 Solution

For the reverse centered polynomial birth move in reversible-jump sampling, recover $z={\beta }_{k+1}^{\prime }$ from the current coefficient vector. Remove that highest-order coefficient and undo the intercept shift:No auxiliary random number is drawn for this death move: the removed coefficient supplies the auxiliary value needed to reconstruct the birth move. The reverse Jacobian determinant is also one. The inverse prediction change is $-zv$, hence the residual in the lower-order model is $r=r^{\prime }+zv$.