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Building a hierarchical Bayes model, and I am interested in Bayesian inference of two parameters $a > 0$ and $b > 0$.

Right now I am using uninformative priors on both $a$ and $b$.

But I do have distribution on $w = ab$. How can I use this information to create an informative prior on a and b?

I am willing to make any distributional assumptions on $a$, $b$, and $w$ that are convenient for modeling purposes. Right now I treat $w$ as normally distributed (though it too cannot technically be negative).

Edit: The motivation is that I can acquire data where I can infer both a and b, but it is expensive and estimates are imprecise (fat posteriors). I can also acquire different data quite easily and cheaply, where a and b are not identifiable but w =a/b is. So I'd like to use the posterior on w to inform a prior on a and b, so when I run the expensive experiment, I know I'll get better inference on a and b.

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  • $\begingroup$ Your problem is not very specified, so there are many possibilities. Given $a>0$, $b>0$, $w=ab$, and your desire for "convenience", a simple possibility would be log-normal for all three. This would certainly be more appropriate than normal, for positive variables. What are your "uninformative priors"? $\endgroup$
    – GeoMatt22
    Commented Oct 11, 2016 at 23:11
  • $\begingroup$ Log-normal is a good idea. Right now the uninformative priors are Cauchy with a big scale parameter. Using Stan and that's kind of the default. $\endgroup$
    – Count Zero
    Commented Oct 11, 2016 at 23:54
  • $\begingroup$ You say first "I do have distribution on $w$", and then later you say "Right now I treat $w$ as normally distributed". For $w$ do you have data? or a "true" expected form? $\endgroup$
    – GeoMatt22
    Commented Oct 12, 2016 at 0:00
  • $\begingroup$ @GeoMatt22 so I have a distribution $f_w(\mu_w, \sigma_w)$, which I assume is log normal. So I would sample like this perhaps? $w \sim f_w(\mu_w, \sigma_w)$ $a \sim \text{lognormal}(\mu_a, \sigma_a)$ Accept a if $log(a) < log(w)$ $b = exp(log(w) -log(a))$ Or something like that? $\endgroup$
    – Count Zero
    Commented Oct 12, 2016 at 0:01
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    $\begingroup$ One thought is that if you know (or are willing to assume) the prior $p(a)$ and the conditional prior $p(b|a)$ for any given value of a, you would actually have a joint prior for $a$ and $b$. Then you could do an analysis, where your posterior $p(a,b|\text{data}) \propto p(a) p(b|a) p(\text{data}|w(a,b))$. Depends on this way of prior specification being possible, of course. $\endgroup$
    – Björn
    Commented Oct 12, 2016 at 6:45

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