Working Paper

Posterior distribution of nondifferentiable functions

Authors

Toru Kitagawa, Jose Luis Montiel Olea, Jonathan Payne

Published Date

9 May 2016

Type

Working Paper (CWP20/16)

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 ).


Latest version

Posterior distribution of nondifferentiable functions
Toru Kitagawa, Jose Luis Montiel Olea, Jonathan Payne
CWP44/17