Posterior distribution of nondifferentiable functions

Toru Kitagawa (), Jose Luis Montiel Olea (), Jonathan Payne and Amilcar Velez Salamanca ()
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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 CWP17/19, 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(?), where ? is a finite-dimensional parameter of either a parametric or semiparametric model. The main assumption is that the distribution of a suitable estimator ?^n, its bootstrap approximation, and the Bayesian posterior for ? all agree asymptotically. It is shown that whenever g is locally Lipschitz, though not necessarily differentiable, the posterior distribution of g(?) and the bootstrap distribution of g(?^n) coincide asymptotically. One implication is that Bayesians can interpret bootstrap inference for g(?) 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(?) cannot be presumed to be approximately valid confidence intervals (even when this relation holds true for ?).

Date: 2019-04-03
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Related works:
Journal Article: Posterior distribution of nondifferentiable functions (2020)
Working Paper: Posterior Distribution of Nondifferentiable Functions (2019)
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