Posterior distribution of nondifferentiable functions

Toru Kitagawa, Jose Luis Montiel Olea and Jonathan Payne

No 44/17, CeMMAP working papers from Institute for Fiscal Studies

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).

Date: 2017-10-03
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Persistent link: https://EconPapers.repec.org/RePEc:azt:cemmap:44/17

DOI: 10.1920/wp.cem.2017.4417

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