I would like to describe a large number of measurements of rotations $\textbf{x}_i$. Each rotation is described by its rotation axis $\textbf{v} =\frac{\textbf{x}}{|\textbf{x}|}$ and the rotation angle $a =|\textbf{x}|$ around this axis.
Distributions like von Mises-Fisher and Bingham only describes how the directions are distributed, but I would also like to describe the rotation angle's dependency on direction. So I am looking to find $P(\textbf{x}) = P(\textbf{x}|\textbf{v})P(\textbf{v})$, where $\textbf{x}\in R^3$ and $\textbf{v}$ is a point on the unit sphere. (edit: I misunderstood the $SO(3)$ notation)
I know $P(\textbf{v})$ can be described with fx. von Mises-Fisher, but what are some usual candidates for the distribution $P(\textbf{x}|\textbf{v})$? Or is there already a combined distribution for direction vectors with magnitude?
I could add that some of my data looks like a 3D hourglass shape, so I cannot just use a 3D normal distribution.
(edit: According to Wikipedia von Mises-Fisher is a probability density function on a random unit vector. So as I understand it only describes the axis of most likely rotation, not the amount rotated.)
