Posterior distribution of nondifferentiable functions

Toru Kitagawa (), Jose Luis Montiel Olea () and Jonathan Payne
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Toru Kitagawa: Institute for Fiscal Studies and University College London
Jose Luis Montiel Olea: Institute for Fiscal Studies and New York University
Jonathan Payne: Institute for Fiscal Studies

No CWP44/17, CeMMAP working papers from Centre for Microdata Methods and Practice, 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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Citations: View citations in EconPapers (1)

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