The example of inferring X→Y from the independence of X and XxorY reminds me of some techniques discussed in Elements of Causal Inference. They discuss a few different techniques for 2-variable causal inference.
One of them, which seems to be essentially analogous to this example, is that if X,Y are real-valued variables, then if the regression error (i.e Y−aX for some constant a) is independent of X, it’s highly likely that Y is downstream of X. It sounds like factored sets (or some extension to capture continuous-valued variables) might be the right general framework to accommodate this class of examples.
This is really cool!
The example of inferring X→Y from the independence of X and XxorY reminds me of some techniques discussed in Elements of Causal Inference. They discuss a few different techniques for 2-variable causal inference.
One of them, which seems to be essentially analogous to this example, is that if X,Y are real-valued variables, then if the regression error (i.e Y−aX for some constant a) is independent of X, it’s highly likely that Y is downstream of X. It sounds like factored sets (or some extension to capture continuous-valued variables) might be the right general framework to accommodate this class of examples.