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I read in serveral books about the definition of a spherical distribution; $x \in \mathbb{R^n}$ has a spherical distribution if and only if $x \stackrel{d}{=}Ox$ for any orthogonal, $n$-dimensional matrix $O$. But now I try to understand that; Let's take the "Von Mises–Fisher distribution"; If you don't know it, it's absolutely no problem, it's just a distribution on the $n$-dimensional unit sphere.

https://en.wikipedia.org/wiki/Von_Mises%E2%80%93Fisher_distribution

Now considering that it's distributed on the sphere, it would make sense that the distribution is a spherical distribution; But if you check the picture from the Wikipedia article and see the red points on it, then you see that the distribution can be very concentrated on a point on the sphere; But if $Qx$ now has the same distribution as $x$, then it would be invariant regarding rotations but it seems unlogical that the distribution stays the same, since the concentration point $\mu$ would change.

So where is my mistake?

  • "Distribution on the sphere" is not the same as "spherical distribution." – angryavian Jun 09 '17 at 22:28
  • So this means that the Mises–Fisher distribution is just not spherical? –  Jun 09 '17 at 22:30
  • But is a spherical distribution always distributed on the sphere? Is at least this direction correct? –  Jun 09 '17 at 22:31

1 Answers1

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A "distribution on the sphere" is a probability distribution such that the probability assigned to the unit sphere is $1$.

Your definition of "spherical distribution" (which I see more often as "rotationally invariant") means that the distribution does not change after any rotation. The definition you have written does not require the distribution to be on a sphere.

angryavian
  • 89,882
  • Okay, so there are spherical distributions not distributed on the sphere? Or is at least every spherical distribution distributed on the sphere? –  Jun 09 '17 at 22:32
  • The standard Gaussian distribution is rotationally invariant, but not supported on a sphere. – angryavian Jun 09 '17 at 22:34
  • Yeah right, that was part of my other question you answered recently... But why is it called spherical? Are at least most spherical distribution distributed on a sphere? :-) –  Jun 09 '17 at 22:37