I had intended to be using the program’s output as a time series of bits, where we are considering the bits to be “sampling” from A and B. Let’s say it’s a program that outputs the binary digits of pi. I have no idea what the bits are (after the first few) but there is a sense in which P(A) = 0.5 for either A = 0 or A = 1, and at any timestep. The same is true for P(B). So P(A)P(B) = 0.25. But clearly P(A = 0, B = 0) = 0.5, and P(A = 0, B = 1) = 0, et cetera. So in that case, they’re not probabilistically independent, and therefore there is a correlation not due to a causal influence.
Just to chip in on this: in the case you’re describing, the numbers are not statistically correlated, because they are not random in the statistics sense. They are only random given logical uncertainty.
When considering logical “random” variables, there might well be a common logical “cause” behind any correlation. But I don’t think we know how to properly formalise or talk about that yet. Perhaps one day we can articulate a logical version of Reichenbach’s principle :)
Yeah, I think I agree that the resolution here is something about how we should use these words. In practice I don’t find myself having to distinguish between “statistics” and “probability” and “uncertainty” all that often. But in this case I’d be happy to agree that “all statistical correlations are due to casual influences” given that we mean “statistical” in a more limited way than I usually think of it.
But I don’t think we know how to properly formalise or talk about that yet.
A group of LessWrong contributors has made a lot of progress on these ideas of logical uncertainty and (what I think they’re now calling) functional decision theory over the last 15ish years, although I don’t really follow it myself, so I’m not sure how close they’d say we are to having it properly formalized.
Just to chip in on this: in the case you’re describing, the numbers are not statistically correlated, because they are not random in the statistics sense. They are only random given logical uncertainty.
When considering logical “random” variables, there might well be a common logical “cause” behind any correlation. But I don’t think we know how to properly formalise or talk about that yet. Perhaps one day we can articulate a logical version of Reichenbach’s principle :)
Yeah, I think I agree that the resolution here is something about how we should use these words. In practice I don’t find myself having to distinguish between “statistics” and “probability” and “uncertainty” all that often. But in this case I’d be happy to agree that “all statistical correlations are due to casual influences” given that we mean “statistical” in a more limited way than I usually think of it.
A group of LessWrong contributors has made a lot of progress on these ideas of logical uncertainty and (what I think they’re now calling) functional decision theory over the last 15ish years, although I don’t really follow it myself, so I’m not sure how close they’d say we are to having it properly formalized.
nice, yes, I think logical induction might be a way to formalise this, though others would know much more about it