Abstract
The theory of two projections is utilized to study two-component Gibbs samplers. Through this theory, previously intractable problems regarding the asymptotic variances of two-component Gibbs samplers are reduced to elementary matrix algebra exercises. It is found that in terms of asymptotic variance, the two-component random-scan Gibbs sampler is never much worse, and could be considerably better than its deterministic-scan counterpart, provided that the selection probability is appropriately chosen. This is especially the case when there is a large discrepancy in computation cost between the two components. The result contrasts with the known fact that the deterministic-scan version has a faster convergence rate, which can also be derived from the method herein. On the other hand, a modified version of the deterministic-scan sampler that accounts for computation cost can outperform the random-scan version.
Funding Statement
The author was supported by NSF Grant DMS-2112887.
Acknowledgments
The author thanks the Editors, an anonymous Associate Editor, and two anonymous referees for their valuable feedback. In particular, the referees suggested studying the modified DG sampler and its parallelization. The author would like to thank Riddhiman Bhattacharya, Austin Brown, James P. Hobert, Galin L. Jones, and Haoxiang Li for their helpful comments.
Citation
Qian Qin. "Analysis of two-component Gibbs samplers using the theory of two projections." Ann. Appl. Probab. 34 (5) 4310 - 4341, October 2024. https://doi.org/10.1214/24-AAP2066
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