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.