Question: The fire / burning house example confuses me because I do not see how it makes a case for using actual causation. Do you mean that F = 1 is not actual cause of B = 0 while F = 0 is?
Evidence: Actual Causes are those events which explain an output in the most general sense of explanation.
There is a lively literature on how to define actual causation using causal models, but for the reasons outlined above, this literature has yet to find its way into the XAI literature. I here aim to set this straight by explicitly connecting actual causation to the notions of explanation that we have come across. A concise counterfactual explanation is good when you can get it, but often you cannot get it, and thus we need a weaker notion of explanation that is more generally applicable. 7 (The causal counterpart of this message is what initiated the formal causation literature some fifty years ago.)
It is clear from the definitions that sufficient explanations are weaker than counterfactual explanations. But sufficient explanations ignore the counterfactual aspect entirely, which means they are of little value for action-guidance in the presence of an already existing observation. Therefore I define actual causes as parts of explanations that sit in between counterfactual and sufficient explanations: they are parts of good sufficient explanations such that there exist counterfactual values which would not have made the explanation better. This is weaker than demanding that the counterfactual values are part of a sufficient explanation of a different output, as we do for counterfactual explanations. Informally, X = x rather than X = x is an actual cause of Y = y if X = x is part of a good sufficient explanation for Y = y that could have not been made better by setting X to x .
First I make precise how changing the values of some variables can turn a sufficient explanation into a better one.
Definition 18 If ((X = x, W = w), N ) is a sufficient explanation of Y = y, we say that X = x can replace X = x if there exists a dominating explanation that includes (X = x , W = w).
The following result ensures that the focus on W (instead of its subsets) in Definition 18 is without loss of generality.
Proposition 19 If ((X = x, W = w), N ) is a sufficient explanation of Y = y and there exists a dominating explanation ((X = x , A = a), B) for some values x and a ⊆ w, then X = x can replace X = x.
Definition 20 X = x rather than X = x is an actual cause of Y = y in (M, u) if it is part of a good sufficient explanation of Y = y in which X = x cannot be replaced by X = x .
If ((X = x, W = w), N ) is the relevant good explanation, we say that X = x rather than X = x is an actual cause of Y = y relative to (W = w, N ).
6. Obviously this holds only if V \ (X ∪ {Y }) consists of at least two elements. 7. Obviously you can always find some counterfactual explanation of an output, for changing all of the variables will always allow you to get a different output. But an explanation that involves too many variables is of little use.
Except for a minor technical difference regarding the implementation of minimality, Definition 20 is equivalent to the definition of causation I developed in previous work. Although I have here arrived at the same notion, I did so along a very different path, for in the previous work I did not draw any connection to explanations nor to action-guidance, but instead argued for the definition by contrasting it to other proposals for defining causation (including Definition 13).
Contrary to counterfactual explanations, actual causes do not guide you towards actions that, under the same conditions, would ensure the output to be different. But they do guide you towards actions that would not ensure the actual output under the same conditions as the actual action. For example, imagine a (very fortunate) applicant for whom X 1 = 250, 000, X 3 = 50, 000, and thus X 2 = 125, 000. Obviously their application is successful, and their income by itself offers a good explanation of this fact: X 1 = 250, 000 is a good direct sufficient explanation of Y = 1. Also, their income does not counterfactually explain the output, for the application would have been approved regardless of income. Yet we do have that X 1 = 250, 000 rather than X 1 = 200, 000 is an actual cause of the output, because we cannot replace X 1 = 250, 000 by X 1 = 200, 000 in our sufficient explanation (i.e., X 1 = 200, 000 is not a sufficient explanation of Y = 1). This is helpful for example if the applicant is considering changing jobs and would like to know whether they can use up their savings and still get their application approved.
One might wonder whether we need a separate definition of actual causation, as opposed to simply considering all parts of good sufficient explanations to be causes. The distinction between these two options lies in the existence of alternative actions that would make the actual explanation worse, and it are precisely those actions which make actual causes good guides towards action. Consider for example a situation in which a short-circuit starts a fire (F = 1) in an office building. The flames set off the sprinklers (S = 1), and those put out the flames, preventing the building from burning down (B = 0) -where all variables are binary. Matching equations for this story are: B = F ∧ ¬S, S = F . In this scenario, the fire offers a good sufficient explanation of the building not burning down relative to the sprinklers functioning as they should. But obviously it would be unwise to conclude from this that we should start fires in order to prevent buildings from burning down! This point can be brought out by noting that the fire is not an actual cause of the building not burning down, because there not being a fire would have offered a better sufficient explanation of this outcome, as it does not rely on the sprinklers functioning properly. (Concretely: F = 0 can replace F = 1 in the good sufficient explanation (F = 1, S) of B = 0.) Since sprinklers can malfunction, or can be made to malfunction by a malicious actor, the best action is to not set fires in a building, in accordance with actual causation.
The following result specifies the claim that actual causes sit in between counterfactual and sufficient explanations: counterfactual explanations always contain actual causes (and obviously not vice versa).
Theorem 21 If X 1 = x 1 rather than X 1 = x 1 is a counterfactual explanation of Y = y in (M, u) (relative to some (W = w, N )) then for some X 2 ⊆ X 1 , X 2 = x 2 rather than X 2 = x 2 is an actual cause of Y = y in (M, u) (where x 2 and x 2 are the relevant restrictions to X 2 ).
An obvious strengthening of actual causation is to replace the existential quantifier over counterfactual values with a universal one, so that the actual values are the optimal values in terms of explanations.
Definition 22 X = x is an optimal cause of Y = y in (M, u) if X = x is part of a good sufficient explanation of Y = y in which X = x cannot be replaced. (I.e., there do not exist values x so that it could be replaced by those.)
Finally, replacing strong sufficiency with direct sufficiency offers a notion of direct causation .
We can now present a final result regarding the limitation of Independence: the different notions of causation also reduce to each other.
Theorem 25 If a causal model M satisfies Independence then the following statements are all equivalent:
• there exist values x so that X = x rather than X = x is an actual cause of Y = y in (M, u).
• X = x is part of a good sufficient explanation of Y = y in (M, u).
(Concretely: F = 0 can replace F = 1 in the good sufficient explanation (F = 1, S) of B = 0.)
Answer: Yes, that is indeed what I mean, and I should mention this explicitly. The structure of this example is a common one in the literature on actual causation.