Approximate risk analysis using numerical integration on sparse grids

Abstract

In risk analysis with prior information, one often needs to evaluate multi-dimensional integrals in order to obtain various characteristics of posterior density functions. Monte Carlo Markov Chain method has been widely used. However, the MCMC method could be computationally intensive. The traditional method for numerical integration requires a full grid evaluation which is computationally intensive when the dimensions are not low. We introduce a novel approach to approximate Bayesian computation by numerical integrations on sparse grids. The number of required grid points for numerical integrations by using sparse grids does not rise exponentially with the dimensions. The proposed method is computationally efficient compared with the traditional numerical integration approach. The posterior density including the normalizing factor can be computed numerically. The posterior mean, median and confidence intervals can then be approximated directly. Both simulated and real data sets are used to evaluate the performance of the proposed method. Numerical experiments suggest that the proposed method could provide fast and efficient approximations with relatively high level of accuracy.