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LARGE SAMPLE JUSTIFICATIONS FOR THE BAYESIAN EMPIRICAL LIKELIHOOD

Published online by Cambridge University Press:  05 December 2022

Naoya Sueishi Open the ORCID record for Naoya Sueishi [Opens in a new window]
Naoya Sueishi*
Affiliation:
Kobe University
*
Address correspondence to Naoya Sueishi, Graduate School of Economics, Kobe University, 2-1 Rokkodai-cho, Nada-ku, Kobe, Hyogo 657-8501, Japan; e-mail: [email protected].
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Abstract

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This study investigates the asymptotic properties of the Bayesian empirical likelihood (BEL), which uses the empirical likelihood as an alternative to a parametric likelihood for Bayesian inference. We establish two asymptotic equivalence results based on the Bernstein–von Mises (BvM) theorem by introducing a new formulation of the moment restriction model. First, the limiting posterior distribution of the BEL is the same as that of a parametric Bayesian method that uses the likelihood of a least favorable model of the moment restriction model. Second, the limiting posterior distribution is also the same as that of a semiparametric Bayesian method that places priors on both a finite-dimensional parameter of interest and an infinite-dimensional nuisance parameter. Because parametric and semiparametric Bayesian methods are legitimate Bayesian procedures, the equivalence results provide a large sample justification for the BEL as a Bayesian inference method. Moreover, the BvM theorem provides a frequentist justification for BEL posterior inference.

Type
MISCELLANEA
Information
Econometric Theory , Volume 40 , Issue 4 , August 2024 , pp. 926 - 956
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Footnotes

I would like to thank Patrik Guggenberger and two anonymous referees for their comments and suggestions. I would also like to thank Mototsugu Shintani, Kohtaro Hitomi, Yoshihiko Nishiyama, and Takahide Yanagi for their comments. This research was supported by JSPS KAKENHI Grant Number 18K01547.

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