Posterior distribution of nondifferentiable functions

Toru Kitagawa, Jose Luis Montiel Olea and Jonathan Payne

No 20/16, 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. The main assumption is that the distribution of the maximum likelihood 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 g(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 the posterior distribution of g( theta) does not coincide with the asymptotic distribution of g(theta_n) at points of nondifferentiability. Consequently, frequentists cannot presume that credible sets for a nondifferentiable parameter g(theta ) can be interpreted as approximately valid confidence sets (even when this relation holds true for theta ).

Date: 2016-05-09
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Persistent link: https://EconPapers.repec.org/RePEc:azt:cemmap:20/16

DOI: 10.1920/wp.cem.2016.2016

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