On the last example with the XOR temporal inference—since the partitions/queries we’re asking about are also possible factors, doesn’t the temporal data in terms of history etc depend on which choice of factorisation we go with?
We have a choice of 2 out of 3 factors each of which corresponds to one of the partitions in question, so surely by factorising in different ways we can make any two of the variables have history of 1 and thus automatically orthogonal?
So we are allowing S to have more than 4 elements (although we dont need that in this case), so it is not just looking at a small number of factorizations of a 4 element set. This is because we want an FFS model, not just a factorization of the sample space.
If you factor in a different way, X will not be before Y, but if you do this it will not be the case that X is orthogonal to X XOR Y. The theorem in this example is saying that X being orthogonal to X XOR Y implies that X is before Y.
On the last example with the XOR temporal inference—since the partitions/queries we’re asking about are also possible factors, doesn’t the temporal data in terms of history etc depend on which choice of factorisation we go with?
We have a choice of 2 out of 3 factors each of which corresponds to one of the partitions in question, so surely by factorising in different ways we can make any two of the variables have history of 1 and thus automatically orthogonal?
So we are allowing S to have more than 4 elements (although we dont need that in this case), so it is not just looking at a small number of factorizations of a 4 element set. This is because we want an FFS model, not just a factorization of the sample space.
If you factor in a different way, X will not be before Y, but if you do this it will not be the case that X is orthogonal to X XOR Y. The theorem in this example is saying that X being orthogonal to X XOR Y implies that X is before Y.