Global Likelihood Sampler for Multimodal Distributions

Abstract

Drawing samples from a target distribution is essential for statistical computations when the analytical solution is infeasible. Many existing sampling methods may be easy to fall into the local mode or strongly depend on the proposal distribution when the target distribution is complicated. In this article, the Global Likelihood Sampler (GLS) is proposed to tackle these problems and the GL bootstrap is used to assess the Monte Carlo error. GLS takes the advantage of the randomly shifted low-discrepancy point set to sufficiently explore the structure of the target distribution. It is efficient for multimodal and high-dimensional distributions and easy to implement. It is shown that the empirical cumulative distribution function of the samples uniformly converges to the target distribution under some conditions. The convergence for the approximate sampling distribution of the sample mean based on the GL bootstrap is also obtained. Moreover, numerical experiments and a real application are conducted to show the effectiveness, robustness, and speediness of GLS compared with some common methods. It illustrates that GLS can be a competitive alternative to existing sampling methods. Supplementary materials for this article are available online.

Keywords:

• Convergency
• GL bootstrap
• Low-discrepancy point set
• Robustness

Supplementary Materials

Code: The supplemental file includes all the programs to reproduce the results in the article. (GLS_CODE_ALL.zip)

Appendix: The supplemental file includes the Appendix which gives all the proofs and additional results. (GLS-appendix.pdf)

Acknowledgments

The authors thank the editor, the associate editor, and two reviewers for their helpful comments. The authors would like to thank Yuchung Wang for his help.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (11871288 and 12131001), the National Ten Thousand Talents Program, the Fundamental Research Funds for the Central Universities, LPMC, and KLMDASR.