Integration of Multimodal Functions by Monte Carlo Importance Sampling

Abstract

Numerical integration of a multimodal integrand f(θ) is approached by Monte Carlo integration via importance sampling. A mixture of multivariate t density functions is suggested as an importance function g(θ), for its easy random variate generation, thick tails, and high flexibility. The number of components in the mixture is determined by the number of modes of f(θ), and the mixing weights and location and scale parameters of the component distributions are determined by numerical minimization of a Monte Carlo estimate of the squared variation coefficient of the weight function f(θ)/g(θ). Stratified importance sampling and control variates are shown to be particularly effective variance reduction techniques in this case. The algorithm is applied to a 10-dimensional example and shown to yield significant improvement over usual integration schemes.

Key Words:

• Control variate
• Mixture
• Numerical integration
• Stratified importance sampling
• Weight function