Abstract
Bayesian statistics has two common measures of central tendency of a posterior distribution: posterior means and Maximum A Posteriori (MAP) estimates. In this paper, we discuss a connection between MAP estimates and posterior means. We derive an asymptotic condition for a pair of prior densities under which the posterior mean based on one prior coincides with the MAP estimate based on the other prior. A sufficient condition for the existence of this prior pair relates to α-flatness of the statistical model in information geometry. We also construct a matching prior pair using α-parallel priors. Our result elucidates an interesting connection between regularization in generalized linear regression models and posterior expectation.
Funding Statement
This work is supported by JSPS KAKENHI (JP19K20222, JP20K23316, JP21H05205, JP21K12067, JP22H00510, JP23K11024), MEXT (JPJ010217), and “Strategic Research Projects” grant (2022-SRP-13) from ROIS (Research Organization of Information and Systems).
Acknowledgments
The authors thank the Editor, the handling editor, and two referees for their constructive comments that have improved the quality of this paper. The authors thank Kaoru Irie for helpful comments to the early version of this work. The authors thank Ryoya Kaneko for sharing his python codes of data preprocessing. The authors thank Takemi Yanagimoto for helpful discussions.
Citation
Michiko Okudo. Keisuke Yano. "Matching Prior Pairs Connecting Maximum A Posteriori Estimation and Posterior Expectation." Bayesian Anal. Advance Publication 1 - 25, 2024. https://doi.org/10.1214/24-BA1500
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