I am writing a Gibbs sampler for data that is Log-Normal (LN) distributed, with unknown mean and variance. There is a wealth of information on inference for LN models when either the mean or variance (precision) are known, but I'm not finding much information on inferring both parameters. I have an idea of "reasonable" bounds on both parameters, but otherwise want to remain mostly uninformative.

This question on priors for LNs seems related, but they simply mention that one should use Jeffrey's prior for the variance. I'm looking for more specific information: which priors and how would one sample from them. Also, would we have to resort to a Metropolis step or can we make some assumptions in the priors such that we can get posteriors in closed form?

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    $\begingroup$ One easy approach: do normal inference on the logs; the inference on the parameters (which in the usual parameterization are the same for both) should be the same as long as the priors correspond. Jeffrey's priors have the nice feature of making the inference not impacted by the parameterization. $\endgroup$ – Glen_b May 28 '14 at 23:20
  • $\begingroup$ Are you ONLY trying to infer these parameters? Because if there is no other aspect to your model then you can just take a log-transform of the data and use a normal-inverse-gamma prior, which is conjugate.. no Gibbs sampling necessary $\endgroup$ – user44764 May 29 '14 at 0:07
  • $\begingroup$ Ah... that makes sense. If either of you want to supply the answer, I'll happily accept it. Note @Matthew -- the log-normal is part of a larger hierarchical Gibbs sampler. $\endgroup$ – Wesley Tansey May 29 '14 at 0:27

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