This seems so easy so I had to convince myself with an example, where I set:

– a /sqrt(2)/2 ~ N(1, 0.5)

– b /sqrt(2)/2 ~ N(1, 0.5)

– sqrt(a^2 + b^2) ~ N(1, 0.1)

Basically, in (a, b) space, the first two priors are centered around (1, pi/4) in polar coordinates and the last prior acts as a mask to take a ring of radius normally distributed around 1.

FYI it works but I was wondering how would you even adjust for the Jacobian in this context?

Like, if you consider the transformation (a, b) -> (a, sqrt(a^2 + b^2)) (we can assume a and b are positive for simplicity) you would do the change of variables for the first and third prior but you would still need to define a prior for b.

Alternatively, if you go to polar coordinates with a = r *cos(phi) and b = r * sin(phi), you could define the third prior for r but you would have to trouble defining the first and second priors, unless you define something like tan(phi) ~ N(1, 0.5) / N(1, 0.5).

Do you have any thoughts on this?

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