log absolute determinant of the Jacobian

[Vishv Jeet]

I am trying to sample a vector y in the model block, y is a non-linear transform of a parameter which is a matrix. So the Jacobian J of the transform is a matrix but non-square. 

Since J is non-square matrix, J' * J is a square matrix; whose determinant would be square of the determinant I need, so I multiply the log_determinant with 0.5. I did the following:

target += 0.5 * log_determinant(J' * J)

This should be correct but I want to make sure. 

STAN reference does mention that log_determinant or determinant takes only a square matrix but I guess that an is obvious requirement.

Regards,

Vishy 

[Vishv Jeet]

Another related question would be how to do this if the parameter matrix that vector y depends upon is ragged, i.e., not every entry in the parameter matrix is defined?

[Michael Betancourt]

This is not correct.

The Jacobian adjustment is for a 1-1 transformation which always

yields a square Jacobian.  For many-to-one or one-to-many

transformations you have to first construct a 1-1 transformation

with auxiliary parameters and then marginalize out the auxiliary

parameters from the resulting joint distribution.  For example,

if you had pi(x, y) and wanted the density for r = sqrt(x^2 + y^2)

then you could compute

pi(r) = \int d(theta) pi(r, theta) = \int d(theta) pi( x(r, theta), y(r, theta) ) | J (r, theta) |

Additionally, computing J and then calling log_determinant will be

viciously inefficient.  You almost always want to (and can) figure 

out the log Jacobian determinant analytically.

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[Vishv Jeet]

Thanks Michael. Not sure if I understood your suggestion entirely. My question is perhaps simpler, say

z = sqrt(x^2 + y^2)

suppose in STAN I am sampling z and x, y are parameters.

The jacobian J = (dz/dx, dz/dy) = (x / (x^2 + y^2)^1.5, y / (x^2 + y^2)^1.5) is a vector. So how do I compute the det(J)?

regards,

Vishy

[andre.p...@googlemail.com]

Vishv,

wolframalpha showed something different:

 http://www.wolframalpha.com/input/?i=log(abs(d%2Fdy++sqrt(x%5E2+%2B+y%5E2)))

For the Determinant, since x and y are independent, the bc part in ad-bc is 0.

Thus its ad. Since STAN needs on log scale its log(ad) = log(a) + log(d).

target += log(a)

target += log(d)

https://en.wikipedia.org/wiki/Determinant

I hope that made it clear and I dont have a typo somewhere.

Andre