Importance Sampling 的权重

\[E_p [f(z)] = \int p(z)f(z) dz = \int \frac {p(z)}{q(z)} q(z) f(z) dz = \int \frac{p(z)}{q(z)} f(z) q(z) dz \approx \frac{1}{N}\sum_{i=1}^Nf(z_i)\frac{p(z_i)}{q(z_i)} \]

\[z_i \sim q(z) , i = 1, \dots ,N \]

用\(q(z_i)\)采样，得到\(z_i\), 然后用\(\frac{1}{N}\sum_{i=1}^Nf(z_i)\frac{p(z_i)}{q(z_i)}\),近似原来的期望。当\(p(z_i)\)和它的近似分布\(q(z_i)\)的比值越大时，表明如果按照\(q(z_i)\)来采样时，采到的\(z_i\)代入\(p(z_i)\)时，概率越大，那么在\(p\)分布中，其权重越大，这种样本点越重要(其权重\(\frac{p(z_i)}{q(z_i)}\)越大)。

Sampling-Importance-Resampling
就是在importance sampling的基础上，根据weight \(\frac{p(z_i)}{q(z_i)}\) 重新采样，weight越大，相应的采到的\(z_i\)越多,也就是说在\(p(z)\)中越难采到的那些点会被更多的采样。