Open Access
2019 Criteria for posterior consistency and convergence at a rate
B. J. K. Kleijn, Y. Y. Zhao
Electron. J. Statist. 13(2): 4709-4742 (2019). DOI: 10.1214/19-EJS1633


Frequentist conditions for asymptotic consistency of Bayesian procedures with i.i.d. data focus on lower bounds for prior mass in Kullback-Leibler neighbourhoods of the data distribution. The goal of this paper is to investigate the flexibility in these criteria. We derive a versatile new posterior consistency theorem, which is used to consider Kullback-Leibler consistency and indicate when it is sufficient to have a prior that charges metric balls instead of KL-neighbourhoods. We generalize our proposal to sieved models with Barron’s negligible prior mass condition and to separable models with variations on Walker’s condition. Results are also applied in semi-parametric consistency: support boundary estimation is considered explicitly and consistency is proved in a model for which Kullback-Leibler priors do not exist. As a further demonstration of applicability, we consider metric consistency at a rate: under a mild integrability condition, the second-order Ghosal-Ghosh-van der Vaart prior mass condition can be relaxed to a lower bound for ordinary KL-neighbourhoods. The posterior rate is derived in a parametric model for heavy-tailed distributions in which the Ghosal-Ghosh-van der Vaart condition cannot be satisfied by any prior.


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B. J. K. Kleijn. Y. Y. Zhao. "Criteria for posterior consistency and convergence at a rate." Electron. J. Statist. 13 (2) 4709 - 4742, 2019.


Received: 1 October 2019; Published: 2019
First available in Project Euclid: 20 November 2019

zbMATH: 07136628
MathSciNet: MR4033683
Digital Object Identifier: 10.1214/19-EJS1633

Primary: 62G07 , 62G20

Keywords: Asymptotic consistency , Bayesian consistency , marginal consistency , posterior consistency , posterior rate of convergence

Vol.13 • No. 2 • 2019
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