Original: Causal Explanations and XAI
Revised: Causal Explanations and XAI

Original: Author names withheld Editor: Under Review for CLeaR
Revised: Editors: Bernhard Sch¨olkopf, Caroline Uhler and Kun Zhang

Original: Abstract
Revised: Abstract

Original: Although standard Machine Learning models are optimized for making predictions about observations, more and more they are used for making predictions about the results of actions.
Revised: Although standard Machine Learning models are optimized for making predictions about observations, more and more they are used for making predictions about the results of actions.

Original: An important goal of Explainable Artiﬁcial Intelligence (XAI) is to compensate for this mismatch by offering explanations about the predictions of an ML-model which ensure that they are reliably action-guiding .
Revised: An important goal of Explainable Artiﬁcial Intelligence (XAI) is to compensate for this mismatch by offering explanations about the predictions of an ML-model which ensure that they are reliably action-guiding .

Original: As action-guiding explanations are causal explanations, the literature on this topic is starting to embrace insights from the literature on causal models.
Revised: As action-guiding explanations are causal explanations, the literature on this topic is starting to embrace insights from the literature on causal models.

Original: Here we take a step further down this path and formally deﬁne causal notions of sufﬁcient explanations , counterfactual explanations, and actual causation .
Revised: Here I take a step further down this path by formally deﬁning the causal notions of sufﬁcient explanations and counterfactual explanations .

Original: We show how these notions relate to (and improve upon) existing work, and motivate their adequacy by illustrating how different explanations are action-guiding under different circumstances.
Revised: I show how these notions relate to (and improve upon) existing work, and motivate their adequacy by illustrating how different explanations are action-guiding under different circumstances.

Revised: Moreover, this work is the ﬁrst to offer a formal deﬁnition of actual causation that is founded entirely in action-guiding explanations.

Original: Although we primarily focus on XAI, our analysis of causal explanationapplies in general, andis the ﬁrst to analyze actual causation as a guide to action.
Revised: Although the deﬁnitions are motivated by a focus on XAI, the analysis of causal explanation and actual causation applies in general.

Original: This work is also relevant for fairness in AI, as we show how actual causation can be used to improve the idea of path-speciﬁc counterfactual fairness.
Revised: I also touch upon the signiﬁcance of this work for fairness in AI by showing how actual causation can be used to improve the idea of path-speciﬁc counterfactual fairness.

Original: 1. Introduction
Revised: 1. Introduction

Revised: Explainable Artiﬁcial Intelligence (XAI) is concerned with offering explanations of predictions made by Machine Learning models.

Revised: Such predictions can serve many different purposes, each of which calls for a different kind of explanation.

Original: Part of the XAI literature is(ﬁnally) beginning to embrace the insight that Pearl’s causal hierarchy is an invaluable tool to specify thedifferent requirements that  explanations need to fulﬁl for achieving different purposes (Pearl and Mackenzie, 2018).
Revised: Part of the XAI literature is beginning to embrace the insight that Pearl’s causal hierarchy is an invaluable tool to specify the requirements that different explanations need to fulﬁl for achieving those purposes (Pearl and Mackenzie, 2018).

Original: This hierarchy consists of a ladder that has three rungs: observations, interventions, and counterfactuals.
Revised: This hierarchy consists of a ladder that has three rungs: observations, interventions, and counterfactuals.

Original: Standard ML-models are optimized solely for making predictions about the ﬁrst rung of the ladder, observations, and their widespread application is due to the fact that they are very successful in doing so.
Revised: Standard ML-models are optimized solely for making predictions about the ﬁrst rung of the ladder, observations, and their widespread application is due to the fact that they are very successful in doing so.

Original: Yet more and more such models are also being used for action-guidance, and that involves predictions about the second rung of the ladder, interventions (Lipton, 2018; 
Revised: Yet there is also a growing interest in using ML-models for action-guidance, and that involves predictions about the second rung of the ladder, interventions (Lipton, 2018; Molnar et al., 2021;

Original: Molnar et al., 2021; Sch¨olkopf et al., 2021).
Revised: Sch¨olkopf et al., 2021).

Original: Moreover, if the action occurs against the background of an already existing observation, then such predictions take place on the third rung of the ladder, counterfactuals.
Revised: Moreover, if the action occurs against the background of an already existing observation, then such predictions take place on the third rung of the ladder, counterfactuals.

Original: Corresponding to these different purposes we have different notions of explanation that are appropriate for each.
Revised: Corresponding to these different purposes we have different notions of explanation that are appropriate for each.

Original: Simply put, purely associational measures – such as feature attribution (Ribeiro, 2016; Lundberg and Lee, 2017) – are appropriate when one requires explanations of the behavior of the ML-model itself : why did the model produce a particular output Y =
Revised: Simply put, purely associational measures – such as feature attribution (Ribeiro, 2016; Lundberg and Lee, 2017) – are appropriate when one requires explanations of the behavior of the ML-model itself: why did the model produce a particular output Y =

Original: y for a particular input
Revised: y for a particular input

Original: I = i ?
Revised: I = i ?

