In statistics, importance sampling is a general technique for estimating properties of a particular distribution, while only having samples generated from a different distribution rather than the distribution of interest. It is related to umbrella sampling in computational physics. Depending on the application, the term may refer to the process of sampling from this alternative distribution, the process of inference, or both.
Importance sampling exists mainly to address one of the greatest problems with Monte Carlo sampling in that many samples are redundant by falling into regions containing very little value. Which does have a purpose in its own right to bring down the global value, but we don't need tons of estimates to do that, if we know that the surrounding areas of what is of interest are values close to zero we can easily in one step recalculate the average value of high density regions over a wider domain.
(High density regions are more important to sample.)
Suppose the density p(x) roughly approximates the density (of interest) q(x),
- q(x): the distribution we wish to sample
- p(x): the distribution we will use the draw samples from
then
This forms the basis for the method of importance sampling, with
where the xi are drawn from the distribution given by p(x).
An alternative formulation of importance sampling is to use
where xi are drawn from the density p(x).
The idea behind importance sampling is that certain values of the input random variables in a simulation have more impact on the parameter being estimated than others. If these "important" values are emphasized by sampling more frequently, then the estimator variance can be reduced. Hence, the basic methodology in importance sampling is to choose a distribution which "encourages" the important values. This use of "biased" distributions will result in a biased estimator if it is applied directly in the simulation. However, the simulation outputs are weighted to correct for the use of the biased distribution, and this ensures that the new importance sampling estimator is unbiased. The weight is given by the likelihood ratio, that is, the Radon–Nikodym derivative of the true underlying distribution with respect to the biased simulation distribution.