A unified theory of counterfactual reasoning

A unified theory of counterfactual reasoning Christopher G. Lucas Charles Kemp cglucas@cmu.edu Department of Psychology Carnegie Mellon University ckemp@cmu.edu Department of Psychology Carnegie Mellon University (a) Abstract A successful theory of causal reasoning should be able to account for inferences about counterfactual scenarios. Pearl (2000) has developed a formal account of causal reasoning that has been highly influential but that suffers from at least two limitations as an account of counterfactual reasoning: it does not distinguish between counterfactual observations and coun- terfactual interventions, and it does not accommodate back- tracking counterfactuals. We present an extension of Pearl’s account that overcomes both limitations. Our model provides a unified treatment of counterfactual interventions and back- tracking counterfactuals, and we show that it accounts for data collected by Sloman and Lagnado (2005) and Rips (2010). A (b) U a (c) C B ′ B B U c C A U a ′ U b U b ′ A ′ B ′ B U c U c C U a A U b U b B A A (d) U a C C U c ′ C ′ Figure 1: (a) A causal chain in which A causes B, which causes C. (b) A functional causal model that captures the causal chain in (a). Exogenous variables U a , U b and U c have been added, and nodes with double edges are deterministic functions of their parents. (c) A twin network where A 0 , B 0 and C 0 represent counterfactual values of A, B, C. (d) An aug- mented twin network which allows for the possibility that the exogenous variables take different values in the counterfac- tual scenario. In addition to reasoning about actual states of affairs, hu- mans find it natural to reason about what might have been. A doctor may ask “if Alice had not been treated with the experimental drug, would she have survived? ” and a parent might tell a child that “if you had been paying attention, you wouldn’t have gotten hurt. ” Researchers from several disci- plines have developed formal models of counterfactual rea- soning, and recent empirical studies have evaluated the psy- chological merits of some of these models (Rips, 2010; De- hghani, Iliev, & Kaufmann, 2012). This paper describes a new model of counterfactual reasoning and evaluates it using data sets from the psychological literature. The problems that we consider can be illustrated using a causal chain over three variables (Figure 1a). For example, suppose that A, B, and C are variables that indicate whether three transponders are active. Transponder A is active about half of the time, and whenever it is active it tends to activate B, which in turn tends to activate C. Suppose that we observe on a certain occasion that all three transponders are active. We can now ask counterfactual questions such as “if B had not been active, would C have been active? ” The formal approach that we present is inspired by the work of Pearl (2000), who developed a model of counterfac- tual reasoning that we refer to as the modifiable structural model, or MSM for short. The MSM assumes that the causal system in question is a functional causal model, where ex- ogenous variables are introduced if necessary so that the vari- ables of primary interest are deterministic functions of their parents. For example, the system in Figure 1a may be repre- sented more precisely by adding exogenous variables U A , U B and U C such that U A determines whether or not node A is ac- tive, and U B and U C capture factors such as atmospheric con- ditions that determine whether the links in the chain operate successfully. Suppose now that A, B and C are all observed to be active, and that we want to know whether C would be active if B were not active. The MSM addresses this ques- tion by using the observations in the actual world to update prior beliefs about the status of the exogenous variables, then modifying the resulting causal model to reflect a counterfac- tual intervention where B is forced to be inactive. Inferences about any other variables can be computed using the modi- fied causal model—for example, if B were inactive, then C would probably also be inactive. Several psychological stud- ies of counterfactual reasoning have evaluated the predictions of the MSM and have found that some counterfactual infer- ences do appear to be treated as inferences about counterfac- tual interventions (Sloman & Lagnado, 2005; Kemp, Shafto, & Tenenbaum, 2012). The approach we present builds on the key ideas behind the MSM, and we refer to it as the doubly-modifiable struc- tural model or DMSM for short. Like the MSM, the DMSM works with causal systems that are represented using func- tional causal models and allows these systems to be modified via counterfactual interventions. In addition, however, the DMSM permits a second kind of modification where exoge- nous variables are altered not because of a counterfactual in- tervention, but simply because the counterfactual world might have turned out differently from the real world. An impor- tant consequence of this difference is that the DMSM alone accounts for backtracking counterfactuals. For example, sup- pose again that all three transponders in A were observed to be active, and we are asked to decide whether A would be ac- tive if B were inactive. The DMSM allows for the possibility that variables upstream of B might explain the counterfactual premise that B is inactive, and thus predicts that A is likely to be inactive. The MSM, however, can only reason about