Réduction de la variance

In mathematics, more specifically in the theory of Monte Carlo methods, variance reduction is a procedure used to increase the precision of the estimates obtained for a given simulation or computational effort. Every output random variable from the simulation is associated with a variance which limits the precision of the simulation results. In order to make a simulation statistically efficient, i.e., to obtain a greater precision and smaller confidence intervals for the output random variable of interest, variance reduction techniques can be used. The main ones are common random numbers, antithetic variates, control variates, importance sampling, stratified sampling, moment matching, conditional Monte Carlo and quasi random variables. For simulation with black-box models subset simulation and line sampling can also be used. Under these headings are a variety of specialized techniques; for example, particle transport simulations make extensive use of "weight windows" and "splitting/Russian roulette" techniques, which are a form of importance sampling. Suppose one wants to compute with the random variable defined on the probability space . Monte Carlo does this by sampling i.i.d. copies of and then to estimate via the sample-mean estimator Under further mild conditions such as , a central limit theorem will apply such that for large , the distribution of converges to a normal distribution with mean and standard error . Because the standard deviation only converges towards at the rate , implying one needs to increase the number of simulations () by a factor of to halve the standard deviation of , variance reduction methods are often useful for obtaining more precise estimates for without needing very large numbers of simulations. The common random numbers variance reduction technique is a popular and useful variance reduction technique which applies when we are comparing two or more alternative configurations (of a system) instead of investigating a single configuration. CRN has also been called correlated sampling, matched streams or matched pairs.