Posterior convergence given the mean

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.