2.1. Strongly Active Paths (H1)

A boulder falls (F) toward a hiker (Hitchcock Reference Hitchcock2001). The hiker sees the boulder in time, and ducks (D), thereby surviving (S). Each variable is binary. We represent the causal variables as nodes in a graph and the causal links as arrows between them. We specify the exact dependencies with a series of simple structural equations, as in figure 1. The first equation gives an initial condition: in the actual world, the boulder falls. The second says that the hiker ducks if the boulder falls. The third says the hiker survives if he ducks or if the boulder does not fall. The causal model is fully specified by the equations and the possible values for each variable.

Figure 1

Boulder example. D = ducks; F = falls; S = survives.

Causal models specify the counterfactuals over the variables in the model, in the same way a circuit diagram specifies the effect of any alteration or failure of its components (Pearl Reference Twardy and Korb2000). Counterfactuals do not backtrack. If they did, ducking would make no difference to survival. But counterfactuals are assessed by interventions, not observations. We do not observe that D is 1. Instead, we replace the structural equation with and reevaluate. The boulder falls , and when , and when , so ducking makes a difference. Therefore, we want to say that ducking caused survival. Conversely, we do not want to say that the boulder falling caused survival. The difference is that only ducking has an active path to survival.

We now define active path actual causation (APAC). A path is a traversal from C to E along the arrows, in the direction of the arrows. Then:

Intuitively, a path is active if it makes a difference. A path is strongly active if c makes a difference to e when all other paths between C and E are blocked by setting variables along those paths to their actual values. “Making a difference” has the usual counterfactual meaning: had C been different, E would have been different, for at least some .

In Boulder, neither of the paths from F to S is strongly active. First, we hold D fixed at its actual value blocking and leaving only . But since , F makes no difference: given that the hiker ducked, the boulder has no effect. The path is not strongly active.

To evaluate for strong activity, we would have to block , which we cannot, as there is no intermediate variable. With nothing to block, we are back to the original case where F makes no difference to S because the paths cancel. We happily conclude that F did not cause S. “Intuitively, the falling boulder does not save Hiker's life because without it, Hiker's life would not have been endangered in the first place” (Hitchcock Reference Hitchcock2001, 295).

When we are considering a particular path, all ancestors of E not on it constitute the background. A straightforward (if inefficient) attempt to block all but one path is to intervene to set the whole background—all ancestors of E except those along our path . Call this background β.

We can now give a formal definition of strong activity. Let , or just , denote that we intervene to set ; similarly for . (We shall also use “set” as an English equivalent for the I notation.) We assume throughout that C is the beginning of the path and E its end. Then:

If we allow only strongly active paths, then APAC yields Hitchcock's first active path account, which Hiddleston (Reference Hiddleston2005) dubs H1. Unfortunately, H1 handles neither symmetric overdetermination nor late cutting. We review redundancy ranges to handle overdetermination and then resetting ranges to avoid late cutting. Afterward, we present our probabilistic reformulation.

2.2. Redundancy Ranges

Symmetric overdetermination happens, given multiple sufficient causes. One such case is Matches. Two lit matches (M ₁ and M ₂) are dropped in the forest. Either would suffice to burn it down, and the forest burns down (FB; fig. 2).

Figure 2

Matches example. FB = forest burns; M = match.

The reader can verify that neither match has a strongly active path. However, intuitively both matches contributed to the fire, although neither alone was necessary. The causal structure allows us to reveal the hidden dependency.

Note, we can let M ₂ be 0 without affecting the result—the forest still burns down. Hitchcock calls the set of such “redundant” settings of background variables the redundancy range.

The redundancy range is the set of all possible ways of fixing the background variables so that they make no difference to E. Of course, the actual values of β are also in the redundancy range.

In Matches, by setting , we reveal a hidden dependency: now makes a difference. We define an active path (simpliciter) as a path that becomes strongly active when we set background variables to some values in their redundancy range (H2).

Both paths in Matches are active, so both matches are causes according to definition 1. But we cannot yet handle late cutting.

2.3. Resetting Ranges (∼H3)

Late cutting is a kind of preemptive overdetermination. It occurs when one of the potential causes preempts the other, precisely by completing first. For example, Bottle:

We want to say that Suzy's throw (ST) caused the bottle to break (BS) but that Billy's throw (BT) did not. First, note that the obvious model (fig. 3) does not work. This is the same model as Matches, so it gives the same answer: both throws caused the bottle to smash. But our intuitions strongly favor Suzy's throw over Billy's: the preemption matters.

Figure 3

Bottle example. BS = bottle shatters; BT = Billy throws; ST = Suzy throws.

Halpern and Pearl (Reference Halpern and Pearl2005a) argue that this model misrepresents the case since the model is symmetric and Bottle is not. They argue that a proper model should show the key asymmetry: when Suzy's throw hits (SH), it prevents Billy's from hitting (BH). Therefore, Halpern and Pearl suggest the model in figure 4.

