Jacobian Adjustment with low to high dimension mapping

[spinkney]

Maybe this answers your question. I remember thinking it was interesting and filed it away:

Nonlinear transformation from R² to R

You can’t transform probability density functions between two real spaces of different dimensions – the Jacobian update rule is valid only when the input space and output space have the same dimension and the Jacobian matrix is square and the Jacobian determinant well-defined. What you really have to do instead is buffer your output space with auxiliary variables to define a one-to-one transformation, compute the corresponding Jacobian, and then marginalize if you can. For your transformation …