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> So in a sense, the huge body of mathematics underlying probability has already failed us at this basic juncture because we cannot speak of how random one part

> So in a sense, the huge body of mathematics underlying probability has already failed us at this basic juncture because we cannot speak of how random one particular outcome of an experiment is. This reminds me of the 'Is 14 a random number?' debate. Part of the problem, though, is that we have terminology that is horribly misleading. First, a random variable is neither random nor a variable. They're not variables, because they're functions that yield non-deterministic values - a huge distinction. You can't have a random number - the idea itself doesn't make any sense. Second, the definition of 'random' is itself problematic - or at least contested. Randomness implies probability, but probability can be defined in two different (and incompatible) ways, one of which is essentially the inverse of the other. Ironically, the one that Keynes proved back in 1921 to be logically inconsistent is the one that is more commonly used today (probably because it is more intuitive and mathematically convenient... even if it is almost always incorrect!). The question 'Is 14 a random number' doesn't really make any sense. 14 cannot be a random number; it can only be a number drawn from a random distribution. That may seem like it simply begs the question, but in fact, this subtle shift is incredibly important - determining whether a number was drawn from a random distribution is much easier to frame in terms of probability, and probability, not randomness, is the language of statistics. Unfortunately, this is one of those cases where the two definitions of probability yield widely different answers. You could tell me that the answer is undefined, in almost exactly the same way that division-by-zero is undefined in mathematics. Or you could construct a model over all possible distributions of numbers, the probabilities associated with each of those distributions, and integrate accordingly to yield some (probably computationally unfeasible) functional answer. In this school of thought, we can speak of how 'random' a particular outcome is - we're essentially partitioning the (potentially infinite) universe of functions f such that our value v is in the range of f into two categories: one designated as 'random' and the other designated as 'not random'. Then, we are determining the probability that our value was generated from one of the former, as opposed to the latter. Either one would be correct ways of answering this second question, but neither one addresses the first question, which is essentially nonsensical. (Well, I guess the answer is 'no, 14 is not a random number, because a number cannot be random', but that's a bit of a cop-out!).

dsr_ 15y ago
You can also think of randomness as a measure of predictability: given our knowledge prior to being presented with this number, how well could we predict this number?
- chimeracoder 15y ago
  You're thinking like a Bayesian - I avoided using those terms, but that's essentially the latter approach. The prior knowledge is built into the model with our choice of the functions and the weights that we assign those functions. (It should be noted that the Bayesian approach doesn't guarantee a defined answer - we can easily have a divide-by-zero/undefined value in that school of thought as well. However, the frequentist approach always leads to an undefined outcome in this question, because it follows from the definition of probability itself, whereas in the Bayesian approach, it follows from insufficient information in the model specification.)
MathProgramming 15y ago
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 15y 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.

dsr_ 15y ago

You can also think of randomness as a measure of predictability: given our knowledge prior to being presented with this number, how well could we predict this number?

chimeracoder 15y ago
You're thinking like a Bayesian - I avoided using those terms, but that's essentially the latter approach. The prior knowledge is built into the model with our choice of the functions and the weights that we assign those functions. (It should be noted that the Bayesian approach doesn't guarantee a defined answer - we can easily have a divide-by-zero/undefined value in that school of thought as well. However, the frequentist approach always leads to an undefined outcome in this question, because it follows from the definition of probability itself, whereas in the Bayesian approach, it follows from insufficient information in the model specification.)

MathProgramming 15y ago

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 15y 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.