>In other words, you don’t need reality to be i.i.d.; you simply need to structure your beliefs in a way that allows an “as if” i.i.d. interpretation.
I think I view exchangeability vs. iid slightly differently. In my view, the “independence” part of iid is just way too strong, and is not required in most of the places people scatter the acronym “iid”.
For example, say you are catching fish in a lake, and you know only bass and carp live in the lake, and that there are a ton of fish in it, but not how many of each, and you’re trying to estimate the proportion of carp as you catch fish.
When I catch a carp, the probability that my next catch is a carp goes up. So my probability is dependent on my previous catches—that’s why I can learn things about the proportion! If they were indeed independent, then I couldn’t learn anything. But happily, the correct requirement is not independence, but exchangeability, so I can still update my beliefs as I see more fish.
However, I may just be confused about “iid” as classicists use it, since I never properly learned classic statistics. Interested in what you think about the difference between the two in this example.
>In other words, you don’t need reality to be i.i.d.; you simply need to structure your beliefs in a way that allows an “as if” i.i.d. interpretation.
I think I view exchangeability vs. iid slightly differently. In my view, the “independence” part of iid is just way too strong, and is not required in most of the places people scatter the acronym “iid”.
For example, say you are catching fish in a lake, and you know only bass and carp live in the lake, and that there are a ton of fish in it, but not how many of each, and you’re trying to estimate the proportion of carp as you catch fish.
When I catch a carp, the probability that my next catch is a carp goes up. So my probability is dependent on my previous catches—that’s why I can learn things about the proportion! If they were indeed independent, then I couldn’t learn anything. But happily, the correct requirement is not independence, but exchangeability, so I can still update my beliefs as I see more fish.
However, I may just be confused about “iid” as classicists use it, since I never properly learned classic statistics. Interested in what you think about the difference between the two in this example.