General multi-dimensional probability integration by directional simulation

Abstract

n-Dimensional integration by Monte Carlo directional importance sampling using analytical or numerical integration for each simulated direction is discussed. A suitable general class of sampling distributions is considered. The class is denoted as the class of mixed ūgm-centred standard Gaussian and D -truncated ūgm-centred standard Gaussian distributions, where ūgm is a position vector and is a pragmatically chosen subset of R n. The relative efficiency of different choices of the sampling distribution from the class is studied by way of examples for probability integration in particular. In addition, the same integrals are formulated in different variable representations by the substitution method before the directional Monte Carlo integration is carried out. Moreover, it is demonstrated how sensitivities of the integrals with respect to parameter variations or integration boundary variations can be obtained during the same Monte Carlo simulation primarily set up for obtaining the integral. The novelty compared with previous reports on the topic of integration in the standard Gaussian space is the demonstration that the combination of simulation and numerical integration can be applied for general integrals without any significant change of efficiency.