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Almost every probability theorem starts with "let's take independent random variables". But in reality almost nothing is independent. Superdeterminism even clai
by deepsun 1y ago
Almost every probability theorem starts with "let's take independent random variables". But in reality almost nothing is independent. Superdeterminism even claims that exactly nothing is independent.
- srean 1y agoYou are right. The degree of dependence matters though. Mutual information is one way to measure that. Thankfully, some theorems remain valid even when independence is violated. The next stop after independence is martingale criteria. Martingale difference sequences can be quite strongly dependent yet allow some of the usual theorems to go through but with worse convergence rates.
- mananaysiempre 1y agoAlmost every elementary probability theorem. There are plenty of theorems conditional on bounds on how weak the dependency should be (or how fast correlations decrease with order, etc.), including CLT-like theorems, it’s just that they are difficult to state, very difficult to prove, and almost impossible to verify the applicability of. In practice you are anyway going to instead use the simpler version and check if the results make sense after the fact.