February 2021 Frequentist validity of Bayesian limits
B. J. K. Kleijn
Ann. Statist. 49(1): 182-202 (February 2021). DOI: 10.1214/20-AOS1952


To the frequentist who computes posteriors, not all priors are useful asymptotically: in this paper, a Bayesian perspective on test sequences is proposed and Schwartz’s Kullback–Leibler condition is generalised to widen the range of frequentist applications of posterior convergence. With Bayesian tests and a weakened form of contiguity termed remote contiguity, we prove simple and fully general frequentist theorems, for posterior consistency and rates of convergence, for consistency of posterior odds in model selection, and for conversion of sequences of credible sets into sequences of confidence sets with asymptotic coverage one. For frequentist uncertainty quantification, this means that a prior inducing remote contiguity allows one to enlarge credible sets of calculated, simulated or approximated posteriors to obtain asymptotically consistent confidence sets.


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B. J. K. Kleijn. "Frequentist validity of Bayesian limits." Ann. Statist. 49 (1) 182 - 202, February 2021. https://doi.org/10.1214/20-AOS1952


Received: 1 November 2017; Revised: 1 January 2020; Published: February 2021
First available in Project Euclid: 29 January 2021

Digital Object Identifier: 10.1214/20-AOS1952

Primary: 62G05 , 62G20
Secondary: 62B15 , 62C10 , 62G10

Keywords: contiguity , coverage , Credible set , frequentist consistency , Model selection , posterior consistency , Posterior odds , posterior rate of convergence , uncertainty quantification

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 1 • February 2021
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