can be used
in the calculation of
, if we
can be rewritten as follows:
It easily to see that the method works well if overlaps well with . Thus, a proper choice of can be made by studying where the probability mass of is concentrated (for example finding the mode of the distribution in a numerical way). Often a Gaussian function is used for , with parameters chosen to approximate in the proximity of the mode, as described in Sect. 5.10. In other cases, other functions can be used which have more pronounced tails, like -Student or Cauchy distributions. Special techniques, into which we cannot enter here, allow independent random numbers to be generated and, subsequently, by proper rotations, turned into other numbers which have a correlation matrix equal to that of the multi-dimensional Gaussian which approximates .
Note, finally, that, contrary to the rejection sampling, importance sampling is not suitable for generate samples of `unweighted events', such as those routinely used in the planning and the analysis of many kind experiments, especially particle physics experiments.