Posterior distribution of nondifferentiable functions

Toru Kitagawa, José Luis Montiel Olea, Jonathan Payne, Amilcar Velez

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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).

Original languageEnglish (US)
Pages (from-to)161-175
Number of pages15
JournalJournal of Econometrics
Issue number1
StatePublished - Jul 2020

All Science Journal Classification (ASJC) codes

  • Economics and Econometrics


  • Bernstein–von Mises theorem
  • Bootstrap
  • Directional differentiability
  • Posterior inference


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