This is coming from a statistics background. I have several distributions defined on the hypersphere (really, the positive orthant of the hypersphere). I'm trying to compare these candidate distributions against the original data I used to fit the distribution.
For this, I'm intending to use something like Hellinger distance. I can calculate the probability of falling into a particular area numerically--using the posterior predictive and empirical, I just compute the number that fell into a said area and divide by total sample size. But for Hellinger distance, I still need the area that I'm integrating across--that means the surface area of an arbitrary subset of the hypersphere.
This $n$-dimensional geometry is somewhat outside my current understanding. Is there any established method for calculating surface area of an arbitrary portion of the hypersphere?
All very well understood:
$$ S_n = \frac{2^{(n+1)/2} \pi^{(n-1)/2)}}{(n-2)!!} $$
for $n$ odd;
$$S_n = \frac{2 \pi^{n/2}}{(n/2 -1)!}$$
for $n$ even.
And the area of the positive orthant is a trivial fraction of this.
(Computer that orthant area as a proportion for $n = 2,3,4...$ and see if you see the rather obvious rule.)