In Stan, if you have an expression that looks like
foo ~ distrib(a,b,c);
Where foo is either a function, or a transformed parameter variable, and distrib is some distribution, Stan will complain about you needing to add a log-Jacobian to the target.
DIAGNOSTIC(S) FROM PARSER:
Warning (non-fatal):
Left-hand side of sampling statement (~) may contain a non-linear transform of a parameter or local variable.
If so, you need to call target += with the log absolute determinant of the Jacobian of the transform.
Left-hand-side of sampling statement:Now, if foo is directly a transformed single parameter, then in fact, Stan is correct. But if foo is the result of applying a function to several parameters together, collapsing an N dimensional space down to 1, then there does not exist a Jacobian
for example:
exp(q/r) ~ distrib(a,b,c)
where q and r are both parameters. There is one function, but two parameters, q,r. So the Jacobian matrix is 1x2 and has no determinant. How can you understand what this means?
Consider the prior on q,r perhaps you've said
q ~ normal(qguess, qerr);
r ~ normal(rguess, rerr);
and you thought to yourself "Ok there's the prior for q and r", but then you wrote down:
exp(q/r) ~ distrib(a,b,c)
and thought to yourself what? Something like "there's the prior for exp(q/r)" which is kind of true. But exp(q/r) is not a parameter, it's a deterministic function of two parameters q,r. On the other hand, we're multiplying them all together, so the actual prior for q,r is
normal_pdf(q,qguess,qerr) * normal_pdf(r,rguess,rerr) * distrib_pdf(exp(q/r),a,b,c)/Z
that is, "exp(q/r) ~ distrib(a,b,c)" declares a weighting function on values of q,r which further downweights those regions of q,r space that cause exp(q/r) to fall outside the high density region of distrib(a,b,c) (and you need the Z to re-normalize after the reweighting, but note that in MCMC you typically don't need the normalization constant so you're ok there).
This "declarative" technique produces an intentional "tilt" on the prior for q,r that helps you create a more complex and nuanced prior. In particular, when there are various ways to get your function to be near the high probability region for distrib, you'll wind up with a non-independent prior over the parameters q,r you might have a ridge along some curve, or several modes or whatever. This is good when you really do have information that q,r should lead your function to be in the high probability region of distrib.
To generate from this model, you'd need to first generate according to the simple prior, and then reject on the basis of the weighting function, so for example
q = normal_rng(qguess,qerr);
r = normal_rng(rguess,rerr);
f = exp(q/r)
w = distrib_pdf(f,a,b,c)/distrib_pdf_max_pdf_value
p = runif(0,1);
if(p < w) accept
else reject