Dirichlet distribution is a conjugate prior for multinomial distribution. I want to impose a non-informative prior over sampling weights $\pi$ for a draw $x=(x\_1,…,x\_N)$ from a multinomial distribution with support $d=(d\_1,…,d\_K)$ (all the possible values that $x\_i$ can take) and sampling weights $\pi=(\pi\_1,…,\pi\_K)$.

I was under impression that $Dir(\alpha)$ with $\alpha\_i=1$ is a right choice. But I've read that (see e.g. [this](https://www.jmlr.org/papers/volume5/lee04a/lee04a.pdf)) $Dir(\alpha)$ with $\alpha\_i=0$ yields an improper non-informative distribution.

Question:

1. Why $Dir(\alpha)$ with $\alpha\_i=0$ is non-informative? Doesn't $\alpha\to 0$ impose higher sampling weight on a single data and zero on all others?
2. Shouldn't a uniform distribution $Dir(\alpha)$ with $\alpha\_i=1$ be non-informative choice for the prior instead?