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Posterior Distribution of Nondifferentiable Functions

Kitagawa, T; Montiel-Olea, J; Payne, J; (2016) Posterior Distribution of Nondifferentiable Functions. (cemmap working paper 20/16). Institute for Fiscal Studies: London, UK. Green open access

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Abstract

This paper examines the asymptotic behavior of the posterior distribution of a possibly nondifferentiable function g(theta), where theta is a finite dimensional parameter. The main assumption is that the distribution of the maximum likelihood estimator theta_n, its bootstrap approximation, and the Bayesian posterior for theta all agree asymptotically. It is shown that whenever g is Lipschitz, though not necessarily differentiable, the posterior distribution of g(theta) and the bootstrap distribution of g(theta_n) coincide asymptotically. One implication is that Bayesians can interpret bootstrap inference for g(theta) as approximately valid posterior inference in a large sample. Another implication—built on known results about bootstrap inconsistency—is that the posterior distribution of g(theta) does not coincide with the asymptotic distribution of g(theta_n) at points of nondifferentiability. Consequently, frequentists cannot presume that credible sets for a nondifferentiable parameter g(theta) can be interpreted as approximately valid confidence sets (even when this relation holds true for theta).

Type: Working / discussion paper
Title: Posterior Distribution of Nondifferentiable Functions
Open access status: An open access version is available from UCL Discovery
DOI: 10.1920/wp.cem.2016.2016
Publisher version: http://dx.medra.org/10.1920/wp.cem.2016.2016
Language: English
Additional information: This version is the version of record. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: Bootstrap, Bernstein-von Mises Theorem, Directional Differentiability, Posterior Inference
UCL classification: UCL
UCL > Provost and Vice Provost Offices > UCL SLASH
UCL > Provost and Vice Provost Offices > UCL SLASH > Faculty of S&HS
UCL > Provost and Vice Provost Offices > UCL SLASH > Faculty of S&HS > Dept of Economics
URI: https://discovery.ucl.ac.uk/id/eprint/1522146
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