Importance sampling is a variance reduction technique that can be used in the Monte Carlo method.
The Monte Carlo method is used to approximate the integral of a function f as the average of the function evaluated at a set of points x1,...,xN.
![\int_a^b f(u)\,du = \mathrm{E}\{f(X)(b-a)\}\quad (X\sim unif(a,b))\approx \left[ \frac{1}{N}\,\sum_{i=1}^N f(x_i) \right] \left( b-a \right),](a/../upload/math/5f31ee7bcf1a6cdc56e96385888a78db.png)
where x1,...,xN were drawn from a uniform distribution over the inter val [a,b), and E{.} denotes the expected value. If we have additional knowledge about what f looks like and can find a function g similar to f, we can rewrite the integral as
![\int_a^b \frac{f(u)}{g(u)} g(u) \,du \approx \left[ \frac{1}{N}\,\sum_{i=1}^N \frac{f(y_i)}{g(y_i)} \right],](a/../upload/math/467333435e8f11484a655e5b3fd884ce.png)
where the yi now follow a g distribution.
A popular method to sample from the g distribution is Metropolis sampling.
If f(u) / g(u) has a smaller variance than f(u), the new sequence will converge faster.