Original: Such  explanations serve to better understand the model, be it to gain trust in the model or to perform model audit (Lipton, 2018; K¨onig et al., 2021).
Revised: Such model explanations serve to better understand the model, be it to gain trust in the model or to perform model audit (Lipton, 2018; K¨onig et al., 2021).

Original: Actions, on the other hand, take place in the real world, and thus acting on one feature might affect the value of another.
Revised: Actions, on the other hand, take place in the real world, and thus acting on one feature might affect the value of another.

Original: Therefore if one desires explanations about the outputs produced as the result of performing an action, then one is looking for causal explanations .
Revised: Therefore if one desires explanations about the outputs produced as the result of performing an action, then one is looking for action-guiding explanations .

Original: To ﬁnd those one needs to go beyond the ML-model and consider the target system that it is a model of.
Revised: To ﬁnd those one needs to go beyond the ML-model and consider the target system that it is a model of.

Original: To do so requires knowledge of an additional model, namely a causal model of the target system.
Revised: Since action-guiding explanations are causal explanations, this requires knowledge of an additional model, namely a causal model of the target system.

Original: Fully speciﬁed causal models are notoriously hard to come by, but there are promising approaches that focus on combining partial causal models with ML-models (Mahajan et al., 2019; Karimi et al., 2020; Sch¨olkopf et al., 2021).
Revised: Fully speciﬁed causal models are notoriously hard to come by, but there are promising approaches that focus on combining partial causal models with ML-models (Mahajan et al., 2019; Karimi et al., 2020; Sch¨olkopf et al., 2021).

Original: In this paper we focus on a conceptual analysis of the relevant notions of causal explanations – as opposed to practical methods for obtaining such explanations – and therefore we assume perfect knowledge of a deterministic causal model of the target system.
Revised: This paper focuses on a conceptual analysis of the relevant notions of causal explanations – as opposed to practical methods for obtaining such explanations – and therefore we assume perfect knowledge of a deterministic causal model of the target system.

Original: The results of weakening this assumption are to be investigated in future work.
Revised: The results of weakening this assumption are to be investigated in future work.

Original: Explanations that are action-guiding can occur in two basic forms.
Revised: Action-guiding explanations can occur in two basic forms.

Original: If one is interested inﬁnding an action that changes an already observed output, then Counterfactual Explanations are called for. If one is interested in explaining under which conditions an action guarantees a particular output, then one needs Sufﬁcient Explanations . 
Revised: If one is interested in explaining under which conditions an action guarantees a particular output, then one needs Sufﬁcient Explanations . If one is interested in ﬁnding an action that changes an observed input to produce a change in an already observed output, then Counterfactual Explanations are called for.

Original: Beyond these basic forms, there exist many circumstances in which one is interested more generally in ﬁnding out which past actions actually explain an already observed output.
Revised: Beyond these basic forms, there exist many circumstances in which one is interested more generally in ﬁnding out which past actions actually explain an already observed output.

Original: Such explanations ask for the Actual Causes of an output, and it will turn out that they sit in between sufﬁcient and counterfactual explanations in roughly the following way: an actual cause is a part of a good sufﬁcient explanation for which there exist counterfactual values that would not have made the explanation better.
Revised: Such explanations ask for the Actual Causes of an output, and it will turn out that they sit in between sufﬁcient and counterfactual explanations in roughly the following way: an actual cause is a part of a good sufﬁcient explanation for which there exist counterfactual values that would not have made the explanation better.

Original: The next two sections provide the background context for our analysis, setting the stage for deﬁning sufﬁcient and counterfactual explanations in Sections 4 and 5.
Revised: The next two sections provide the background context for our analysis. Sections 4 and 5 introduce various deﬁnitions of sufﬁcient and counterfactual explanations by relating existing accounts of these notions to their causal counterparts.

Original: These are used for deﬁning actual causation in Section 6, where we also point out its relevance for deﬁning fairness.
Revised: Section 6 offers a deﬁnition of actual causation, and concludes by presenting a novel deﬁnition of fairness based on actual causation.

Original: 2. Structural Equations Modeling
Revised: 2. Structural Equations Modeling

Original: For the purposes of notational consistency across related work, this section reviews the deﬁnition of causal models along the lines of Halpern (2016) with little change.
Revised: For the purposes of notational consistency across related work, this section reviews the deﬁnition of causal models along the lines of Halpern (2016) with little change.

Original: Deﬁnition 1 A signature S is a tuple ( U , V , R ) , where U is a set of exogenous variables, V is a set of endogenous variables, and R a function that associates with every variable Y ∈
Revised: Deﬁnition 1 A signature S is a tuple ( U , V , R ) , where U is a set of exogenous variables, V is a set of endogenous variables, and R a function that associates with every variable Y ∈

Original: U ∪ V a nonempty set R ( Y ) of possible values for Y (i.e., the set of values over which Y ranges ).
Revised: U ∪ V a nonempty set R ( Y ) of possible values for Y (i.e., the set of values over which Y ranges ).

Original: If
Revised: If

Original: X =
Revised: X =

Original: ( X 1 , . . .
Revised: ( X 1 , . . .

Original: , X n ) , R ( X ) denotes the crossproduct R ( X 1 ) × · · · × R ( X n ) .
Revised: , X n ) , R ( X ) denotes the crossproduct R ( X 1 ) × · · · × R ( X n ) .

