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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).
Authors

Research Associate University College London and Brown University
Toru is a Research Associate of the IFS, a Professor of Economics at UCL and an Associate Professor in the Department of Economics at Brown University

Jose Luis Montiel Olea

Jonathan Payne

Amilcar Velez
Working Paper details
- DOI
- 10.1920/wp.cem.2019.1719
- Publisher
- The IFS
Suggested citation
Kitagawa, T et al. (2019). Posterior distribution of nondifferentiable functions. London: The IFS. Available at: https://ifs.org.uk/publications/posterior-distribution-nondifferentiable-functions-1 (accessed: 8 February 2025).
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