Question about eqution(4)(5)(6)

[SpiritBear000]

Hi
Great work! Your paper gave me a lot of inspiration. I encountered a problem during reading. I would like to ask you to help me understand the Eq.(4)(5)(6). Specifically, why (4) can be directly derived (5) and (6). I guess it should be because I don't know much about the characteristics of the Jacobian. Since I am just getting started, I don't know what basic knowledge should be searched to further understand this Eq, so I would like to ask for your help.

[weihaox]

The absolute value of the partial derivative term is the determinant of the jacobian matrix. This can be obtained by considering the derivative of the CDF $P(\cdot)$.
Let X be a random variable ($\tilde{z}^a$ $\tilde{z}^b$ in the paper) and $z=T^{-1}(x)$, if T is a monotonically increasing function (similar in the decreasing case), then
$\begin{align} P(Z \leq z) = P(T^{-1}(X) \leq z) = P(X \leq T(z)) \end{align}$.
Deriving both sides of the equation,
$\begin{align} p(z) = \pm p(T(z)) \frac{\partial T(z)}{\partial z} = p(x) \left\lvert \frac{\partial x}{\partial z}\right\rvert = \left\lvert T^{\prime}(x)\right\rvert p(T(x)) \end{align}$.