Counterfactual Inferences

Causal Bayes nets theory not only provides computational mechanisms to infer the consequences of hypothetical interventions but also models counterfactual actions.

Counterfactual interventions refer to interventions that deviate from the factual course of events in the world. For example, “If my friend had stopped me from driving too fast then I would still have my driver’s license” is an inference based on a counterfactual statement, because the events “driving too fast” and “losing my driving license” are true in the actual world. Similar to the predictions for hypothetical interventions, counterfactual inferences are not invariant against causal structure.

Counterfactual inferences combine observations and interventions. A counterfactual intervention is defined as an action that alters a factual state of the world. Thus, inferences of this type refer to states in a counterfactual world. Causal Bayes nets allow to formalize questions such as “Today no signal fire was observed on tower C – what is the (counterfactual) probability of a fire on tower D if a fire on tower C had been lit by lightning?” Such counterfactuals comprise two pieces of information: a factual observation and a counterfactual action altering this state. For example, the counterfactual probability P(d | ¬c. Do c) is read as “the probability of D = d given that C was observed to be absent but counterfactually generated”. The first piece of information is that C was observed to be absent, a statement referring to the actual world. This information provides the basis for updating the probabilities of C’s Markovian parents (i.e., variable A). The second piece of information posits the counterfactual generation of C. This action refers not to the actual world (in which event C was absent) but to a counterfactual world in which C has been generated by external intervention. From these two pieces of information we can derive the state of D in a counterfactual world (which might or might not correspond to the factual world).

Thus, the basic logic is that the probabilities of C’s causes are updated in accordance with the event’s factually observed state and estimates for C’s effects are computed conditional on the implications of the counterfactual action. We then have a three-step procedure for computing the consequences of counterfactual interventions

(cf. Pearl, 2000). First, the probabilities of the observed variable’s causes are updated in accordance with the event’s state in the actual world. Second, the causal graph is modified according to the counterfactual intervention, that is, graph surgery is performed. The crucial point is that graph surgery is performed in the updated model, not on the original one. Otherwise, the intervention would render the variable targeted by the intervention independent of its causes which then could not be updated. Finally, the updated and truncated model is used to predict the consequences of the counterfactual intervention. Figure 10 illustrates how causal Bayes nets model counterfactual interventions.

Figure 10. Modeling counterfactual interventions.

For example, in the diamond-shaped model we might observe C to be present (e.g., a signal fire is observed on tower C) but ask for the probability of a certain variable in the system conditional on a counterfactual prevention of event C (i.e., an intervention that would have prevented the signal fire on tower C). Again, I start with the simple diagnostic inference from C to its cause A. Since intervening in an effect will not influence its cause, the probability of A is determined by the factually observed state of C alone. Thus, the probability of A given we observe C to be present but counterfactually remove C is given by

( | . Do ) ( | )

P a c ¬ =c P a c (21)

Conversely, the probability of A given that C is observed to be absent but counterfactually generated is given by

C_AUSAL B_AYES N_ETS T_HEORY 54 ( | . Do ) ( | )

P a c¬ c =P a c¬ (22)

Whereas for hypothetical interventions the probability of C’s cause A was given by A’s base rate, in the case of counterfactual interventions the probability of event A is updated in accordance with the observed state of C. Since the counterfactual intervention implies the alteration of a factual state, the probability of A is higher in the case of a counterfactual inhibition of C (which logically entails that C has been observed to be present) than in the case of a counterfactual generation of C (which implies that C has been observed to be absent in the actual world). For example, the probability of a signal fire on tower A is higher when a fire on tower C is observed to be present but counterfactually undone than in the case of a hypothetical intervention preventing the fire on tower C.

It is also possible to compute the probability of the final effect D conditional on counterfactual interventions in C. For this inference, it is necessary to integrate both the observed values of C and the counterfactual intervention in C. This is an interesting case since the counterfactual intervention will affect the direct link C→D, but the observed value of C provides information about A, and, therefore, also about the probability with which D’s alternative cause, B, occurs. Consider the counterfactual probability P(d | c. Do ¬c) which asks for the probability of D given that C is observed to be present but counterfactually removed. The counterfactual prevention of C will break the causal path C→D therefore D is not any longer influenced by C in the counterfactual world.

However, observing C to be present indicates that its cause A has been present, too, which, in turn, raises the probability of B occurring. Similar to the hypothetical intervention Do ¬c, event D is then completely determined by the backdoor path.

However, there is one crucial difference: whereas in case of the “normal”(hypothetical) intervention the probability of event A is given by its base rate, in case of a counterfactual intervention A is conditionalized on the observed value of C. Thus, the probability of D is higher in case of a counterfactual prevention of C than in case of a hypothetical prevention of C, provided P(a) < P(a | c). The corresponding formula is

( | . Do ) ( | ) ( | ) ( | . ) Conversely, the probability of D given that C is observed to be absent but is counterfactually generated is given by

( | . Do ) ( | ) ( | ) ( | . )

( | )· ( | )· ( | . ) ( | )· ( | )· ( | . ) ( | )· ( | )· ( | . ) ( | )· ( | )· ( | . ) P d c c P A c P B A P d B c

P a c P b a P d b c P a c P b a P d b c P a c P b a P d b c P a c P b a P d b c

¬ = ¬ ⋅ ⋅

= ¬ + ¬ ¬ ¬ +

¬ ¬ ¬ + ¬ ¬ ¬ ¬ ¬

∑

(24) Note that in these formulas event A (C’s cause) is conditionalized on the observed state of C but the probability of D (C’s effect) is then computed from the counterfactually altered state of C. Again only parameters are involved which can be derived from observational learning.