$X$ is a uniform distribution on the standard simplex $L$. For any point $\vec{x}$ on $X$, all entries of $\vec{x}$ are between 0 and 1 and $\vec{x}\cdot \vec{1} = 1$.
$S$ is the standard simplex scaled by a positive $w_{i}$ along each dimension. That means each point $(x_{1}, x_{2}, ...)$ on $X$ becomes $(w_{1}x_{1}, w_{2}x_{2}, ...)$ on $S$.
The scaling of all axes is a one to one transformation $T$ from $L$ to $S$.
If I apply transformation $T$ to $X$, do I get a uniform distribution over non-standard simplex S?
Is this still true if some of the $w_i = 0$?
Is the following reasoning correct if all $w_i \ne 0$?
- Determinant of the Jacobian for the change of coordinate is constant
- By change of coordinate formula in multi-variable calculus, area of any infinitesimal area over the standard simplex is scaled by the same constant when the area is transformed by $T$.
- Therefore, the probability density is scaled by the same constant upon the transformation.
- Consequently, I get a uniform distribution over $S$.
Thank you.