# Any criterion to determine if a probability distribution over vectors is induced by a distribution over orthogonal bases?

Let $$\mathcal{O}$$ be the set of all orthogonal bases of the vector space $$\mathbb{R}^d$$. That is, every element in $$\mathcal{O}$$ is a single orthogonal basis of $$\mathbb{R}^d$$. To make this more precise, I consider two orthogonal bases $$o_1$$ and $$o_2$$ as equal if and only if they consist of exactly the same vectors (irrespective of order). If at least one vector belongs to one and not the other, then I consider them different.

Notice that I am talking about orthogonal bases, not orthonormal bases, which means that I don't restrict the vectors to be unit length. I don't even restrict all the vectors of a basis to have equal length. I only restrict the vectors to be orthogonal and to form a basis of $$\mathbb{R}^d$$.

Now, let's look at some probability distribution $$f'(o)$$ over $$\mathcal{O}$$. In other words, $$f'(o)$$ assigns a probability density to every orthogonal basis $$o$$.

Let's look at the following two step process:

(1) sample a basis $$o$$ from the distribution $$f'(o)$$.

(2) sample a vector $$v$$ out of the basis $$o$$ at uniform (each vector with probability $$\frac{1}{d}$$).

This process induces a probability distribution $$f(v)$$ over $$\mathbb{R}^d$$.

Let $$g(v)$$ be a probability distribution over $$\mathbb{R}^d$$. I say that $$g(v)$$ is Induced by an Orthogonal Basis Distribution (IOBD) if there exists a probability distribution $$g'(o)$$ over $$\mathcal{O}$$ such that $$g(v)$$ can be represented by the above two-step process using $$g'(o)$$.

Now suppose that I am given a probability distribution $$g(v)$$ over $$\mathbb{R}^d$$, and I want to deremine whether it is IOBD.

My question is: what are some simple criteria, that can be applied to such a $$g(v)$$, to determine whether it is IOBD?

(My only idea is that isotropy of $$g(v)$$ implies IOBD. But this is not a neccesary condition. Ideally, I would like to have a sufficient and neccesary condition that can be applied to $$g(v)$$).

• It might be helpful to frame this process in terms of the Stiefel manifold – Omnomnomnom Apr 18 at 16:49