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I am trying to determine what the most appropriate non-informative priors are for the two parameters of a log-normal distribution (for an accelerated failure time model). I had been working with a normal prior on the mean of the logged data (or log-normal prior on the median), and an independent normal prior on the log of the sd of logged data (or log-log normal prior on the dispersion). It seems to me that the parameterization of the second parameter is quite informative on the natural scale of the parameter but I am pretty new to this. Any ideas on better ways to approach this?

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up vote 5 down vote accepted

When I'm trying to be relatively uninformative, I have tended to use a uniform prior on $\ln \sigma$ and specify an upper bound, which corresponds to $p(\sigma) \propto 1/\sigma$ over a finite range—relatively uninformative, and equal to Jeffreys' prior over the range (not equal to what the Jeffreys prior would be if you knew there was an upper bound on $\sigma$ and what it was.) If the posterior piles up against your upper bound, you can increase it and rerun, unless you have some strong reason for choosing that upper bound. This was suggested by Andrew Gelman in the Prior distributions for variance parameters paper here. (Some of the other articles in this issue of Bayesian Analysis are possibly relevant too, hence the link to the journal page.)

However, recently I've tried the beta-prime prior suggested in the first response to Weakly informative prior distributions for scale parameters and that worked out well for me also. Importance sampling on the output of the MCMC indicated that the differences between the posteriors of the parameters of interest using the two priors were trivial, which, after all, is what you want when you're trying to be relatively uninformative - and it gets you away from that annoying specification of an upper bound on $\sigma$.

This question and answer may also be relevant:

Random effect on scale parameter

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@Scortchi - you would think that after all these years I would reflexively write "Jeffreys'" instead of "Jeffrey's", thanks for covering up my embarrassing mistake! And my mother was a professor of English! – jbowman Nov 17 '13 at 2:12

In the absence of a particular desired form of prior, often people use conjugate priors; the conjugate prior for $\sigma^2$ in the normal would be inverse gamma. You can choose anything from highly uninformative (an improper prior like $1/\sigma^2$ is often used and is a limiting case of the Inverse Gamma), through mildly informative, to a fully informative prior (such as one based on a previous study).

I believe the Jeffreys prior is $1/\sigma^2$. The nice thing about Jeffreys priors is they don't depend on your parameterization (if you transform the parameter, the Jeffreys prior 'follows' it so that everything corresponds).

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Jeffreys in his book on probability describes what noninformative priors he would use for scale parameters. He uses improper priors (i.e. prior density functions that do not have a finite integral). Here is an amazon link to the book.

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