Can you give some more intuitions as to why allowing finite support is among your criteria?
I can imagine a definition which, lacking this criterion, is still useful, and requiring to have infinite support might be a useful reminder that 0 and 1 are not probabilit(y densities). Further, whereas requiring infinite support might risk analyzing absurd outcomes, it may also allow us to consider, and thus reach maximally great futures.
Reality is inherently bounded—I can confidently assert that there is no possible risk today that would endanger a trillion lives, because I am confident the number of people on the planet is well below that.
The point you are making—that distributions with infinite support may be used to represent model error—is a valid one.
And in fact I am less confident about that one that point relative to others.
I still think that is a nice property to have, though I find it hard to pinpoint exactly what is my intuition here.
One plausible hypothesis is because I think it makes a lot of sense to talk about frequency of outliers in bounded contexts. For example, I expect that my beliefs about the world are heavy tailed—I am mostly ignorant about everything (eg, “is my flatmate brushing their teeth right now?”), but have some outlier strong beliefs about reality which drives my decision making (eg, “after I click submit this comment will be read by you”).
Thus if we sample the confidence of my beliefs the emerging distribution seems to be heavy tailed in some sense, even though the distribution has finite support.
One could argue that this is because I am plotting my beliefs in a weird space, and if I plot them with a proper scale like odd-scale which is unbounded the problem dissolves. But since expected value is linear with probabilities, not odds, this seems a hard pill to swallow.
Another intuition is that if you focus on studying asymptotic tails you expose yourself to Pascal’s mugging scenarios—but this may be a consideration which requires separate treatment (eg Pascal’s mugging may require a patch from the decision-theoretic side of things anyway).
As a different point, I would not be surprised if allowing finite support requires significantly more complicated assumptions / mathematics, and ends up making the concept of heavy tails less useful. Infinites are useful to simplify unimportant details, as with complexity theory for example.
TL;DR: I agree that infinite support can be used to conceptualize model error. I however think there are examples of bounded contexts where we want to talk about dominating outliers—ie heavy tails.
Can you give some more intuitions as to why allowing finite support is among your criteria?
I can imagine a definition which, lacking this criterion, is still useful, and requiring to have infinite support might be a useful reminder that 0 and 1 are not probabilit(y densities). Further, whereas requiring infinite support might risk analyzing absurd outcomes, it may also allow us to consider, and thus reach maximally great futures.
Consider that the number of animal lives is probably greater than one trillion, and you didn’t specify *human* lives. You could also consider future lives, or abstruse moral realism theories. Your definition of personhood (moral personhood?) could change. Having finite support considered harmful (?).
The point you are making—that distributions with infinite support may be used to represent model error—is a valid one.
And in fact I am less confident about that one that point relative to others.
I still think that is a nice property to have, though I find it hard to pinpoint exactly what is my intuition here.
One plausible hypothesis is because I think it makes a lot of sense to talk about frequency of outliers in bounded contexts. For example, I expect that my beliefs about the world are heavy tailed—I am mostly ignorant about everything (eg, “is my flatmate brushing their teeth right now?”), but have some outlier strong beliefs about reality which drives my decision making (eg, “after I click submit this comment will be read by you”).
Thus if we sample the confidence of my beliefs the emerging distribution seems to be heavy tailed in some sense, even though the distribution has finite support.
One could argue that this is because I am plotting my beliefs in a weird space, and if I plot them with a proper scale like odd-scale which is unbounded the problem dissolves. But since expected value is linear with probabilities, not odds, this seems a hard pill to swallow.
Another intuition is that if you focus on studying asymptotic tails you expose yourself to Pascal’s mugging scenarios—but this may be a consideration which requires separate treatment (eg Pascal’s mugging may require a patch from the decision-theoretic side of things anyway).
As a different point, I would not be surprised if allowing finite support requires significantly more complicated assumptions / mathematics, and ends up making the concept of heavy tails less useful. Infinites are useful to simplify unimportant details, as with complexity theory for example.
TL;DR: I agree that infinite support can be used to conceptualize model error. I however think there are examples of bounded contexts where we want to talk about dominating outliers—ie heavy tails.