Original: Exogenous variables represent factors whose causal origins are outside the scope of the causal model, such as background conditions and noise.
Revised: Exogenous variables represent factors whose causal origins are outside the scope of the causal model, such as background conditions and noise.

Original: The values of the endogenous variables, on the other hand, are causally determined by other variables within the model.
Revised: The values of the endogenous variables, on the other hand, are causally determined by other variables within the model.

Original: Deﬁnition 

Revised: Deﬁnition

Original: A causal model M is a pair ( S , F ) , where S is a signature and F deﬁnes a function that associates with each endogenous variable X a structural equation F X giving the value of X in terms of the values of other endogenous and exogenous variables.
Revised: A causal model M is a pair ( S , F ) , where S is a signature and F deﬁnes a function that associates with each endogenous variable X a structural equation F X giving the value of X in terms of the values of other endogenous and exogenous variables.

Original: Formally, the equation F X maps
Revised: Formally, the equation F X maps

Original: R ( U ∪ V − { X } ) to R ( X ) , so F X determines the value of X , given the values of all the other variables in U ∪ V .
Revised: R ( U ∪ V − { X } ) to R ( X ) , so F X determines the value of X , given the values of all the other variables in U ∪ V .

Original: We call a setting u ∈ R ( U ) of values of exogenous variables a context .
Revised: We call a setting u ∈ R ( U ) of values of exogenous variables a context .

Original: The value of X may depend on the values of only a few other variables.
Revised: The value of X may depend on the values of only a few other variables.

Original: X depends on Y in context u if there is some setting of the endogenous variables other than X and Y such that if the exogenous variables have value u , then varying the value of Y in that context results in a variation in the value of X ; that is, there is a setting z of the endogenous variables other than X and Y and values y and y (cid:48) of Y such that F X ( y, z , u ) (cid:54) =
Revised: X depends on Y in context u if there is some setting of the endogenous variables other than X and Y such that if the exogenous variables have value u , then varying the value of Y in that context results in a variation in the value of X ; that is, there is a setting z of the endogenous variables other than X and Y and values y and y (cid:48) of Y such that F X ( y, z , u ) (cid:54) =

Original: F X ( y (cid:48) , z , u ) .
Revised: F X ( y (cid:48) , z , u ) .

Original: We then say that Y is a parent of X and X is a child of Y .
Revised: We then say that Y is a parent of X and X is a child of Y .

Original: In this paper we restrict attention to strongly recursive models, that is, models where there is a partial order (cid:22) on variables such that if X depends on Y , then Y ≺ X .
Revised: In this paper we restrict attention to strongly recursive models, that is, models where there is a partial order (cid:22) on variables such that if X depends on Y , then Y ≺ X .

Original: In a strongly recursive model, given a context u , the values of all the remaining variables are determined (we can just solve for the value of the variables in the order given by (cid:22) ).
Revised: In a strongly recursive model, given a context u , the values of all the remaining variables are determined (we can just solve for the value of the variables in the order given by (cid:22) ).

Original: An intervention has the form X ← x , where X is a set of endogenous variables.
Revised: An intervention has the form X ← x , where X is a set of endogenous variables.

Original: Intuitively, this means that the values of the variables in X are set to the values x .
Revised: Intuitively, this means that the values of the variables in X are set to the values x .

Original: The structural equations deﬁne what happens in the presence of interventions.
Revised: The structural equations deﬁne what happens in the presence of interventions.

Original: Setting the value of some variables X to x in a causal model M =
Revised: Setting the value of some variables X to x in a causal model M =

Original: ( S , F ) results in a new causal model, denoted M X ← x , which is identical to
Revised: ( S , F ) results in a new causal model, denoted M X ← x , which is identical to

Original: M , except that F is replaced by F X ← x : for each variable Y / ∈ X , F X ← x Y = F Y (i.e., the equation for Y is unchanged), while for each X (cid:48) in X , the equation F X (cid:48) for X (cid:48) is replaced by X (cid:48)
Revised: M , except that F is replaced by F X ← x : for each variable Y / ∈ X , F X ← x Y = F Y (i.e., the equation for Y is unchanged), while for each X (cid:48) in X , the equation F X (cid:48) for X (cid:48) is replaced by X (cid:48)

Original: =
Revised: =

Original: x (cid:48) (where x (cid:48) is the value in x corresponding to X (cid:48) ).
Revised: x (cid:48) (where x (cid:48) is the value in x corresponding to X (cid:48) ).

Original: Given a signature S
Revised: Given a signature S

Original: =
Revised: =

Original: ( U , V , R ) , an atomic formula is a formula of the form X = x , for X ∈ V and x ∈ R
Revised: ( U , V , R ) , an atomic formula is a formula of the form X = x , for X ∈ V and x ∈ R

Original: ( X ) .
Revised: ( X ) .

Original: A causal formula (over S ) is one of the form [ Y 1 ← y 1 , . .
Revised: A causal formula (over S ) is one of the form [ Y 1 ← y 1 , . .

