Perhaps of further use(fulness/lessness) is a cartesian product interpretation of screening-off in the V-Y model[&]. If we consider each stage to be a set of *observations* and functions between them as relating *evidence*, we can interpret *cause* as the epimorphisms, those functions that are right cancellable (screening off earlier stages) and whose domain observations fully account for the codomain observations[s]. Correlation ultimately sneaks in whenever we coequalize (also an epimorphism and so causal) functions from the product.
The Bayesian interpretation, as far as I can tell, gives criteria for when this modding out should occur (distal causes?) and how it is to be handled. My hope for this approach is to elucidate when one can infer linkages safely in a causal network and when one cannot, the distinction being that while evidence ought to compose without side-effects, causality can not. From a high level, the *screen-breaking* condition is effectively summarized as 'no functions on products without modding out'. Now, given any product data: (π1: X -> R1, π2: X -> R2)[𝝥] we can look at how maps from earlier stages relate to the triple (X, π1, π2). It follows that any pair of functions with a common domain: (a: S -> R1, b: S -> R2) have a unique interpretation through X, as X is a product as-well-as a cause. The functions (a or b, say) can come in two varieties, causal or not, the latter perhaps contributing evidence. In the case that a and b are causal, we are not guaranteed 'classical' screening-off of S from R1 and R2, via X. As stated above, X being a product guarantees a unique representation (r1, r2): S -> X, such that we can recover a: S -> R1 as π1∘(r1, r2) and b by π2∘(r1, r2). Now in the case that the map (r1, r2) is epic, we either have just as much information as is carried into the projection, or something unnecessary is lost. Otherwise, S is 'smaller' than X and cannot be a cause of X. This being said, we can now return to the Markov interpretation: For S causal on X, S is in 'V', X is in DE(S) and so nothing can be said about the truth of P(R1 & R2 ∣ PA(S) & X) = P(R1 & R2 ∣ PA(S)). For S non- causal on X, S is neither a parent nor descendent cause and so classically, P(R1 & R2 ∣ PA(X) & S) = P(R1 & R2 ∣ PA(X)). I continued to sketch out a handful of other ideas, but they were much sketcher than even that above. Let me stop here for now. [&] Cartesian product is what I think of whenever we invoke 'AND'. [s] I wish to connect the question of *choice* in variation partitioning with the idea of *section* for epimorphisms, further suggesting that the collection of these sections may give a presheaf category. It is not yet at all clear to me that this intuition is correct, but hey. [𝝥] The projection maps (π1, π2) are epimorphic by design and so are directly interpretable as causal. -- Sent from: http://friam.471366.n2.nabble.com/ - .... . -..-. . -. -.. -..-. .. ... -..-. .... . .-. . FRIAM Applied Complexity Group listserv Zoom Fridays 9:30a-12p Mtn GMT-6 bit.ly/virtualfriam un/subscribe http://redfish.com/mailman/listinfo/friam_redfish.com FRIAM-COMIC http://friam-comic.blogspot.com/ archives: http://friam.471366.n2.nabble.com/
