Posterior distribution of nondifferentiable functions

Toru Kitagawa (t.kitagawa@ucl.ac.uk), Jose Luis Montiel Olea (montiel.olea@nyu.edu) 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 CWP20/16, 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. 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).

Keywords: Distribution; nondifferentiable functions (search for similar items in EconPapers)
Date: 2016-05-09
New Economics Papers: this item is included in nep-ecm and nep-net
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