# Why is infinite data required to verify statistical independence

I've been reading about Independent Component Analysis and the FastICA algorithm. The wiki page for FastICA states:

FastICA is an efficient and popular algorithm for independent component analysis invented by Aapo Hyvärinen at Helsinki University of Technology.[1][2] Like most ICA algorithms, FastICA seeks an orthogonal rotation of prewhitened data, through a fixed-point iteration scheme, that maximizes a measure of non-Gaussianity of the rotated components. Non-gaussianity serves as a proxy for statistical independence, which is a very strong condition and requires infinite data to verify.

I understand that non-gaussianity can serve as a proxy for independence due to the central limit theorem, but why is it that independence itself requires infinite data to verify?

## 1 Answer

Independence, like many other concepts in statistics, is a feature of a population rather than a sample. You can't conclude much with certainty about a population from only a finite random sample, since you could always be arbitrarily unlucky in your choice of sample.

• Ah, didn't realize the answer was that simple. Great, thanks. Dec 5, 2017 at 18:42
• Yes, but by that reasoning everything requires "infinite data." (except when sampling from finite populations). Thus, either the quotation is a meaningless triviality or else something else was meant.
– whuber
Dec 5, 2017 at 18:42
• @whuber I tend towards thinking "meaningless triviality", but it's possible that the author meant something else. Dec 5, 2017 at 18:45