Why does the von Mises-Fisher distribution need two parameters? The von Mises-Fisher distribution has two parameters: the mean $\mu \in \mathbb{R}^p$ and concentration $\kappa \geq 0$, where $\mu$ is constrained to have unit norm.
Why not instead define the distribution for one unconstrained parameter i.e. $y := \kappa \mu \in \mathbb{R}^p$?
What is gained (conceptually or computationally or statistically) by thinking in terms of two parameters rather than one?

In geometry we represent directions with a unit norm vector, so $\mu$ represents the direction. That's why its norm is $||\mu||=1.$ The other parameter is similar to variance in that it represents how directions are concentrated around $\mu.$
You can pack $\kappa$ into $\vec\mu$ as its magnitude $\vec\mu'=\kappa\vec\mu$, then $||\vec\mu'||\ne 1$, but this will create uninterpretable artifacts: concentration components in different directions $\kappa_x,\kappa_y,\kappa_z$. This is different from a technically similar trick in physics: packing a magnitude $a$ of acceleration into a length of a vector $\vec a$ with $a=||\vec a||\ne 1$. The acceleration components $a_x,a_y,a_z$ have a clear physical interpretation here.
The distribution in question, although technically operates in nD space, concerns with directions on (n-1)D sphere. So, trying to use up that extra dimension may look like a clever way to parsimony, but it in fact doesn't correspond to the mature of the phenomenon modeled.
