Whether an event may be called a cause of a second event, obviously depends on how the influence of the associated first variable on the sec- ond variable behaves. In particular, one node in the diagram should be seen as a cause of a second node if assigning a specific value to the first node evokes a difference in the evaluation of the second one (in the vo- cabulary of Bayesian networks). Pearl’s approach thus centers around the notion of causal effect. Such a causal effect may be tested in analogy with a controlled experiment in the laboratory: The scenario is manip- ulated locally, certain conditions of the setting are modified and fixed in such a manner that occurring changes in the values of observed variables can be measured. Now, quite in agreement with this procedure, in the causal model the value of a specific structural function will be modified and fixed, therebycutting the links between the respective variables and their parents. As a formal expression of this intervention, of this manip- ulation from outside, Pearl introduces a new operator which does not become effective within a model but precisely converts one causal model into a second. The so-called do(·)-operator, which may very well be read imperatively, thus induces atransformation of the model under consider- ation, unambiguously. In doing so, it explicitly breaks the Closed World Assumption, on which in particular probabilistic models rest.
From Lewis to Pearl 59
A causal effect can now be expressed asprobabilistic quantity which may be calculated from a probability distribution upon transformation: Definition 2.8.1 (Pearl’s Causal Effect)40
Given two disjoint sets of variables, X andY, the causal effect of X on
Y, denoted either as P(y|xˆ) or as P(y|do(x)), is a function from X
to the space of probability distributions on Y. For each realization x of
X, P(y|xˆ) gives the probability of Y = y induced by deleting from the structural causal model all equations corresponding to variables inX and substituting X=x in the remaining equations.
This definition precisely expresses that the variable X does not depend functionally on any other variables any more. It will be assigned its value
from outside by an intervention external to the model. This process of assigning is not encoded in the model itself, but is part of just such a transformation symbolized by the do(·)-operator.
Our Californian sprinkler example may be consulted for illustration, once more. The notion of external intervention becomes more transpar- ent if one sets out to examine the causal influence of the sprinkler on the slipperiness of the pavement: In our list of structural equations (2.10) the value of the random variable X2 is set to ‘switched on’, i. e., ‘true’. The corresponding equation thus becomes inoperative, and the value x2 in the equation for X4 will as well be fixed to ‘true’. Any possible alter- native for the value of X5 is eliminated – it shall be remarked here, that is was certainly possible for X5 to assume alternative values before the intervention, i. e., ‘true’ or ‘false’. The unblocked causal flow from X2 to
X5 now ultimately brings about the actual slipperiness of the pavement, of course modulo obstructive exogenous influences as plastic covers and such.
In the corresponding graph the modification of the structural equations becomes evident if for any variable the elimination of functional depen- dencies, graphically interpreted, means the elimination of influent edges. In the sprinkler example this means in particular that the transfer of degrees of personal belief between X1 and X2 becomes blocked. Before the intervention the modeling traces the mere observation of the setting. As soon as the running sprinkler is observed, one can infer with great certainty that is is summer or spring, due to the underlying positive cor- relation of X1 and X2. The dry seasons are finally responsible for the sprinkler being switched on, as was our modeling intention.
40
X4
X2 X3
X1
X5
Season
do(Sprinkler = on) Rain
Pavement wet
Pavement slippery
Fig. 2.6: The sprinkler is “switched on by intervention” – applyingdo(·) trans- forms the graph and breaks the link betweenX1 andX2.
An intervention external to the model can be understood as deliberate manipulation of the setting, not influenced by any conditions within the model. In the transformed model one cannot infer the dry season from the observation of the running sprinkler anymore. This deliberate manipulation of X2 does not depend on the value of the variableX1, in particular, which is marked by eliminating the connecting arrow: The sprinkler may now be switched on and off in all seasons virtually if the causal connection with the slipperiness of the pavement is to be tested.
On the basis of these structural local modifications, together with de- pendence tests and quantitative comparison,Pearlestablishes the fine- grained formal representation of statements about causes, direct causes, indirect causes, and potential causes.41 In short: The variable X is a cause of Y in this framework if (given the values of all background vari- ables) there exist two possible values x and x′ such that the choice of a
value for X (either in favor of x or in favor ofx′) makes a difference in
the evaluation of the variable Y.
David Lewis shall be consulted once more for comparison: To de- termine in his possible worlds semantics counterfactually if an event P was causally responsible for a second event Qto occur, one had to stride through a metaphysically existent similarity space with great mental effort to test for metaphysically existent alternative worlds of various
41
From Lewis to Pearl 61
similarity distance, whether the statements ¬P and¬Qdescribe the re- spective settings there correctly – or not. Even when restricting ourselves to dichotomous variables, i. e., bivalent logic, respectively, depending on the number of propositional constants we only obtain a semi-decidable procedure for the identification of causes in the worst case, since we uni- versally quantify over possible worlds.
i ¬p b b w ¬q? P =p Q=q i do(P =¬p) Q=¬q? w
Fig. 2.7: On the search for alternative testing environments forQmoving from thesetting i to the setting w – as proposed by Lewis (on the left)
andPearl(on the right).
In direct comparison: When moving away from the actual world i along the similarity relation in Lewis’ framework, we have to check in the closest ¬pworlds w, if we also find ¬q to hold there. When model- ing generic relationships in the actual world iinPearl’s formalism, we obtain a graph G mirroring the mere observation of correlations. The transition to an alternative world w where ¬p is to be determined can be achieved qua intervention by means of the do(·)-operator: The vari- able P is set to ‘false’, influent edges are eliminated. The question is now, whether the assignment of the P value also leads to a measurable difference in the evaluation of the variable Q. If the causal effect of P on Q in the model is identifiable (and Pearlgives algorithmic criteria for determining if it is or not), then it can be calculated uniquely and efficiently on the basis of the stable functional mechanisms. Identify- ing causes inPearl’s formalism can thus be understood both asnatural andoperationally effective at the same time, because the invariant mech- anisms representintuitively obvious basal assumptions, and because the external interventions are limited to local surgeries of the graph.