Original: , Y k ← y k ] φ , where
Revised: , Y k ← y k ] φ , where

Original: • φ is a Boolean combination of atomic formulas, • Y 1 , . . .
Revised: • φ is a Boolean combination of atomic formulas, • Y 1 , . . .

Original: , Y k are distinct variables in V , and • y i ∈R
Revised: , Y k are distinct variables in V , and • y i ∈

Original:  ( Y i ) for each 1 ≤ i ≤ k .
Revised: R ( Y i ) for each 1 ≤ i ≤ k .

Original: Such a formula is abbreviated as [ Y ← y ] φ .
Revised: Such a formula is abbreviated as [ Y ← y ] φ .

Original: The special case where k = 0 is abbreviated as φ .
Revised: The special case where k = 0 is abbreviated as φ .

Original: Intuitively, [ Y 1 ← y 1 , . . .
Revised: Intuitively, [ Y 1 ← y 1 , . . .

Original: , Y k ← y k ] φ says that φ would hold if Y i were set to y i , for i = 1 , . . .
Revised: , Y k ← y k ] φ says that φ would hold if Y i were set to y i , for i = 1 , . . .

Original: , k .
Revised: , k .

Original: A causal formula ψ is true or false in a causal setting , which is a causal model given a context.
Revised: A causal formula ψ is true or false in a causal setting , which is a causal model given a context.

Original: As usual, we write ( M, u ) | = ψ if the causal formula ψ is true in the causal setting ( M, u ) . 
Revised: As usual, we write ( M, u ) | = ψ if the causal formula ψ is true in the causal setting ( M, u ) . (If ψ is true for all contexts u we write M | = ψ .)

Original: The | = relation is deﬁned inductively.
Revised: The | = relation is deﬁned inductively.

Original: ( M, u ) | =
Revised: ( M, u ) | =

Original: X = x if the variable X has value x in the unique (since we are dealing with recursive models) solution to the equations in M in context u (i.e., the unique vector of values that simultaneously satisﬁes all equations in M with the variables in U set to u ).
Revised: X = x if the variable X has value x in the unique (since we are dealing with recursive models) solution to the equations in M in context u (i.e., the unique vector of values that simultaneously satisﬁes all equations in M with the variables in U set to u ).

Original: The truth of conjunctions and negations is deﬁned in the standard way.
Revised: The truth of conjunctions and negations is deﬁned in the standard way.

Original: Finally, ( M, u ) | =
Revised: Finally, ( M, u ) | =

Original: [ Y ← y ] φ if ( M Y ← y , u ) | = φ .
Revised: [ Y ← y ] φ if ( M Y ← y , u ) | = φ .

Original: 3. Background
Revised: 3. Background

Original: Assume that we have an ML-model which implements the overall function h : R ( I ) → R ( Y ) , where I consists of all the input variables and Y is the single output variable.
Revised: Assume that we have an ML-model which implements the overall function h : R ( I ) → R ( Y ) , where I consists of all the input variables and Y is the single output variable.

Original: As mentioned, we assume knowledge of a causal model M of the target domain.
Revised: As mentioned, we assume knowledge of a causal model M of the target domain (i.e., that part of the world which h is a model of).

Original: A ﬁrst requirement is that such a causal model is consistent with the ML-model, meaning that they at least agree on how to classify all observations.
Revised: A ﬁrst requirement is that such a causal model is consistent with the ML-model, meaning that they at least agree on how to classify all observations.

Original: A second requirement that is helpful given the conceptual nature of our analysis is that the endogenous variables I sufﬁce to obtain deterministic causal knowledge of the target domain.
Revised: A second requirement that is helpful given the conceptual nature of our analysis is that the endogenous variables I sufﬁce to obtain deterministic causal knowledge of the target domain.

Original: Technically this means that there are only two kinds of endogenous variables: those which are determined directly by the unobserved exogenous variables, and those which are determined entirely by members of I .
Revised: Technically this means that there are only two kinds of endogenous variables: those which are determined directly by the unobserved exogenous variables, and those which are determined entirely by members of I .

Original: In future work both requirements can be loosened by adding probabilities to each.
Revised: In future work both requirements can be loosened by adding probabilities to each.

Original: For the ﬁrst, we can demand that both models are likely to agree on observations.
Revised: For the ﬁrst, we can demand that both models are likely to agree on observations.

Original: For the second, we can add new exogenous variables and a probability distribution over them, which give us probabilistic causal models.
Revised: For the second, we can add new exogenous variables and a probability distribution over them, which give us probabilistic causal models.

Original: These allow for probabilistic generalizations of all the deﬁnitions here presented.
Revised: These allow for probabilistic generalizations of all the deﬁnitions here presented.

Original: Deﬁnition 

Revised: Deﬁnition

Original: A causal model M =
Revised: A causal model M =

Original: (( V , U , R ) , F ) agrees with h : R ( I )
Revised: (( V , U , R ) , F ) agrees with h : R ( I )

Original: → R ( Y ) if V
Revised: → R ( Y ) if V

Original: =
Revised: =

Original: I ∪{ Y } , each V i ∈ V either has a unique single exogenous parent U
Revised: I ∪{ Y } , each V i ∈ V either has a unique single exogenous parent U

Original: i ∈
Revised: i ∈

Original: U or no exogenous parents, and for all i ∈ R ( I ) , y ∈ R ( Y ) :
Revised: U or no exogenous parents, and for all i ∈ R ( I ) , y ∈ R ( Y ) :

Original: M | =
Revised: M | =

Original: I =
Revised: I =

Original: i → Y =
Revised: i → Y =

Original: y iff h
Revised: y iff h

Original: ( i ) =
Revised: ( i ) =

Original: y .
Revised: y .

