# 重要性采样

## 原理

${\displaystyle {\widehat {\mathbf {E} }}_{n}[X;P]={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}$

${\displaystyle \operatorname {var} [{\widehat {\mathbf {E} }}_{n};P]=\operatorname {var} [X;P]/n}$

${\displaystyle \mathbf {E} [X;P]=\mathbf {E} \left[{\frac {X}{L}};P^{(L)}\right].}$

X在Ω上不变号时，最优的L${\displaystyle L^{*}={\frac {X}{\mathbf {E} [X;P]}}\geq 0}$。此时X/L*即为要估计的E[X;P]，只需一个样本便可得到该值。然而由于L*与要估计的E[X;P]有关，在实际操作中我们无法取到理论上最优的L*。不过，我们仍可以采用如下方式逼近该理论值：

{\displaystyle {\begin{aligned}\forall a\in \mathbb {R} ,\;P^{(L^{*})}(X\in [a;a+da])&=\int _{\omega \in \{X\in [a;a+da]\}}{\frac {X(\omega )}{E[X;P]}}dP(\omega )\\&={\frac {1}{E[X;P]}}\;a\,P(X\in [a;a+da])\end{aligned}}}

${\displaystyle E[X;P]=\int _{a=-\infty }^{+\infty }a\,P(X\in [a;a+da])}$

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