Note quite, because the probability density changes as you shift between coordinates. The core idea here is they are *densities*, and so equivalent differential elements need to have the same total power contained within them. E.g., if I have some constrained parameter x that converts to unconstrained parameter z, I need:

P_x(x) dx = P_z(z) dz

Which of course rearranges to:

P_x(x) \cdot\frac{dx}{dz} = P_z(z) \\
\ln \vert P_x(x) \vert + \ln \vert \frac{dx}{dz}\vert = \ln \vert P_z(z)\vert

I.e. to go between the two log-densities I need to add a log of the derivative.

If I have many parameters being transformed, like:

x_1 \rightarrow z_1(x_1) \\
x_2 \rightarrow z_2(x_2) \\
\vdots \\

I need to correct by the product of all of these:

\ln \vert P_z(z)\vert = \ln \vert P_x(x) \vert + \ln \vert \frac{dx_1}{dz_1} \frac{dx_2}{dz_2} \dots \vert

And in the more general case where the transformed parameters “mix”, i.e.:

x_1 \rightarrow z_1(x_1, x_2 \dots) \\
x_2 \rightarrow z_2(x_1, x_2 \dots) \\
\vdots \\

The correction factor is the determinant of the Jacobian of all of these transformations:

J = \left[ \matrix{{\frac{\partial z_1}{\partial x_1},\frac{\partial z_1}{\partial x_2}}\\{\frac{\partial z_2}{\partial x_1},\frac{\partial z_2}{\partial x_2}}} \right]

\ln \vert P_z(z)\vert = \ln \vert P_x(x) \vert + \ln \vert det(J) \vert

Throw in a negative or two to account for inverses and the difference between density and potential energy, and you have a clearer picture