3.2.1
Directionality and information transfer
In networks literature the references to ‘causality’ take many guises. The term directional- ity, information transfer and sometimes even independence can possibly refer to some sort of ‘causality’ in line with our previously defined concept. ‘Causality’ plays a main role when [76] discusses flow of information in Bayesian systems and when [66] expounds on way to formalize information transfer on fully known dynamical systems. Our definition of ‘causality’ is based on how well a variable helps the prediction of another variable. Now let us assume thatY causes X.
We would expect the relationship between X and Y to be asymmetric and that the
information flows in a direction from Y to X. [68, 66, 52] highlights the importance of asymmetry in information transfer. When it comes to directionality it is paramount to point out that the main reason correlation is not equal to causation is due to the fact that causation has direction and thus essentially asymmetric [44]. When one variable causes another variable obviously the affected variable depends on the causal variable. Therefore, one could also say that our prediction based definition of ‘causality’ is equivalent to looking for dependencies between the variables at a certain causal lag.
Information transfer needs a source and target. The source where the information is from and the target where it is transferred to. Thus in the case of ‘causality’ the source will be the causal variable and the target is the affected one. One can assume that this informa- tion transfer is the unique information provided by the causal variable to the affected one. However this does not mean that the causal variable has complete control over the affected variable.
3.2 Issues in ‘causality’ 44
3.2.2
Deterministic variables and instantaneous casuality
If a variable has complete control of another variable then it is deterministically determined by the control variable and thus indistinguishable from it. A purely deterministic variable cannot be said to have any other causal influence other than its own past and a simple example of a deterministic case given by [44] illustrates this. Let there be variables X
and Y where Xt = bt and Yt−1 = c(t + 1). Then X can exactly be predicted by the
equationXt = b + Xt−1 or equally by the equationXt = bcYt−1. The predictions using
bothXt−1 andYt−1 are exactly the same hence indistinguishable. Moreover, one can also
expressYtin terms ofXt−1through the formulaYt= cb(Xt−1+ 2b) which is equivalent to
Yt = Yt+1+c. Consequently it seems that ‘causality’ requires the variables to be stochastic.
With the uncertainty one is able to measure the ‘causal’ element and the directionality. The ‘causal’ and affected variable needs to have an independent source of variation [50].
The notion of instantaneous causality is discussed in [46]. The idea that two variables can instantaneously cause each other with no causal lag at all has been said to be impossible. Granger maintains that true instantaneous causality can never occur [46] and if anything appears to be like it, then the ‘causality’ is either not measured at the correct time scale (the causal lag is smaller than the measured time scale) or there is another variable jointly (or indirectly) causing it which is not observed (not incorporated in the model).
3.2.3
Indirect ‘causality’ and independence
Granger pointed out that apparent instantaneous causality could be caused by variables that were not incorporated in the model. He also brought to attention in [46], that any two variables that are independent may not be conditionally independent. Referring back to subsection (2.3.1), if two variables are statistically independent then it means that their joint distribution is the product of their marginal distributions. Therefore what Granger is implying is that variablesX and Y may be independent but at the same time variables X|Y
andX|Z may not be independent.
Thus one can say thatZ brought about the dependency between X and Y . And since we
have defined ‘causality’ as a sort of dependency over a certain causal lag, then one can also expect that there will be cases whereZ brings about a causal effect between X and Y . [17]
Chapter 3. The question of ‘causality’ 45 speaks about latent variables (hidden variables not directly incorporated in the model) that might give rise to correlations in a model where ‘causality’ is supposed to be inferred from. Dufour [31] mentions indirect causality that might be induced by an auxiliary variable Z
on the ‘causal’ relationship betweenX and Y . Pearl talks about mediation in his paper [80] where the term direct effect refers to an effect that is not mediated by other variables in the model and by saying this he acknowledges that there exist mediation [16, 83] needed for some (or most) type of causal relationship.
The previous discussions clearly indicates that there is a need to include more than just two variables in a ‘causal’ analysis. Take lightning and thunder for example [45], we now know that the reason we usually observe lightning before thunder is because light is much faster than sound. We also know the fact that lightning and thunder are both essentially the same event manifested at different times and caused by the same electrical discharge. Let
X be thunder, Y be the lightning and Z be the electrical discharge. If we only look at X
andY we will mistakenly say that Y causes X i.e. lightning causes thunder. However if
we includeZ then we will be able to infer that Z (the electrical discharge) is the real cause
ofX and Y as well as the fact that the very existence of a relationship between X and Y
depends completely onZ.
This kind of ‘causality’ is what we shall refer to as indirect ‘causality’ whereZ indi-
rectly causes the relationship betweenX and Y . Whereas the relationship between Z and X as well as the relationship between Z and Y can be said to be a direct cause. By that
definition, the electrical discharge directly causes the thunder and it also directly causes the lightning but the electrical discharge indirectly causes the thunder to be related to the lightning. The condition that a causal relation cannot be due to a common cause is referred to as causal sufficiency and some philosophers claim that only direct ‘causality’ can be considered to be a real ‘causality’ [50]. Indirect ‘causality’ is a problem in many fields [45] and we believe that the brain is no exception. To incorporate this indirect ‘causality’ is very much a problem when the ‘causality’ measure is not model-free since we have to incorporate all the right variables into the model to begin with. This is one of the main reasons why we will mainly be in favor of model-agnostic ‘causality’ measures.