real X_div_Z = X / Z is two parameters in, one parameter out, so it doesn’t fit into the Jacobian thing. Same with the Q -> X thing. This comes up with transforms from unconstrained space to polar coordinates. The argument against is what does (r, theta) = (0, 0) map to? The argument for is, regardless, it seems to be doing the expected thing.
I searched a bit and didn’t find it, but it’s come up a few times here. If this is relevant, search around a bit. If you can’t find it let me know and I’ll dig more.
Just because you can’t add a Jacobian for your transform doesn’t mean you can’t do this sort of thing in your model, it just means you’re inventing a prior that nobody else is using but that’s not a bad thing necessarily (assuming it normalizes – and it might not, and assuming it better encodes prior information about your model, and assuming it isn’t too hard to communicate).
You’d probably want to try to check the normalization thing on paper.
Write out your log density in terms of the parameters Z and Q and see if you can integrate over everything. If integrals aren’t analytical, try to bound them with things that are and integrate those. It should give you some insight into what the problems might be, at least.
If it normalizes, then it’s up to you to communicate what you’re doing. Since it’s something non-standard, you should probably put together a couple slides to convince yourself and anyone else coming along in the future that it’s a reasonable thing to do (what are the parameters of this prior, how does it change, etc.).
Ofc., I don’t ever remember how a gamma distribution is parameterized so I gotta go read about them whenever they pop up, but the disadvantage you have here is Wikipedia isn’t gonna have a webpage for what you’re doing.