Section 16 The chaining rule, functional form (zooming out / black-box rule)
Given two mini-maps containing the paths \(E = f(B)\) and \(O = g(E)\) respectively, the direct influence of B on O is \(O = g(f(B))\).
This rule tells us how to calculate downstream effects in a map which has been created using the chaining rule.
It is an elementary application of the “substitution rule” from secondary school maths.
This problem is familiar from statistics. We can ask, if B affects E and E affects O, what is the direct effect of B on O? If we know \(f\) and \(g\), we can trivially calculate \(g(f(B))\) simply by applying \(g\) to the result of \(f\).
This rule, although it seems trivial, has wide practical applications. For one thing, it allows us to simplify causal maps by “zooming out” and omitting some of the detail.
So if you know the information in this reasonably complicated network:
..… you can in some sense “zoom out”, create a black box, subsuming the functional content of the pink boxes, to conclude this:
… where the function \(g\) can in principle be constructed if you know \(f1\), \(f2\) and \(f3\).
16.1 Two special cases
What’s quite weird about this rule is that there are two extreme extensions of it which are in a sense opposites. These special cases are important because they are possible interpretations of the arrows in an arbitrary causal map in which no other information is given about the function involved.
16.1.1 Bare influence
Consider the very simple function \(H\) such that \(E = H(B)\) which just means, B has some influence, a bare influence, on E. Then from
\(E = H(B)\)
and
\(O = H(E)\)
we can conclude, using the rule, not only
\(O = H(H(B))\)
but we can also argue
\(O = H(B)\)
.. this does not follow from the rule but it should be possible to prove it: if B sometimes has some effect on E, and E sometimes has some effect on O, then B sometimes has some effect on O.
We can generalise this argument to cases with more than one influence variable. So \(E = H(C, D)\) means that there are some values of D such that making some difference to C makes a difference to E (it might be that most of the time, tweaking C does nothing to E), and similarly there are some, any, values of C such that tweaking D makes a difference to E, and so on.
16.1.2 Total control
Whereas in the second extreme version, the interpretation of the function \(H\) is “the influence variables all have total causal control of the consequence variable”. There is no room for E to do anything not dictated by B (and possibly, simultaneously by C, D etc).
So, if
B completely controls E
and
E completely controls O
we can conclude that
B completely controls O
16.2 Technical note
It is a bit naive to assume that if we know \(f(x)\) and we know \(g(x)\) we can always easily calculate \(g(f(x))\). In fact, the mere fact that we know f(x) does not mean that it is easily calculable either. There are important caveats about computability and decidability which have practical consequences.
The zooming-out rule brings us to the edge of what seems like a debate about philosophical realism. For example, we could assert that in all practical applications, we can assume the reverse of the zooming-out rule: that with a more detailed investigation we could “zoom in” to any atomic relationship and find other, previously hidden, variables mediating the relationships. Or:
What counts as a “black box”, what counts as “revealing the underlying mechanism” is relative and context-dependent. You can always zoom in a bit further or zoom out a bit further.
Arguably this is counter to some interpretations of Realist approaches in evaluation and social science according to which there is a hard-and-fast distinction between the two: there is a superficial, black-boxy way of looking at things and a correct, Realist, zoomed-in view; and according to which there is a “fact of the matter” about mechanisms, which are real things waiting to be discovered, not constructions. (Of course there are other differences too between the two views, according to the Realist approach.)
In fact, the approach I am presenting here is not extensionalist. I do not claim that the zoomed-out causal map is equivalent to the zoomed-in version, even though by definition the designated upstream variables produce the same effects on the designated downstream variables. For one thing, as Pearl points out, a genuinely extensionalist large cognitive map would be computationally impossible. No brain and no device could construct a single global causal map from all possible inputs to all possible outputs. Mini-maps and small aggregations of them are in fact the key to acquiring and using knowledge.