Original: As knowledge of the causal model is hard to come by many approaches to action-guiding explanations ignore it entirely and assume that the input variables are causally independent (Sharma et al., 2020; Ustun et al., 2019; Poyiadzi et al., 2020; Kommiya Mothilal et al., 2021).
Revised: As knowledge of the causal model is hard to come by many approaches to action-guiding explanations ignore it entirely and assume that the input variables are causally independent (Sharma et al., 2020; Ustun et al., 2019; Poyiadzi et al., 2020; Kommiya Mothilal et al., 2021).

Original: Deﬁnition 4 A causal model M that agrees with h satisﬁes Independence if for each I a , I b ∈
Revised: Deﬁnition 4 A causal model M that agrees with h satisﬁes Independence if for each I a , I b ∈

Original: I ,
Revised: I ,

Original: I a is not a parent of I b .
Revised: I a is not a parent of I b .

Original: Obviously Independence rarely holds, and thus any account of action-guiding explanation that depends on it is limited.
Revised: Obviously Independence rarely holds, and thus any account of action-guiding explanation that depends on it is limited.

Original: Although this limitation is recognized (Mahajan et al., 2019; Karimi et al., 2021b; Molnar et al., 2021), this paper offers several results that formally characterize how extreme this limitation is.
Revised: Although this limitation is recognized (Mahajan et al., 2019; Karimi et al., 2021b; Molnar et al., 2021), this paper offers several results that formally characterize how extreme this limitation is.

Original: Simply put, under Independence a variety of very different causal notions become indistinguishable from each other.
Revised: Simply put, under Independence a variety of very different causal notions become indistinguishable from each other.

Original: This holds for notions of sufﬁcient explanation, counterfactual explanations, and various notions of actual causation.
Revised: This holds for notions of sufﬁcient explanation, counterfactual explanations, and various notions of actual causation.

Original: Because of this limitation, the literature on  XAI has failed to take up the most relevant lessonfor XAI that the literature on causation has to offer, namely that to give a causal explanation of an outcome is to give actual causes of that outcome (Woodward, 2003; Halpern and Pearl, 2005).
Revised: Because of this limitation, work on action-guiding explanations in XAI has failed to take up the most relevant lesson that the literature on causation has to offer, namely that to give a causal explanation of an outcome is to give actual causes of that outcome (Woodward, 2003; Halpern and Pearl, 2005).

Original: An additional reason for this oversight is that the precise relation between explanation and prediction has been the subject of much debate in the history of philosophy of science.
Revised: An additional reason for this oversight is that the precise relation between explanation and prediction has been the subject of much debate in the history of philosophy of science.

Original: Prediction in its most natural application is a forward-looking notion, meaning one predicts an event before it takes place.
Revised: Prediction in its most natural application is a forward-looking notion, meaning one predicts an event before it takes place.

Original: Explanation on the other hand is a backward-looking notion, meaning that one explains an event after it has happened.
Revised: Explanation on the other hand is a backward-looking notion, meaning that one explains an event after it has happened.

Original: Yet as many papers on XAI clearly illustrate, explanations about past events are often required precisely to inform predictions about future events.
Revised: Yet as many papers on XAI clearly illustrate, explanations about past events are often required precisely to inform predictions about future events.

Original: Therefore a suitable notion of causal explanation, and thus also of actual causation, needs to specify how it relates to predictions.
Revised: Therefore a suitable notion of causal explanation, and thus also of actual causation, needs to specify how it relates to predictions.

Original: Given the tumultuous history that this relation has in the philosophy literature, it has been duly neglected in the philosophical work on causation, thereby obscuring the importance of causation for the practical goals that XAI is concerned with.
Revised: Given the tumultuous history that this relation has in the philosophy literature, it has been duly neglected in the philosophical work on causation, thereby obscuring the importance of causation for the practical goals that XAI is concerned with.

Revised: (Hitchcock (2017) forms a notable exception!)

Original: This paper corrects this by developing an account of causal explanation that shows both how it is connected to actual causation and how it can lead to action-guiding predictions.
Revised: This paper corrects this by developing an account of causal explanation that shows both how it is connected to actual causation and how it can lead to action-guiding predictions.

Original: Simply put, the goal of this paper is to upgrade the formalization of Woodward’s inﬂuential philosophical account of causal explanation described below with the most recent insights from the causation literature, whilst also keeping track of the action-guiding demands that are prevalent in
Revised: Simply put, the goal of this paper is to upgrade the formalization of Woodward’s inﬂuential philosophical account of causal explanation described below with the most recent insights from the causation literature, whilst also keeping track of the action-guiding demands that are prevalent in

Original: XAI (Woodward, 2003, p. 11).
Revised: XAI (Woodward, 2003, p. 11).

