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I think I made the point to say that random only applies to things being drawn from a distribution, but that's really the whole point of the theory: we _want_ t

I think I made the point to say that random only applies to things being drawn from a distribution, but that's really the whole point of the theory: we _want_ to be able to talk about how random a number is! We just call it 'Kolmogorov random' now, and by luck or insight, numbers chosen uniformly at random are almost always Kolmogorov random.

chimeracoder 14y ago
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.
- MathProgramming 14y ago
  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#Expected_Time.2FSpace_Complexity_of_Algorithms_under_the_Universal_Distribution 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.

chimeracoder 14y ago

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.

MathProgramming 14y ago
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#Expected_Time.2FSpace_Complexity_of_Algorithms_under_the_Universal_Distribution 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.