🪆 Importance Sampling

Importance sampling is a form of 🤔 Monte Carlo Sampling that uses an alternate distribution in case requires too many samples for an accurate estimate. Since we can write

we can compute the expectation over instead. Sampling , we have

However, we need to choose a good distribution . The variance of our estimate

If the fraction is large, the variance of our estimate can increase; this occurs if is too small and neither nor are small enough. Since we’re sampling from , we still need to be a simple distribution.

Info

Typically, the discrepancy between and leads to common underestimations of with rare cases of extreme overestimation.

Biased Importance Sampling §

If we allow for a biased sample, we can use more complicated distributions for without requiring normalization. Specifically,

where and are unnormalized forms of and and are drawn from .