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Posterior distribution of nondifferentiable functions

Authors: Toru Kitagawa , Jose Luis Montiel Olea , Jonathan Payne and Amilcar Velez
Date: 03 April 2019
Type: cemmap Working Paper, CWP17/19


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 θ^n, 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(θ^n) 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).

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Previous version:
Toru Kitagawa, Jose Luis Montiel Olea and Jonathan Payne October 2017, Posterior distribution of nondifferentiable functions, cemmap Working Paper, CWP44/17, The IFS

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