Original: Put differently, my idea is that one ought to be able to associate with any successful explanation a hypothetical or counterfactual experiment that shows us that and how manipulation of the factors mentioned in the explanation (the explanans , as philosophers call it) would be a way of manipulating or altering the phenomenon explained (the explanandum ).
Revised: Put differently, my idea is that one ought to be able to associate with any successful explanation a hypothetical or counterfactual experiment that shows us that and how manipulation of the factors mentioned in the explanation (the explanans , as philosophers call it) would be a way of manipulating or altering the phenomenon explained (the explanandum ).

Original: Put in still another way, an explanation ought to be such that it can be used to answer what I call a what-if-things-had-been-different question : the explanation must enable us to see what sort of difference it would have made for the explanandum if the factors cited in the explanans had been different in various possible ways.
Revised: Put in still another way, an explanation ought to be such that it can be used to answer what I call a what-if-things-had-been-different question : the explanation must enable us to see what sort of difference it would have made for the explanandum if the factors cited in the explanans had been different in various possible ways.

Original: A crucial novel element of my account is the addition that a successful explanation must also be explicit about those factors that may not be manipulated for the explanation to hold, i.e., it must state which variables are to be safeguarded from interventions.
Revised: A crucial novel element of my account is the addition that a successful explanation must also be explicit about those factors that may not be manipulated for the explanation to hold, i.e., it must state which variables are to be safeguarded from interventions.

Original: Importantly, this is distinct from stating which variables must be held ﬁxed at their actual values, for to hold variables ﬁxed in fact means to intervene on them.
Revised: Importantly, this is distinct from stating which variables must be held ﬁxed at their actual values, for to hold variables ﬁxed in fact means to intervene on them.

Original: The following example (modiﬁed from (Karimi et al., 2021b)) is helpful for illustrating the various purposes that explanations can serve.
Revised: The following example (modiﬁed from (Karimi et al., 2021b)) is helpful for illustrating the various purposes that explanations can serve.

Original: Example 1 Consider a system for loan applications that is captured by a causal model such that
Revised: Example 1 Consider a system for loan applications that is captured by a causal model such that

Original: Y =
Revised: Y =

Original: ( X 1 + 5 · X 2 − 225 , 000) > 0 , where Y is a binary variable representing whether the loan is granted, X 1 is the applicant’s income, and X 2 is the applicant’s savings.
Revised: ( X 1 + 5 · X 2 − 225 , 000) > 0 , where Y is a binary variable representing whether the loan is granted, X 1 is the applicant’s income, and X 2 is the applicant’s savings.

Original: Further, assume that the applicant’s savings are determined by their initial savings X 3 and their income in the following manner:
Revised: Further, assume that the applicant’s savings are determined by their initial deposit X 3 and their income in the following manner:

Original: X 2 = 3 / 10 · X 1 + X 3 .
Revised: X 2 = 3 / 10 · X 1 + X 3 .

Original: It is also the case that people with high savings take out a safety deposit box ( X 4 ) at the bank:
Revised: It is also the case that people with high savings take out a safety deposit box ( X 4 ) at the bank:

Original: X 4 =
Revised: X 4 =

Original: X 2 > 1 , 000 , 000 .
Revised: X 2 > 1 , 000 , 000 .

Original: Standard predictions in ML take place on the ﬁrst rung of Pearl’s causal ladder, namely that of observations.
Revised: Standard predictions in ML take place on the ﬁrst rung of Pearl’s causal ladder, namely that of observations.

Original: For example, an ML-model might pick up on the fact that all observed loan applicants who have a safety deposit box ( X 4 = 1 ) also get a loan ( Y = 1 ), and thus could predict that an applicant who has a safety deposit box will get a loan, i.e., it might learn a function so
Revised: For example, an ML-model might pick up on the fact that all observed loan applicants who have a safety deposit box ( X 4 = 1 ) also get a loan ( Y = 1 ), and thus could predict that an applicant who has a safety deposit box will get a loan, i.e., it might learn a function so

Original: that h ( x 1 , x 2 , x 3 , 1) = 1 for all values x 1 , x 2 , x 3 .
Revised: that h ( x 1 , x 2 , x 3 , 1) = 1 for all values x 1 , x 2 , x 3 .

Original: In terms of the causal model, it is indeed the case that
Revised: In terms of the causal model, it is indeed the case that

Original: M | =
Revised: M | =

Original: X 4 = 1 → Y = 1 .
Revised: X 4 = 1 → Y = 1 .

Original: Here X 4 = 1 is what Ribeiro et al.
Revised: Here X 4 = 1 is what Ribeiro et al.

Original: (2018) refer to as an anchor , which they interpret as a sufﬁcient explanation of the outcome.
Revised: (2018) refer to as an anchor , which they interpret as a sufﬁcient explanation of the outcome.

Original: Yet clearly such observational explanations are not a good guide towards action, for it would be absurd to recommend to someone to take out a safety deposit box so that their loan application is approved.
Revised: Yet clearly such observational explanations are not a good guide towards action, for it would be absurd to recommend to someone to take out a safety deposit box so that their loan application is approved.

