = 1 n ∑ i = 1 n Y i 1 A i = a p ^ n ( A i ∣ X i ) {\displaystyle {\hat {\mu }}_{a,n}^{\mathrm {IPWE} }={\frac {1}{n}}\sum _{i=1}^{n}\,Y_{i}{\frac {\mathbf {1} _{A_{i}=a}}{{\hat {p}}_{n}(A_{i}\mid X_{i})}}}
μ ^ a , n A I P W E = 1 n ∑ i = 1 n ( Y i 1 A i = a p ^ n ( A i ∣ X i ) − 1 A i = a − p ^ n ( A i ∣ X i ) p ^ n ( A i ∣ X i ) Q ^ n ( X i , a ) ) = 1 n ∑ i = 1 n ( Q ^ n ( X i , a ) ) + 1 n ∑ i = 1 n 1 A i = a p ^ n ( A i ∣ X i ) ( Y i − Q ^ n ( X i , a ) ) {\displaystyle {\begin{aligned}{\hat {\mu }}_{a,n}^{\mathrm {AIPWE} }&={\frac {1}{n}}\sum _{i=1}^{n}\left({\frac {Y_{i}\,1_{A_{i}=a}}{{\hat {p}}_{n}(A_{i}\mid X_{i})}}-{\frac {1_{A_{i}=a}-{\hat {p}}_{n}(A_{i}\mid X_{i})}{{\hat {p}}_{n}(A_{i}\mid X_{i})}}{\hat {Q}}_{n}(X_{i},a)\right)\\&={\frac {1}{n}}\sum _{i=1}^{n}\left({\hat {Q}}_{n}(X_{i},a)\right)+{\frac {1}{n}}\sum _{i=1}^{n}{\frac {1_{A_{i}=a}}{{\hat {p}}_{n}(A_{i}\mid X_{i})}}\left(Y_{i}-{\hat {Q}}_{n}(X_{i},a)\right)\end{aligned}}}