重要性采样与方差控制Importance Sampling & Variance Control
从另一种分布抽样时，怎样加权才能估计目标平均值？How can samples from another distribution be weighted to estimate a target expectation?
前置知识：Prerequisites: B20：抽样误差与蒙特卡洛估计。B20: sampling error and Monte Carlo estimation.
复习前置：Review prerequisites: B20 · 抽样误差与蒙特卡洛估计Sampling Error & Monte Carlo Estimation
学完后你能：After this lesson you can: 计算重要性权重，并区分缺失覆盖与高方差。Calculate importance weights and distinguish missing coverage from high variance.
动手观察Explore
改变提议分布中稀有类别的概率，抽样并比较加权估计与理论方差；检查缺失覆盖。Change the proposal probability of a rare category, sample and compare the weighted estimate and theoretical variance; inspect missing coverage.
从观察到解释From observation to explanation
先算一个有限例子Start with a finite example
目标分布有稀有与常见两类，稀有概率为 0.1；令稀有类 f=10、常见类 f=0，目标期望为 1。The target distribution has rare and common categories, with rare probability 0.1. Let f=10 for rare and f=0 for common; the target expectation is 1.
改变抽样分布Change the sampling distribution
若提议分布以 q 的概率抽到稀有类，每个稀有样本的加权值为 (0.1/q)×10=1/q，常见样本贡献 0。把加权值取平均，期望仍为 1。If the proposal draws the rare category with probability q, each rare sample contributes (0.1/q)×10=1/q and common samples contribute 0. Their average still has expectation 1.
覆盖与方差是不同问题Coverage and variance are different issues
本例单次加权值的方差为 1/q−1；q 太小时，少量极大权重可能造成高方差。q=0 时稀有类别缺失，无法估计其非零贡献；完整重加权通常要求提议分布覆盖目标分布的支持。Here the variance of one weighted value is 1/q−1; very small q produces rare, large weights and high variance. At q=0 the rare category is missing, so its nonzero contribution cannot be estimated. General reweighting typically requires proposal support to cover target support.
公式与符号Formula and symbols
p 是目标分布，q 是提议分布，f 是要平均的数值；p(X)/q(X) 是权重。相关非零项必须由 q 覆盖，且期望要存在。p is the target distribution, q the proposal, and f the quantity to average. p(X)/q(X) is the weight. The proposal must cover relevant nonzero terms and the expectation must exist.
留意这个区别Watch for this
覆盖不足与高方差不同；裁剪权重通常引入偏差，不能免费消除方差。Missing coverage and high variance are distinct. Clipping weights generally introduces bias and does not eliminate variance for free.
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