Figure 4

Modified bottle example. BH = Billy hits; BS = bottle shatters; BT = Billy throws; SH = Suzy hits; ST = Suzy throws.

This seems a good model, and it works for Suzy, whose path is strongly active. Unfortunately, Billy's path is also active. Let : is in the redundancy range, and under , BT makes a difference.

Hitchcock's H2 solves that problem by requiring that not change anything on the path , whereas implies a change to BH. But Halpern and Pearl showed that this strategy was vulnerable to a voting machine example (Twardy and Korb Reference Twardy and Korb2011). Their solution is to allow resetting of variables along . We now present a somewhat simplified version of resetting variables.

Note that resetting is only done for ; otherwise, no cause could make a difference.

Condition A amounts to saying that there is a background state where c can make a difference to e. Condition B ensures that the background alone is insufficient to force : it means that is in the redundancy range for e. The “even if” clause about resetting variables in to their actual values is only necessary in cases of late cutting.

In the extended bottle-smashing model, Suzy's throw remains an actual cause of using a redundancy range to reveal the dependency. Choosing as our background, and setting , we see that ST makes a difference. Resetting has no effect because, actually, .

Billy's throw would similarly make a difference, save for the new “resetting” clause. Let the background be . Setting , Billy's throw makes a difference. However, we can reset BH to its original value when . Of course, then Billy's throw will no longer make a difference, and , in violation of clause B of definition H3AP. Therefore, this path is not an active path, and BT is not an actual cause. We turn now to probabilistic extensions.

3.1. Soft Interventions

The final step will be to introduce resetting ranges, dealing with late cutting. However, for assessing active paths probabilistically, we turn to “soft interventions.” A soft intervention (Korb et al. Reference Korb, Hope, Nicholson, Axnick, Zhang and Güsgen2004) fixes a variable to a distribution, instead of a single value. For example, fixing (i.e., a 70% chance of Suzy hitting) is a soft intervention. The proper probabilistic analog of “setting X back to the value it would have under c” is “setting X back to the distribution it would have under c.”

Soft interventions are easily dealt with formally. A soft intervention on X is implemented by adding a new parent , such that when I is true, takes the desired distribution, and when I is false, X has its previous distribution. How this might be done experimentally (outside of simulation) is less clear. Nevertheless, many actual interventions are soft rather than hard (Korb et al. Reference Korb, Hope, Nicholson, Axnick, Zhang and Güsgen2004). In any case, we are not here concerned with implementation but with using soft interventions as a calculating tool.

The soft interventions of interest here freeze X at its current distribution. Because the distribution is unchanged, such interventions have no downstream effect. However, all back paths are severed: changes in parent values (other than I) have no effect on X; changes in X can only then tell us about I and not its other parents or, through them, any cousin variables. This gives us exactly what we need: the ability to distinguish causes and effects by the asymmetries of intervention. In this article, we denote such distributional (soft) interventions with a vector symbol —or —denoting that we have simply replaced the distribution for C. (When we do not specify that new distribution, by default we shall mean the current distribution.)

3.2. Resetting Ranges and Probabilistic Active Paths

Let there be n variables along path , not counting C and E. There are possible combinations for clamping (or not). Let denote the set of possible “resetting ranges,” the power set listing all possible combinations of these variables. Then for , let denote the intervention resetting those variables in to their current distribution. We can now define the probabilistic active paths (PAP) criterion:

Condition 3 requires that condition 2 holds for all combinations of variables in ; that is, resetting them to their original distributions leaves in the redundancy range. As noted, we add our clamp to the interventions already done for . A brute force search over the power set of resetting ranges will naturally be expensive, but we are not concerned with that here. For the small models treated in this literature, the problem is not serious.

4.1. Probabilistic Bottle

For a more realistic version of Bottle, we allow that Suzy and Billy sometimes miss. In particular, given knowledge only of ST and BT, we use table 2. We use the extended model of Halpern and Pearl to capture the timing asymmetry as before, yielding figure 6.

Table 2Conditional Probability.

ST	BT
0	0	0
0	1	.8
1	0	.7
1	1	.94

Note BS = bottle shatters; BT = Billy throws; ST = Suzy throws.

Figure 6

Probabilistic version of bottle with results of intervening to fix the probability distributions: a, original state; b, clamping SH has no effect on BS; c, clamping BH affects BS. BH = Billy hits; BS = bottle shatters; BT = Billy throws; SH = Suzy hits; ST = Suzy throws.

Let , so the redundancy ranges are degenerate for both Suzy and Billy: we have to use the actual background or . The network is shown in three states in figure 6. The original distribution is in figure 6a. As before, and for both Suzy and Billy. Next, we see whether resetting the intermediate “hits” variables violates condition 3.

