## The Annals of Statistics

### Locally uniform prior distributions

J. A. Hartigan

#### Abstract

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

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

Dates
First available in Project Euclid: 26 September 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1033066204

Digital Object Identifier
doi:10.1214/aos/1033066204

Mathematical Reviews number (MathSciNet)
MR1389885

Zentralblatt MATH identifier
0853.62008

Subjects
Primary: 62A15

#### Citation

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

#### References

• 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.
• NEW HAVEN, CONNECTICUT 06520