Absolutely! I often think that we use the infinite to approximate the very large but finite, and I think that is a good way to think about the de Finetti theorem for finite sequences. In particular, every finite sequence of exchangeable random variables is equivalent to a mixture over random sampling without replacement. As the length grows, the difference between i.i.d and sampling without replacement gets smaller (in a precise sense). This paper by Diaconis and Freedman on finite de Finetti looks at the variation distance between the mixture of i.i.d. distributions and sampling without replacement in the context of finite sequences, as the length of the finite sequences grows. I’m also aware of work that tries to recover an exact version of a de Finetti style result in the finite context. This paper by Kerns and Székely extends the notion of mixture and shows that with respect to that notion you get a de Finetti style representation for any finite sequence.
Great post! Probably worth noting explicitly that this one of de Finetti’s theorems does not hold in the finite case.
Absolutely!
I often think that we use the infinite to approximate the very large but finite, and I think that is a good way to think about the de Finetti theorem for finite sequences. In particular, every finite sequence of exchangeable random variables is equivalent to a mixture over random sampling without replacement. As the length grows, the difference between i.i.d and sampling without replacement gets smaller (in a precise sense). This paper by Diaconis and Freedman on finite de Finetti looks at the variation distance between the mixture of i.i.d. distributions and sampling without replacement in the context of finite sequences, as the length of the finite sequences grows.
I’m also aware of work that tries to recover an exact version of a de Finetti style result in the finite context. This paper by Kerns and Székely extends the notion of mixture and shows that with respect to that notion you get a de Finetti style representation for any finite sequence.