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Finite Factored Sets

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?

Finite Factored Sets

I’m confused what necessary work the Factorisation is doing in these temporal examples - in your example A and B are independent and C is related to both - the only assignment of “upstream/downstream” relations that makes sense is that C is downstream of both.

Is the idea that factorisation is what carves your massive set of possible worlds up into these variables in the first place? Feel like I’m in a weird position where the math makes sense but I’m missing the motivational intuition for why we want to switch to this framework in the first place

Finite Factored Sets

What would such a distribution look like? The version where X XOR Y is independent of both X and Y makes sense but I’m struggling to envisage a case where it’s independent of only 1 variable.