Distribution of a Mean and Variance Say we have observations $x_1 \dots x_n$ and we have some sort of Bayesian framework where we would like to estimate a distribution for the mean $\mu$ of our observations and the variance $\sigma^2$ of our observations (these distributions will then be updated based on further observations). I have 2 questions:
1) By the central limit theorem, for large $n$, $\hat{\mu} \sim N(\mu, \frac{\sigma^2}{n})$. I was therefore thinking of using this as my prior distribution for $\mu$. If I was going to do the same  thing for the variance, what would the limiting distribution be? 
2) Would it perhaps be better to choose many different size $k$ subsamples of my observations, calculate the sample mean in each case and use this as an empirical prior distribution for $\mu$ (and do the same thing for the variance)?
P.S. Any links to further information on the topic would be appreciated.
 A: To the first question, I'm not sure how one might incorporate the limiting distribution of the standard error, since in the limit it should approach 0. (Unless I'm wrong, in the limit the sample mean approaches the actual, finite, mean and the denominator grows, so the standard error should approach 0.) To the second, that sounds much like a bootstrap approach to estimating the mean and standard error.
For a strictly Bayesian approach, given your interest in using $x_1...x_n$ to inform subsequent analysis of $y_1...y_m$, consider using vaguely informative priors for parameters of the normal distribution. After using those to analyze $x_1...x_n$, the posterior from that analysis can serve as a prior for the analysis of $y_1...y_m$. One method of analyzing unknown mean and variance is detailed here. That method has the benefit of using a conjugate prior, making it solvable analytically and straightforward to incorporate subsequent data. (Whether the dependence of the mean on precision is sensible for your problem of interest is another story.) I'm not sure if there are strictly non-informative priors available for that model, but as $n$ grows the data will quickly overwhelm a diffuse prior.
It may be enlightening to compare your posterior after analyzing $x_1...x_n$ to bootstrap estimates. If your prior isn't very informative, I would wager that the two would give comparable results.
