## The Annals of Statistics

### Posterior convergence given the mean

#### Abstract

For various applications one wants to know the asymptotic behavior of $w(\theta | \overline{X})$, the posterior density of a parameter $\theta$ given the mean $\overline{X}$ of the data rather than the full data set. Here we show that $w(\theta | \overline{X})$ is asymptotically normal in an $L^1$ sense, and we identify the mean of the limiting normal and its asymptotic variance. The main results are first proved assuming that $X_1, \dots, X_n, \dots$ are independent and identical; suitable modifications to obtain results for the nonidentical case are given separately. Our results may be used to construct approximate HPD (highest posterior density) sets for the parameter which is of use in the statistical theory of standardized educational tests. They may also be used to show the covariance between two test items conditioned on the mean is asymptotically nonpositive. This has implications for constructing tests of item independence.

#### Article information

Source
Ann. Statist., Volume 23, Number 6 (1995), 2116-2144.

Dates
First available in Project Euclid: 15 October 2002

https://projecteuclid.org/euclid.aos/1034713650

Digital Object Identifier
doi:10.1214/aos/1034713650

Mathematical Reviews number (MathSciNet)
MR1389868

Zentralblatt MATH identifier
0854.62023

Subjects
Primary: 62F15: Bayesian inference
Secondary: 62E20: Asymptotic distribution theory

#### Citation

Clarke, B.; Ghosh, J. K. Posterior convergence given the mean. Ann. Statist. 23 (1995), no. 6, 2116--2144. doi:10.1214/aos/1034713650. https://projecteuclid.org/euclid.aos/1034713650

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