# What prior distributions could/should be used for the variance in a hierarchical bayesisan model when the mean variance is of interest?

In his widely cited paper Prior distributions for variance parameters in hierarchical models (916 citation so far on Google Scholar) Gelman proposes that good non-informative prior distributions for the variance in a hierarchical Bayesian model are the uniform distribution and the half t distribution. If I understand things right this works well when it is the location parameter (e.g. the mean) is of the main interest. Sometimes the variance parameter is of the main interest however, for example when analyzing human response data from timing tasks mean timing variability is often the measure of interest. In those cases it is not clear to me how variability could be modeled hierarchical with, for example, uniform distributions, as I after the analysis want to get the credibility of the mean variance both on the participant level and on the group level. In order to do that I need to use prior distributions that can be parametrized with the mean variance or similar, right?

My question is then: What distribution's are recommended when building a hierarchical Bayesian model when the variance of the data is of the main interest?

I know that the gamma distribution can be reparametrized to be specified by mean and SD. For example, the hierarchical model below is from Kruschke's book Doing Bayesian Data Analysis. But Gelman outlines some problems with the gamma distribution in his article and I would be grateful for suggestions of alternatives, preferably alternatives that are not to difficult to get working in BUGS/JAGS.

Shortly, Gelman outlines problems in using Gamma distributions as vague (he uses the word noninformative) priors for the variance. On the contrary, your problem (and the Kruschke's example) seems to refer to the case where some knowledge about the variance exists. Also notice that the picture of the distribution of the variance $\tau_i$ is not flat at all.