# Importance Simulation¶

Let us note
.
The goal is to estimate the following probability:

This is a sampling-based method. The main idea of the Importance
Sampling method is to replace the initial probability distribution of
the input variables by a more “efficient” one. “Efficient” means that
more events will be counted in the failure domain and
thus reduce the variance of the estimator of the probability of
exceeding a threshold. Let be a random vector
such that its probability density function
almost everywhere in the
domain ,

The estimator built by Importance Sampling method is:

where:

is the total number of computations,

the random vectors are independent, identically distributed and following the probability density function

**Confidence Intervals**

With the notations,

The asymptotic confidence interval of order associated to the estimator is

where is the quantile from the standard distribution .

This method could also be found under the name “Strategic Sampling”, “Weighted Sampling” or “Biased Sampling” (even if this estimator is not biased as it gives exactly the same result).