Posterior distribution of nondifferentiable functions

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Abstract

This paper examines the asymptotic behavior of the posterior distribution of a possibly nondifferentiable function g(θ), where θ is a finite-dimensional parameter of either a parametric or semiparametric model. The main assumption is that the distribution of a suitable estimator θ̂n, its bootstrap approximation, and the Bayesian posterior for θ all agree asymptotically.

It is shown that whenever g is locally Lipschitz, though not necessarily differentiable, the posterior distribution of g(θ) and the bootstrap distribution of g(θ̂n) coincide asymptotically. One implication is that Bayesians can interpret bootstrap inference for g(θ) as approximately valid posterior inference in a large sample. Another implication—built on known results about bootstrap inconsistency—is that credible intervals for a nondifferentiable parameter g(θ) cannot be presumed to be approximately valid confidence intervals (even when this relation holds true for θ).

Keywords

Bootstrap
Bernstein–von Mises theorem
Directional differentiability
Posterior inference

Cited by (0)

We would like to thank Gary Chamberlain, Tim Cogley, Han Hong, Hiroaki Kaido, Oliver Linton, Ulrich Müller, Mikkel Plagborg-Möller, Andres Santos, Quang Vuong, and three anonymous referees for detailed comments and suggestions on an earlier draft of this paper. We would also like to thank Ruby Steedle for excellent research assistance. All errors remain our own. Financial support from the ESRC through the ESRC Centre for Microdata Methods and Practice (CeMMAP) (Grant Number RES-589-28-0001) and from the ERC through the ERC starting grant (Grant Number EPP-715940) is gratefully acknowledged. Velez thanks the Macro Modeling Division, Central Reserve Bank of Peru, where part of this work was written. First version: July 25th, 2016. This draft: October 31st, 2019.