What's the posterior for mutivariate lognormal with covar known?

Question

I know the univariate case but not the multivariate case.

Suppose we have a multivariate lognormal dist:

$$ \boldsymbol{X} \sim \text{lognormal }(\boldsymbol{\mu}, \boldsymbol{\Sigma}) $$

where $\boldsymbol{\Sigma}$ is known and we have a multivariate Normal as the prior:

$$ \boldsymbol{\mu} \sim \mathcal{N}(\boldsymbol{\theta}, \boldsymbol{\Delta}) $$

Then what's the posterior with observation $\boldsymbol{x_0}$?

Answer 1

If you observe $\boldsymbol{X}$, you've got the same information as if you've observed $\log\boldsymbol{X}$. So that reduces it to a normal distribution with a normal prior on the mean.