Jacobian adjustment of sigmoid (decentered)

[Bob_Carpenter]

@stemangiola

If I have

gamma = logit(beta^ ( log( 0.5 ) / log( 0.2 ) ) ); // moves the 0 on 0.2 instead of 0.5.

( I need it because I want to map a simplex to a real space, then a simplex of size 5 will have it’s null state at [0.2, 0.2, 0.2, 0.2, 0.2])

What would be the Jacobian. According to wolfram alpha a pretty big formula, but maybe I am missing something.

There’s a chapter of the manual that goes over exactly how Stan does the centered stick-breaking transform for the simplex and it works through the Jacobian.