Importance Sampling

Also called biased sampling , this is one of the variance-reducing techniques in Monte Carlo methods. A key issue in order to achieve small errors on the obtained result (for a given number of samplings) is a suitable strategy of sampling the available multidimensional space. If the volume to be sampled is large, but is characterized by small probabilities over most parts, one achieves importance sampling by approximating the probability distribution by some function P(x), and generating randomly x according to P, weighting each result at the same time by . If a Monte Carlo calculation is visualized as a numerical integration in one dimension, say, importance sampling translates into a change of integration variable (interpret f(x) as a probability density function and use the transformation rule):