# 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.

• In reading your first question, I gather that you wish to use $x_1...x_n$ to build a prior distribution to be used in the analysis of further observations. Is that correct? Commented Apr 26, 2014 at 16:43
• Yes that is correct. To make it more clear, I want to build a prior with $x_1 \dots x_n$ and that prior is to be used in some further analysis with other observations $y_1 \dots y_m$ say, that are not directly related to the $x_i$. Commented Apr 26, 2014 at 18:06

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.