The Annals of Statistics

Asymptotic normality with small relative errors of posterior probabilities of half-spaces

R. M. Dudley and D. Haughton

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Let $\Theta$ be a parameter space included in a finite-dimensional Euclidean space and let $A$ be a half-space. Suppose that the maximum likelihood estimate $\theta_n$ of $\theta$ is not in $A$ (otherwise, replace $A$ by its complement) and let $\Delta$ be the maximum log likelihood (at $\theta_n$) minus the maximum log likelihood over the boundary $\partial A$. It is shown that under some conditions, uniformly over all half-spaces $A$, either the posterior probability of $A$ is asymptotic to $\Phi(-\sqrt{2\Delta}\,)$ where $\Phi$ is the standard normal distribution function, or both the posterior probability and its approximant go to 0 exponentially in $n$. Sharper approximations depending on the prior are also defined.

Article information

Ann. Statist., Volume 30, Number 5 (2002), 1311-1344.

First available in Project Euclid: 28 October 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F15: Bayesian inference
Secondary: 60F99: None of the above, but in this section 62F05: Asymptotic properties of tests

Bernstein-von Mises theorem gamma tail probabilities intermediate deviations Jeffreys prior Mills' ratio


Dudley, R. M.; Haughton, D. Asymptotic normality with small relative errors of posterior probabilities of half-spaces. Ann. Statist. 30 (2002), no. 5, 1311--1344. doi:10.1214/aos/1035844978.

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