Locally uniform prior distributions. (English) Zbl 0853.62008

Summary: Suppose that \(X_\sigma \mid \theta \sim N (\theta, \sigma^2)\) and that \(\sigma \to 0\). For which prior distributions on \(\theta\) is the posterior distribution of \(\theta\) 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 \(\theta\) 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.


62A01 Foundations and philosophical topics in statistics
62F15 Bayesian inference
Full Text: DOI


[1] BARRON, A. R. and COVER, T. M. 1991. Minimum complexity density estimation. IEEE Trans. Inform. Theory 37 1034 1054. Z. · Zbl 0743.62003 · doi:10.1109/18.86996
[2] 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. JSTOR: · Zbl 0555.65100 · doi:10.2307/2981672
[3] JEFFREy S, H. 1936. Further significance tests. Proceedings of the Cambridge Philosophical Society 32 416 445. Z.
[4] RISSANEN, J. 1978. Modeling by shortest data description. Automatica 14 465 471. Z. · Zbl 0418.93079 · doi:10.1016/0005-1098(78)90005-5
[5] RISSANEN, J. 1983. A universal prior for integers and estimation by minimum description length. Ann. Statist. 11 416 431. Z. · Zbl 0513.62005 · doi:10.1214/aos/1176346150
[6] RISSANEN, J. 1987. Stochastic complexity. J. Roy. Statist. Soc. Ser. B 49 223 239. Z. JSTOR: · Zbl 0718.62008 · doi:10.1080/07474938708800126
[7] RISSANEN, J. 1989. Stochastic Complexity in Statistical Enquiry. World Scientific Publishers, NJ.Z. · Zbl 0800.68508
[8] SCHWARZ, G. 1978. Estimating the dimension of a model. Ann. Statist. 6 461 464. Z. · Zbl 0379.62005 · doi:10.1214/aos/1176344136
[9] WALKER, A. M. 1969. Asy mptotic behaviour of posterior distributions. J. Roy. Statist. Soc. Ser. B 31 80 88. Z. JSTOR: · Zbl 0176.48901
[10] WALLACE, C. S. and BOULTON, D. M. 1968. An information measure for classification. Comput. J. 11 185 194. Z. Z · Zbl 0164.46208 · doi:10.1093/comjnl/11.2.185
[11] 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. JSTOR: · Zbl 0653.62005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.