terns
One of the criticisms the purely probabilistic account of causation has to acknowledge is the charge of reducing causal relations to probabilistic ones where there may be no grounds for this simplistic transition, as summed up by Williamson:
[P]robabilistic dependencies may be attributable to other kinds of relationships between the variables. AandBmay be dependent not because they are causally related but because they are related logi- cally (e.g. where an assignment toAis logically complex and logi- cally implies an assignment toB), mathematically (e. g. mean and variance variables for the same quantity are connected by a mathe- matical equation), or semantically (e. g. AandB are synonymous or overlap in meaning), or are related by non-causal physical laws or by domain constraints. [. . . ] To take a simple example, ifalog- ically implies b thenP(b|a) = 1while P(b)may well be less than
1. In such a case variables A andB (whereA takes assignments
a and¬aandB takes assignments band ¬b) are probabilistically dependent; however it is rarely plausible to say thatAcausesB or vice versa, or that they have a common cause.15
The embedding of these relations has not been available so far, since the arrows in the graphical portion of Pearl’s causal models had to be interpreted causally, the network structure had to thoroughly obey the causal Markov condition, and all events represented by random variables had to be sufficiently distinct to allow for causal inference at all. Nev- ertheless, I want to argue here that the way we infer causal knowledge from more basic assumptions relies to a large extent also on non-causal
knowledge (of the sort referred to by Williamson above), which quite substantially helps arranging and connecting subnet structures of ac- tual causal purport. Amongst the most important relations serving this purpose are node connections representing deterministic, non-directional knowledge, i. e., links that strictly correlate certain variables and along which information may be transferred instantaneously. These find no place in the Bayes net causal models defined above (especially if those are understood as sub-portions of the all-embracive net built on physical laws), but they can be introduced in the epistemically interpreted vari- ant of those very causal models as carefully restricted augmentations. Of course, a new type of edge is necessary for representing this idea, since directed edges are already reserved for directional, asymmetrical
15
causal knowledge. So-called epistemic contours (ECs) shall enrich the graphical part of Pearl’s causal models – however, integrating these
epistemic contours into Bayes net causal graphs turns these graphs into semi-DAGs with undirected subnets, so-called EC cliques:
Definition 3.3.1 (EC Clique)
An EC clique is a subnet in a semi-DAG (of a causal knowledge pat- tern as defined below) that is exclusively connected by undirected edges (representing epistemic contours). EC cliques are defined as transitively closed under the EC relation.
This new kind of edge bars causal inference in the above Bayes net frame- work. The desideratum remains, namely the unification of causal and non-causal knowledge in structures that allow consistent computation of causal claims. This leads to the formulation ofcausal knowledge patterns (CKPs) targeted at facilitating the prediction of future events, the ex- planation of past events, and the choice of suitable actions for efficient achievement of intended goals on the basis of causalandnon-causal data. Gaps between levels of abstraction or even between different disciplines can be bridged by making knowledge explicit in CKPs:
Definition 3.3.2 (Causal Knowledge Pattern)
A causal knowledge pattern is a quadruple
K =hU, V, F, Ci
such that M =hU, V, Fi is a causal model16 where
(i) U is a set backgroundvariables (exogenousvariables), that are set from outside the model;
(ii) V is a set {V1, V2, . . . , Vn} of n endogenous variables, that are de-
termined by variables in the model – i. e., by variables in U ∪V; (iii) F is a set of causal mechanisms for V, i. e., n functions
{f1, f2, . . . , fn} (determining the value of each variable Vi in V)
such that vi =fi(pai, ui)17 with fi : QkRan(PAik) ×Ran(Ui)→Ran(Vi) (1≤i≤ |V|), 16
Also see definition 2.7.1 ofcausal model on p. 56.
