3 ms·

If you're trying to say that the uniform distribution represents an average of the uncountably infinite set of possible distributions on an interval, I don't se

If you're trying to say that the uniform distribution represents an average of the uncountably infinite set of possible distributions on an interval, I don't see how you can justify that. "Generate a random distribution on an interval" is not an obviously well-defined procedure, so I don't think you can say anything about what the distribution produced by "averaging" every possible distribution would look like. Which matches the fact that you have no information on what the distribution of X looks like, other than that it is zero outside of the interval [0,100]. Edit: In any case, the process you've described is just begging the question (i.e. it is circular reasoning). In order to prove that the uniform distribution is a natural choice for X, you assume that choosing uniformly from the set of all possible distributions is natural. Even if there was a finite number of possible distributions to choose from such that choosing randomly from that set according to a uniform distribution was a well-defined operation, you're still assuming that it's natural to assign equal probability to all options in order to prove that it's natural to assign equal probability to all options.

skissane 9y ago
> If you're trying to say that the uniform distribution represents an average of the uncountably infinite set of possible distributions on an interval, I don't see how you can justify that. So, is this X actually an arbitrary real number? Because real numbers are not known to exist in the physical world. In the physical world, we will take some measurement to a finite number of digits, and also produce an estimate of the error in that measurement. The measurement and the error bounds are all described by rational numbers, of which there are countably many. We don't know whether the set of possible "real values" (as opposed to measurements – if that concept is even meaningful) is actually uncountable – for all we know, all the "real values" could be rational, or could be drawn from some countable subset of the reals. So, if we are dealing with some kind of measured physical quantity, we actually know that the interval [0,100] is countable. We also know that the value could only possibly be measured to some finite degree of accuracy, so we actually know that the interval [0,100] is a finite set. > In any case, the process you've described is just begging the question (i.e. it is circular reasoning). Maybe it is just an axiom? At some point we just have to accept some things as axiomatic, so why not accept that? It can't be used to derive a contradiction, and it feels self-evident to me (even if not to you), so I feel justified in accepting it as such.
- rcthompson 9y ago
  I think it's probably reasonable to say it's an axiom. That's compatible with my claim that it's an arbitrary choice that is not inherently more correct nor better supported by any information than any other choice, except perhaps in the sense that it feels better or more intuitive.
  - skissane 9y ago
    Well, what happens if we adopt alternative axioms (e.g. some sort of assumption of a normal distribution instead of a uniform one)? Are those alternative axioms consistent? Maybe we might be able to show that agents adopting the uniform axiom make better (more rewarding) decisions in practice than those who adopt a competing axiom do? I don't have a proof, but my gut says yes.

skissane 9y ago

> If you're trying to say that the uniform distribution represents an average of the uncountably infinite set of possible distributions on an interval, I don't see how you can justify that. So, is this X actually an arbitrary real number? Because real numbers are not known to exist in the physical world. In the physical world, we will take some measurement to a finite number of digits, and also produce an estimate of the error in that measurement. The measurement and the error bounds are all described by rational numbers, of which there are countably many. We don't know whether the set of possible "real values" (as opposed to measurements – if that concept is even meaningful) is actually uncountable – for all we know, all the "real values" could be rational, or could be drawn from some countable subset of the reals. So, if we are dealing with some kind of measured physical quantity, we actually know that the interval [0,100] is countable. We also know that the value could only possibly be measured to some finite degree of accuracy, so we actually know that the interval [0,100] is a finite set. > In any case, the process you've described is just begging the question (i.e. it is circular reasoning). Maybe it is just an axiom? At some point we just have to accept some things as axiomatic, so why not accept that? It can't be used to derive a contradiction, and it feels self-evident to me (even if not to you), so I feel justified in accepting it as such.

rcthompson 9y ago
I think it's probably reasonable to say it's an axiom. That's compatible with my claim that it's an arbitrary choice that is not inherently more correct nor better supported by any information than any other choice, except perhaps in the sense that it feels better or more intuitive.
- skissane 9y ago
  Well, what happens if we adopt alternative axioms (e.g. some sort of assumption of a normal distribution instead of a uniform one)? Are those alternative axioms consistent? Maybe we might be able to show that agents adopting the uniform axiom make better (more rewarding) decisions in practice than those who adopt a competing axiom do? I don't have a proof, but my gut says yes.