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Posterior distribution of nondifferentiable functions

Authors: Toru Kitagawa , Jose Luis Montiel Olea and Jonathan Payne
Date: 03 October 2017
Type: cemmap Working Paper, CWP44/17

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 of either a parametric or semiparametric model. The main assumption is that the distribution of a suitable 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 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 credible sets for a nondifferentiable parameter g(theta) cannot be presumed to be approximately valid confidence sets (even when this relation holds true for theta).

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Previous version:
Toru Kitagawa, Jose Luis Montiel Olea and Jonathan Payne May 2016, Posterior distribution of nondifferentiable functions, cemmap Working Paper, CWP20/16, IFS

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