Use of the important sample in Monte Carlo method

The problem of the calculation of the normalize constant of density using the given sample is considered. The analog of the importance of sampling algorithm with the important (in the sense of minimization of variance of the standard Monte Carlo method) stochastic values is constructed and investigated. It is shown that in the case, when an integral is calculated by the Monte Carlo method and the important sample is not given beforehand, it is better to use not exactly important but close to important sample values. For this case, the new algorithm (called the double-sided geometric method) is proposed. This algorithm allows to reduce the cost of calculations.