[FRIAM] PM-2017-MethodologicalBehaviorismCausalChainsandCausalForks(1).pdf

[jon zingale]

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.




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