Table 1.

Overview of the transformations used in the Warp-III implementation. 𝜃_i denotes a parameter and ω_i denotes the corresponding new parameter that is obtained after having transformed 𝜃_i to the real line. l denotes a parameter lower bound and u denotes an upper bound. Φ(⋅) denotes the cumulative distribution function and ϕ(⋅) the probability density function of the normal distribution. The table displays the parameter type, the corresponding transformation, inverse-transformation, and the relevant Jacobian contribution

Type	Transformation	Inv.-Transformation	Jacobian contribution
unbounded	ωi = 𝜃i	𝜃i = ωi	$\left|\frac{\partial {𝜃}_i}{\partial {\omega }_i}\right|=1$
lower-bounded	${\omega }_i=\log \left({𝜃}_i-l\right)$	${𝜃}_i=\exp \left({\omega }_i\right)+l$	$\left|\frac{\partial {𝜃}_i}{\partial {\omega }_i}\right|=\exp \left({\omega }_i\right)$
upper-bounded	${\omega }_i=\log \left(u-{𝜃}_i\right)$	${𝜃}_i=u-\exp \left({\omega }_i\right)$	$\left|\frac{\partial {𝜃}_i}{\partial {\omega }_i}\right|=\exp \left({\omega }_i\right)$
double-bounded	${\omega }_i={\mathrm{\Phi }}^{-1}\left(\frac{{𝜃}_i-l}{u-l}\right)$	${𝜃}_i=\left(u-l\right)\mathrm{\Phi }\left({\omega }_i\right)+l$	$\left|\frac{\partial {𝜃}_i}{\partial {\omega }_i}\right|=\left(u-l\right)\varphi \left({\omega }_i\right)$