# Bayesian model with unknown mean and variance with lognormal prior

For $i=1, \ldots, K$ and $j=1, \ldots,n$, assume the following model. \begin{align} X_{ij} \mid \mu_i, \sigma^2 & \stackrel{_\text{iid}}{\sim} N(\mu_i, \sigma^2) \nonumber \\ \mu_i & \stackrel{_\text{iid}}{\sim} N(\mu, \tau^2) \nonumber \\ \ln(\sigma^2) & \sim N(\mu_v, \tau_v^2) \end{align} First assume all the prior parameters are known. Since this is not the conjugate form, is there a way to estimate the posterior $\pi(\mu_i \mid \mathbf{X_i})$ using a closed form density?

Thank you very much.

• I can't answer your question, but this comment by Gelman has a nice discussion of priors for variance parameters in the hierarchical setup you have here. In particular it seems like you could put a non-informative prior $p(\mu, \log \sigma^2) \propto 1$. – user44764 Jun 20 '14 at 15:47