Integration of multimodal functions by Monte Carlo importance sampling

Man Suk Oh, James O. Berger

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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.

Original languageEnglish
Pages (from-to)450-456
Number of pages7
JournalJournal of the American Statistical Association
Issue number422
StatePublished - Jun 1993

Bibliographical note

Funding Information:
* Man-Suk Oh is Assistant Professor, Department of Statistics, Ewha Woman's University, Seoul, Korea. James O. Bergeris Richard M. Brumfield Distinguished Professor of Statistics, Department of Statistics, Purdue University, West Lafayette, IN 47907. This research was supported by National Science Foundation Grants DMS-87 17799 and DMS-8923071. Parts of the work weredone whilethe firstauthor was visitingthe University ofCalifornia, Berkeley.


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


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