Original: This point can be brought out by making use of the second rung of the ladder, namely that of interventions: an applicant who takes out a safety deposit box as the result of being advised to do so will not necessarily get a loan:
Revised: This point can be brought out by making use of the second rung of the ladder, namely that of interventions: an applicant who takes out a safety deposit box as the result of being advised to do so will not necessarily get a loan:

Original: M (cid:54)| =
Revised: M (cid:54)| =

Original: [ X 4 ← 1] Y = 1 .
Revised: [ X 4 ← 1] Y = 1 .

Original: In the same manner, interventions can be used to offer advice that is a good guide towards action, for example by telling an applicant that if they manage to obtain savings of 45 , 001 then they are guaranteed to get a loan:
Revised: In the same manner, interventions can be used to offer advice that is a good guide towards action, for example by telling an applicant that if they manage to obtain savings of 45 , 001 then they are guaranteed to get a loan:

Original: M | =
Revised: M | =

Original: [ X 2 ← 45 , 001] Y = 1 .
Revised: [ X 2 ← 45 , 001] Y = 1 .

Original: 4. Sufﬁcient Explanations
Revised: 4. Sufﬁcient Explanations

Original: Sufﬁcient Explanations generalize the previous point by offering settings X = x that causally sufﬁce for Y =
Revised: Sufﬁcient Explanations generalize the previous point by offering settings X = x that causally sufﬁce for Y =

Original: y .
Revised: y .

Original: Existing approaches come in two ﬂavors.
Revised: Existing approaches come in two ﬂavors.

Original: Some approaches (Galhotra et al., 2021; Watson et al., 2021) deﬁne sufﬁcient explanations using Pearl’s notion of “probability of sufﬁciency”, the deterministic version of which can be informally stated as: if we set the variables in X to x and do not intervene on any of the other variables, then Y takes on the values y .
Revised: Some approaches (Galhotra et al., 2021; Watson et al., 2021) deﬁne sufﬁcient explanations using Pearl’s notion of “probability of sufﬁciency”, the deterministic version of which can be informally stated as: if we set the variables in X to x and do not intervene on any of the other variables, then Y takes on the values y .

Original: A formal deﬁnition of this interpretation of sufﬁciencyhas recently been given by Beckers (2021), who calls it weak sufﬁciency .
Revised: Elsewhere I have called this interpretation of sufﬁciency weak sufﬁciency (Beckers, 2021).

Original: Deﬁnition 5 X
Revised: Deﬁnition 5 X

Original: = x is weakly sufﬁcient for Y =
Revised: = x is weakly sufﬁcient for Y =

Original: y in M if for all u ∈ R ( U ) we have that ( M, u ) | =
Revised: y in M if for all u ∈ R ( U ) we have that ( M, u ) | =

Original: [ X ← x ] Y = y .
Revised: [ X ← x ] Y = y .

Original: Other recent approaches take inspiration from logic and deﬁne sufﬁcient explanations of an output as its prime implicants , giving so-called PI-explanations (Shih et al., 2018; Darwiche and
Revised: Other recent approaches take inspiration from logic and deﬁne sufﬁcient explanations of an output as its prime implicants , giving so-called PI-explanations (Shih et al., 2018; Darwiche and

Original: Hirth, 2020; Koopman and Renooij, 2021).
Revised: Hirth, 2020; Koopman and Renooij, 2021).

Original: Informally: if we set the variables in X to x , then Y takes on the values y , regardless of the values of all other variables.
Revised: Informally: if we set the variables in X to x , then Y takes on the values y , regardless of the values of all other variables.

Original: In the context of causal models this has been coined  direct sufﬁciency by Beckers (2021).
Revised: In the context of causal models I have coined this direct sufﬁciency (Beckers, 2021).

Original: Deﬁnition 6 X = x is directly sufﬁcient for Y =
Revised: Deﬁnition 6 X = x is directly sufﬁcient for Y =

Original: y in M if for all c ∈ R (  V − ( X ∪ Y )) and all u ∈ R ( U ) we have that ( M, u ) | =
Revised: y in M if for all c ∈ R ( C ) , where C = V − ( X ∪ Y ) , and all u ∈ R ( U ) we have that ( M, u ) | =

Original: [ X ← x , C ← c ] Y = y .
Revised: [ X ← x , C ← c ] Y = y .

Original: Obviously the second form of sufﬁcient explanations is stronger.
Revised: Obviously the second form of sufﬁcient explanations is stronger.

Original: Coming back to our example,
Revised: Coming back to our example,

Original: X 2 = 45 , 001 is a sufﬁcient explanation for Y = 

Revised: X 2 = 45 , 001 is a sufﬁcient explanation for Y =

Original: according to both readings.
Revised: according to both readings.

Original: But ( X 1 = 50 , 000 , X 3 = 25 , 000) is a sufﬁcient explanation of Y = 1 only on the ﬁrst reading, for although it is weakly sufﬁcient for Y = 

Revised: But ( X 1 = 50 , 000 , X 3 = 25 , 000) is a sufﬁcient explanation of Y = 1 only on the ﬁrst reading, for although it is weakly sufﬁcient for Y =

Original: it is not directly sufﬁcient.
Revised: it is not directly sufﬁcient.

