# Infinite exchangeability

An infinite sequence of random variables $$(X_1, X_2, ..., X_n, ...)$$ is infinitely exchangeable under probability measure $$\mathbb{P}$$ if the joint distribution for each subsequence $$(X_{n_1},...,X_{n_k})$$ satisfies $(X_{n_1},...,X_{n_k}) =^d (X_{\tau(n_1)},...,X_{\tau(n_k)})$

for all permutations $$\tau$$ over $$[k]$$.

## Thoughts

• Timo Koski, in his notes on the advanced statistical inference course, mentions that the notion of exchangeability involves a judgement of complete symmetry among all observations under consideration - and the de Finetti representation theorem is a statement from the point of view of subjectivist modelling, and that it is as if there is no true parameter, only data and judgement.

Created: 2022-03-13 Sun 21:44

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