Because many common misunderstandings (especially in some of the external links) have been passed along in the thread let me try to clarify a few very important points.
I know everyone hates probability theory but this is one of those cases where it can’t be ignored. A prior model is specified by a probability distribution. In practice we often specify that distribution with a probability density function, but the relationship between the prior distribution and the prior density function is complex. Specifically if we reparameterize the model the prior distribution transforms naturally but the prior density function does not. Indeed the weird transformation properties of a probability density functions are where Jacobian corrections arise. See for example https://betanalpha.github.io/assets/case_studies/probability_theory.html#42_probability_density_functions.
This tenuous relationship becomes even more subtle when we consider pushforward distributions which is the technical name for the induced distribution over a generated quantity. Prior distributions pushforward in a natural way, but prior density functions do not.
Evaluating prior pushforward distributions is a critical step in prior modeling because these distributions encode the, often unintended, consequences of a prior model built up form independent prior distributions over the nominal parameters. In other words when considering only the nominal parameters the prior model seems reasonably compatible with our domain expertise, but if we consider its consequences further down the generative model then we start to see inconsistencies with other domain expertise that we had not yet utilized. For much more see https://betanalpha.github.io/assets/case_studies/principled_bayesian_workflow.html#11_domain_expertise_consistency and I.J. Good’s Good thinking.
The challenge then is implementing prior pushforward checks, and resolving any inconsistencies, in a mathematically self-consistent way. When evaluating prior pushforward checks the awkward transformation properties of prior density functions can be avoided entirely but utilizing samples, which do transform more naturally; again see the above case study for examples.
So from this context we can say that @Guido_Biele built an initial prior model for a and b only to notice that the prior pushforward model for a / b was inconsistent with the available domain expertise. This prior pushforward check could have been implemented with generative sampling or by working out an analytic probability density function for a / b which would involve an intermediate transformation, Jacobian correction, and then marginalization (for example if a and b were given independent normal densities then a/b would end up having a Cauchy density; when a and b themselves are derived this kind fo analytic calculation becomes much, much more difficult).
Now comes the hard part – how do we built a prior model that’s compatible with the available domain expertise on a, b, and a / b?
One option discussed it to try to “invert” a prior density function on a / b to define “corrections” to the prior density functions on a and b. The math, however, is very clear that this won’t work. For one the mapping (a, b) \mapsto a / b is not invertible, as @hhau noted, so there are infinitely many prior density functions on a and b that could achieve the desired pushforward behavior for a / b. For two, however, most of these inversions would not be compatible with the original domain expertise about a and b!
The problem with many of the heuristics suggestions based on the “priors over generated quantities are fine” is that they ignore this consistency problem. A prior over generated quantities changes the prior over the input variables, often in undesirable ways. Critically the mathematical effects of this change are moderated by that Jacobian correction which consequently cannot be ignored!
If one does want to attempt a heuristic prior model of this form then one has to rely heavily on prior pushforward checks, in particular pushing the heuristic prior model forward to each variable of interest (including parameter and intermediate quantities) to verify sufficient consistency. The challenge with this approach is that it’s difficult to resolve any inconsistencies that might be found – because the prior model doesn’t correspond to a well-defined generative structure it’s not clear how the different behaviors relate and hence how to change modify the initial heuristic prior model.
Instead I strongly recommend taking advantage of the existing generative structure and using any inconsistencies in the prior pushforward distributions to motive changes to the priors on each parameter. For example if the pushforward distribution of a / b is too diffuse then one could be more careful about the prior modeling for a and b, working out narrower independent prior density functions or even a positively-correlated prior density function that suppresses fluctuations of a up when b also fluctuates down (that then induce large changes in the ratio).