In figure 6b, we consider Suzy, resetting to . This leaves unchanged, so we conclude that Suzy Throws is an actual cause. In figure 6c, we consider Billy. Resetting to , changes, so , and BT is not actually a cause on PAP. This happens because the intervention on BH blocks the backward flow of probability otherwise relating BS and BH through their common cause SH. Before resetting, the distribution arose because Billy could hit only when Suzy did not. Suzy missed 30% of the time, and . Now, however, Billy never hits more than 24% of the time, so drops from 94% to 77%.

The original probabilistic Boulder can be resolved without resetting. However, if we add noise at S, so that ducking no longer guarantees survival, we find an influence for F, which we argue is appropriate. (For further analysis, see Twardy and Korb Reference Twardy and Korb2011, the online companion piece.)

4.2. The Trainee Assassin

The deterministic version is that a trainee assassin T shoots a victim V, but Supervisor will shoot if Trainee loses nerve. Being deterministic, Victim survives only when neither shot is fired. In the actual case, the trainee shoots, the supervisor does not, and the victim dies. As with Boulder, there is no marginal effect, but any path-based account works because the path is strongly active. However, Hiddleston (Reference Hiddleston2005) alleges that the analysis will not survive a probabilistic extension, which leads us to the stochastic assassin.

4.3. The Stochastic Assassin

In the probabilistic version, Trainee firing lowers the overall chance that Victim dies, by preventing the more accurate Supervisor from shooting. Once again, a path-based analysis of actual causation does fine here. Let and . For completeness, .

Since Supervisor does not shoot, Trainee is strongly active with . Similarly, Supervisor's (vs. ) has in the actual background where Trainee shoots, and so is a strongly active preventer (of death), having the same status as the falling boulder. That is, Supervisor is ineluctably causally active.

Hiddleston recommends adding an extra variable for Supervisor taking Aim (A). The supervisor is unlikely to fire without aiming and is always more likely to fire if she aims. Avoiding extremes, let Supervisor be 90% likely to fire if Trainee loses nerve and 10% likely even if Trainee also fires. Let those numbers average over Aiming and not-Aiming, with a 5050 prior over each. Furthermore, let aiming improve chances 10% over not aiming in both cases. The model and conditional probability table are shown in figure 7.

Figure 7

Stochastic assassin example, with supervisor's conditional probability table. A = aiming; S = supervisor; T = trainee; V = victim.

As before, let T and S be independent causes of V (in a noisy-or model). Consider whether is an actual cause of . In the actual background, , and . Aiming appears to be an actual cause, provided . Clamping S at the distribution induced by has no effect on . Therefore, is on an active path and is a cause.

Hiddleston thinks this is wrong because in actual fact Supervisor did not shoot, so Aiming made no difference. We disagree. As our analysis has made clear, when considering the probabilistic effect of C on E, one must unfix the path between C and E. If we fixed S, we would cut the active path and get Hiddleston's answer, but in a probabilistic context that procedure implies that only direct causes are actual causes. Hiddleston is correct that such probabilistic accounts cannot work, but that is just a misconstrual of probabilistic accounts. Hiddleston's superfluous antidote presents more of a problem.

4.4. Superfluous Antidote

Fred injects himself with a harmless antidote in the false belief that he has been (or is about to be) poisoned (P). The model is simple: , and . Fred survives when . In fact there is no poison , so the antidote is irrelevant. However, because , the redundancy range includes . Were there poison, the antidote would make a difference; hence, the path is active. As Hiddleston says, “That is the wrong answer. H2 [and H3] counts an ‘actual cause’ merely because it would have been a cause if the circumstances had been different” (Reference Hiddleston2005, 32).

This is a good counterexample because it targets redundancy ranges, an epicycle added to handle symmetric cases. H1, which lacks redundancy ranges, correctly says is not a cause. But it also says is not a cause, leaving uncaused, as it does with Matches. Several responses might mitigate the damage:

• • Pick one's poison, trading off cases like Matches for those like Superfluous Antidote. This is unsatisfactory since nothing is mitigated.

• • Make the problem probabilistic. For many reasonable values of ε, in the actual background , or if we try to set . That solves the problem. However, the point of the example involves extreme probabilities, which this response ignores.

• • Hiddleston recommends employing Cheng's account of causal power (Cheng Reference Cheng1997).

• • Argue that the model structure is wrong. Hitchcock (at PSA 2010) noted that antidote and poison have to mix in the body; if we represent that aggregating variable, the problem disappears. While solving the problem, we should like an independent account of when a model is adequate. Handfield et al. (Reference Handfield, Twardy, Korb and Oppy2008) and Korb and Nicholson (Reference Korb, Nicholson, Holmes and Jain2008) sketch an account using the idea of “possible processes.”

We think that some combination of the last two responses—a mixture of an analysis of causal strength with a metaphysical analysis of causal processes—is the most promising framework for completing the story of actual causation.