Yes I can flip two independent coins a finite number of times and get strings that appear to be correlated. But in the asymptotic limit the probability they are the same (or correlated at all) goes to zero. Hence, two causally unrelated things can appear dependent for finite sample sizes. But when we have infinite samples (which is the limit we assume when making statements about probabilities) we get P(a,b) = P(a)P(b).
Thanks for commenting! This is an interesting question and answering it requires digging into some of the subtleties of causality. Unfortunately the time series framing you propose doesnt work because this time series data is not iid (the variable A = "the next number out of program 1" is not iid), while by definition the distributions P(A), P(B) and P(A,B) you are reasoning with are assuming iid. We really have to have iid here, otherwise we are trying to infer correlation from a single sample. By treating non-iid variables as iid we can see correlations ...
In the example of the two programs, we have to be careful with what we mean by statistical correlation v.s. more standard / colloquial use of the term. Im assuming here when you say `the same program running on opposite ends of the universe, and their outputs would be the same’ that you are referring to a deterministic program (else, there would be no guarantee that the outputs were the same). But, if the output of the two programs is deterministic, then there can be no statistical correlation between them. Let A be the outcome of the first program an...
This is such a good deep dive into our paper, which I will be pointing people to in the future. Thanks for writing it!
Agree that conditioning on the intervention is unnatural for agents. One way around this is to note that adapting to an unknown distributional shift given only sensory inputs Pa_D is strictly harder than adapting to a known distributional shift (given Pa_D and sigma). It follows that any agent capable of adapting given only its sensory inputs must have learned a CWM (footnotes, p6).