I'm assuming you're the author of the original post. My comment wasn't meant to invalidate your main argument, since it runs mostly orthogonal to the bulk of what you're saying. If anything, it actually supports it slightly (see below).
I understand what you meant by that example; I was just intrigued by seeing it in this context, and it reminded me of the related example that I brought up. And not to belabor the point with pedantry, but a tiny bit of rephrasing illustrates that your statement can be viewed as completely compatible with my latter definition:
> numbers chosen from a random uniform distribution function appear to be indistinguishable from numbers chosen from a Kolmogorov random function
From the looks of it, it appears that the Kolmogorov example is really just a special case of the distributional viewpoint, in which case your system of partitioning the universe of possible distributions revolves around the Kolmogorov criterion. (And while I made it seem like the partitioning is binary in my previous post, the principle can easily be generalized, so that's not a problem). And we may even be able to equate this statement with an alternate form based on the distributional difference between uniform and Kolmogorov.
I'm familiar with Kolmogorov, though not enough to be confident about this last hypothesis - I'll have to think about it some more.
I'm not super familiar with the various theories of probability, but if you're interested in thinking of these kinds of things, you might enjoy reading about the Universal Distribution, which is as far as I know defined in terms of Kolmogorov complexity. http://www.scholarpedia.org/article/Algorithmic_probability#...
I know the book Gems of Theoretical Computer Science has a good introduction to this topic as well.
I understand what you meant by that example; I was just intrigued by seeing it in this context, and it reminded me of the related example that I brought up. And not to belabor the point with pedantry, but a tiny bit of rephrasing illustrates that your statement can be viewed as completely compatible with my latter definition:
> numbers chosen from a random uniform distribution function appear to be indistinguishable from numbers chosen from a Kolmogorov random function
From the looks of it, it appears that the Kolmogorov example is really just a special case of the distributional viewpoint, in which case your system of partitioning the universe of possible distributions revolves around the Kolmogorov criterion. (And while I made it seem like the partitioning is binary in my previous post, the principle can easily be generalized, so that's not a problem). And we may even be able to equate this statement with an alternate form based on the distributional difference between uniform and Kolmogorov.
I'm familiar with Kolmogorov, though not enough to be confident about this last hypothesis - I'll have to think about it some more.