The Annals of Statistics

Locally uniform prior distributions

J. A. Hartigan

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Suppose that $X_{\sigma} | \mathbf{\theta} \sim N(\mathbf{\theta}, \sigma^2)$ and that $\sigma \to 0$. For which prior distributions on $\mathbf{\theta}$ is the posterior distribution of $\mathbf{\theta}$ given $X_{\sigma}$ asymptotically $N(X_{\sigma}, \sigma^2)$ when in fact $X_{\sigma} \sim N(\theta_0, \sigma^2)$? It is well known that the stated convergence occurs when $\mathbf{\theta}$ has a prior density that is positive and continuous at $\theta_0$. It turns out that the necessary and sufficient conditions for convergence allow a wider class of prior distributions--the locally uniform and tail-bounded prior distributions. This class includes certain discrete prior distributions that may be used to reproduce minimum description length approaches to estimation and model selection.

Article information

Ann. Statist., Volume 24, Number 1 (1996), 160-173.

First available in Project Euclid: 26 September 2002

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Zentralblatt MATH identifier

Primary: 62A15

Discrete prior distributions penalized likelihood minimum description length


Hartigan, J. A. Locally uniform prior distributions. Ann. Statist. 24 (1996), no. 1, 160--173. doi:10.1214/aos/1033066204.

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  • BARRON, A. R. and COVER, T. M. 1991. Minimum complexity density estimation. IEEE Trans. Inform. Theory 37 1034 1054. Z.
  • DAWID, A. P. 1984. Present position and potential developments: Some personal views, statistical theory, the prequential approach. J. Roy. Statist. Soc. Ser. A 147 278 292.
  • JEFFREy S, H. 1936. Further significance tests. Proceedings of the Cambridge Philosophical Society 32 416 445. Z.
  • RISSANEN, J. 1978. Modeling by shortest data description. Automatica 14 465 471. Z.
  • RISSANEN, J. 1983. A universal prior for integers and estimation by minimum description length. Ann. Statist. 11 416 431. Z.
  • RISSANEN, J. 1987. Stochastic complexity. J. Roy. Statist. Soc. Ser. B 49 223 239. Z.
  • RISSANEN, J. 1989. Stochastic Complexity in Statistical Enquiry. World Scientific Publishers, NJ.Z.
  • SCHWARZ, G. 1978. Estimating the dimension of a model. Ann. Statist. 6 461 464. Z.
  • WALKER, A. M. 1969. Asy mptotic behaviour of posterior distributions. J. Roy. Statist. Soc. Ser. B 31 80 88. Z.
  • WALLACE, C. S. and BOULTON, D. M. 1968. An information measure for classification. Comput. J. 11 185 194. Z. Z
  • WALLACE, C. S. and FREEMAN, P. R. 1987. Estimation and inference by compact coding with. discussion. J. Roy. Statist. Soc. Ser. B 49 240 265.