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== Application to simulation ==
'''Importance sampling''' is a [[variance reduction]] technique that can be used in the [[Monte Carlo method]]. 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 test|likelihood ratio]], that is, the [[Radon–Nikodym derivative]] of the true underlying distribution with respect to the biased simulation distribution.
The fundamental issue in implementing importance sampling simulation is the choice of the biased distribution which encourages the important regions of the input variables. Choosing or designing a good biased distribution is the "art" of importance sampling. The rewards for a good distribution can be huge run-time savings; the penalty for a bad distribution can be longer run times than for a general Monte Carlo simulation without importance sampling.
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