17
The boldfacepaicollects for each variableVithe values of its parent variables in
Causality as epistemic principle 93
where for every i: 1 ≤ k ≤ |PAi|, Ui ∈ U (possibly combining
multiple contributing and/or preventative disturbance factors into one complex variable), and PAi ⊆V\{Vi}.18
(iv) C is a set of epistemic contours, i. e., a set of 1-1 functions ci,j
such that
1. 1≤i, j≤n k i6=j, 2. ci,j :Ran(Vi)−→Ran(Vj),
3. di, j, k(ci,j ∈Ckcj,k ∈C⇒ci,k∈C), 4. di, j(ci,j ∈C ⇒cj,i∈C), and
5. cj,i =c−i,j1.
Clause 3 says that the set of functions C is euclidean, while 4 and 5 define C to be closed under inversion.19
(v) A variable X being connected to a second variable Y by an epis- temic contour but possessing no causal mechanisms (i. e., influent arrows in the semi-DAG) is treated like an endogenous variable (since its value is determined by the value of Y) with one excep- tion: If no variable in X’s EC clique receives its value through a causal mechanism (but only via epistemic contours), all variables in X’s EC clique are said to be simultaneously exogenous. One single variable in this EC clique receiving its value from outside conditions suffices to determine the values of all other variables in the EC clique.
The graph DK of a causal knowledge pattern K can be understood
as an augmentation of the graph DM pertaining to the causal model M = hU, V, Fi, which in turn is a sub-structure of K. The set C of deterministic epistemic contours is represented in the graph DK as
undirected edges: The pair of contours {ci,j, cj,i}is graphically rendered as the undirected edge connecting the node with the label Vi to the node with the label Vj. Such an undirected edge will in the following also simply be called contour – although it actually represents a pair of underlying inter-definable functions – since the edge in the graph sym- metrically represents both corresponding functions, and context always disambiguates what is formally referred to.
18
Also see the definition ofcausal mechanisms, 2.7.2, on p. 57.
19
Clauses 3–5 of def. 3.3.2 are listed here w. l. o. g., since allc∈Care 1-1 functions; especially 3 can be loosened to much rather express thepotential expansion ofC.
Now, epistemic contours thus defined satisfy the very desiderata listed above. They represent non-directional knowledge, thereby being capable of bridging different frameworks of description (maybe vertically on different levels or horizontally in different disciplines). Epistemic con- tours deterministically transfer knowledge by virtue of their definition as bijective functions – in a way marking variables that cannot be decou- pled, i. e., variables that cannot be modified separately. In particular, epistemic contours are not to be deactivated by interventions, which re- main defined only for directed edges (i. e., only for causal mechanisms). An epistemic contour ci,j between two variables marks these variables as dependent but not connected causally – a third common cause can be excluded, because intervening on either variable directly (and simulta- neously, i. e., at the same stage of computation) changes the value of the other variable as well. In other words, Vi and Vj are bound intrinsically in such a way that there exists no suitable intervention to detect the direction of any “causal flow.”
X1 X2 X3 X4 Y c3,4 X1 X2 X3 X4 Y c3,4 do(X3=x3) x4=c3,4(x3) y=fY(x4)
Fig. 3.1: Intervening on the variable X3 lifts it from the (causal) influence of
variablesX1 andX2 but does not clip the link betweenX3 andX4.
Consider the left graph of figure 3.1 where two directed acyclic sub- nets X1 A X3 B X2 and X4 A Y are connected by the epistemic contour c3,4 (in the graphical part of the model also denoted by the name ‘c4,3’). Now, the intervention on the variable X3 is expressed in the graph by removing the arrows connecting X3 to its parents X1 and
X2 but upholding the link betweenX3 and X4. This epistemic contour represents deterministic transfer of knowledge and does its job as soon as the interventiondo(X3 =x3) (i. e., the assignment of the valuex3 to the
Causality as epistemic principle 95
variable X3) is performed. X4 receives its valuex4 through the function c3,4 and subsequently passes its value on to the causal mechanism fY which takes c3,4(x3) as the only argument and uniquely computes the outcome y. This example illustrates with the structure X3 n X4 what it means to be anEC clique, as defined above in def. 3.3.1. To allow for consistent inference from causal knowledge patterns, the formulation of suitable restrictions on the construction and the manipulation of these structures is in order.