Section 11 The mini-map coding rule – functional form

The mini-map coding rule said that Information like “the influence variables B, C and D all have some kind of causal influence on the consequence variable E” can be coded with a mini-map in which one or more variables are shown with arrows leading to another.

This information can also be expressed in a functional form: \(E = f(B, C, D)\) which says, given the values of B, C and D, the influence rule or function tells us how to work out the value of E.

More generally, the function will in real-life cases not completely determine the value of E. Other, mostly unknown, factors are involved. With classical, linear functions we can just add the influence of \(f\) to the prior value of E. (Or conversely we can express the value of E as being a main prediction made by the function \(f\) together with additional “error” influences which can likewise be added to the prediction.) As my approach will be broader and will cover non-parametric and non-numerical cases, we cannot rely on this convenient convention. So we should more generally express the function like this: \[E_{posterior} = f(B, C, D, E_{prior})\]. What this says is that \(f\) shifts the value of E away from our previous or prior best guess about E to a value determined not only by the influence variables but also that prior value.

This generalisation might seem very abstract but we will see that it is necessary to adequately present some quite basic functions like necessary and sufficient conditions.

The function statement, the written information and the map are all equivalent.

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Technical note

It is important to note that the = sign in these functional expressions is not the usual equality sign. For one thing, it is not symmetric. See Pearl (xx).