Original: Which one of these notions is correct here?
Revised: Which one of these notions is correct here?

Original: Imagine that a prospective applicant with these values is told that their income and initial savings sufﬁce for getting a loan.
Revised: Imagine that a prospective applicant with these values is told that their income and initial deposit sufﬁce for getting a loan.

Original: As a result the applicant concludes that there is no need to have such high savings and decides to spend 20 , 000 , so that their loan application is denied.
Revised: As a result the applicant concludes that there is no need to have such high savings and decides to spend 20 , 000 , so that their loan application is denied.

Original: The applicant would be quite right to be upset about this!
Revised: The applicant would be quite right to be upset about this!

Original: Such misunderstandings cannot occur when using direct sufﬁciency, as that gives us settings whose explanatory value is immune to the inﬂuence of interventions on other variables.
Revised: Such misunderstandings cannot occur when using direct sufﬁciency, as that gives us settings whose explanatory value is immune to the inﬂuence of interventions on other variables.

Original: Instead of concluding from this that one should always rely on direct sufﬁciency, I propose a generalization of sufﬁcient explanations that adds an element to inform us explicitly as to which variables are assumed to be safeguarded from interventions.
Revised: Instead of concluding from this that one should always rely on direct sufﬁciency, I propose a generalization of sufﬁcient explanations that adds an element to inform us explicitly as to which variables are assumed to be safeguarded from interventions.

Original: Concretely, in addition to specifying which variables need to be set to particular values, a sufﬁcient explanation should also specify a set of variables N that are not to be manipulated for the explanation to be action-guiding.
Revised: Concretely, in addition to specifying which variables need to be set to particular values, a sufﬁcient explanation should also specify a set of variables N that are not to be manipulated for the explanation to be action-guiding.

Original: Informally, if we set the variables in X to x and the variables in N are safeguarded from interventions, then Y takes on the value y , regardless of the values of all remaining variables.
Revised: Informally, if we set the variables in X to x and the variables in N are safeguarded from interventions, then Y takes on the value y , regardless of the values of all remaining variables.

Original: The relevant notion of causal sufﬁciency at workis called strong sufﬁciency by Beckers (2021), which can be formally deﬁned as follows:
Revised: I call the relevant notion of causal sufﬁciency at work strong sufﬁciency (Beckers, 2021), which can be formally deﬁned as follows:

Original: Deﬁnition 7 X = x is strongly sufﬁcient for Y =
Revised: Deﬁnition 7 X = x is strongly sufﬁcient for Y =

Original: y in M along N if Y ⊆ N and X = x is directly sufﬁcient for N = n for some values n ⊇ y .
Revised: y in M along N if Y ⊆ N and X = x is directly sufﬁcient for N = n for some values n ⊇ y .

Original: Note that in this deﬁnition N cannot just be any set, but rather we require that it is itself entirely determined by X = x .
Revised: Note that in this deﬁnition N cannot just be any set, but rather we require that it is itself entirely determined by X = x .

Original: This is because the variables in N can be thought of as a network that transmits the causal inﬂuence of X to Y , and the idea of safeguarding this network is that it can continue fulﬁlling this role even when intervening on X .
Revised: This is because the variables in N can be thought of as a network that transmits the causal inﬂuence of X to Y , and the idea of safeguarding this network is that it can continue fulﬁlling this role even when intervening on X .

Original: (We refer the reader to (Beckers, 2021) for an elaborate discussion of this deﬁnition as well as an equivalent alternative formulation.)
Revised: (I refer the reader to (Beckers, 2021) for an elaborate discussion of this deﬁnition as well as an equivalent alternative formulation.)

Original: The following straightforward result shows the relative strengths of the above three notions of sufﬁciency.
Revised: The following straightforward result shows the relative strengths of the above three notions of sufﬁciency.

Original: Proposition 8 If X = x is directly sufﬁcient for Y =
Revised: Proposition 8 If X = x is directly sufﬁcient for Y =

Original: y then X
Revised: y then X

Original: = x is strongly sufﬁcient for
Revised: = x is strongly sufﬁcient for

Original: Y =
Revised: Y =

Original: y along some N , and if X = x is strongly sufﬁcient for Y =
Revised: y along some N , and if X = x is strongly sufﬁcient for Y =

Original: y along some N then X
Revised: y along some N then X

Original: = x is weakly sufﬁcient for Y =
Revised: = x is weakly sufﬁcient for Y =

Original: y .
Revised: y .

Original: Using strong sufﬁciency we can deﬁne a sufﬁcient explanation so that it speciﬁes all the required elements for it to be action-guiding.
Revised: Using strong sufﬁciency we can deﬁne a sufﬁcient explanation so that it speciﬁes all the required elements for it to be action-guiding.

Original: Deﬁnition 9 A pair ( X = x , N ) is a sufﬁcient explanation of Y =
Revised: Deﬁnition 9 A pair ( X = x , N ) is a sufﬁcient explanation of Y =

Original: y if X = x is strongly sufﬁcient for Y =
Revised: y if X = x is strongly sufﬁcient for Y =

Original: y along N .
Revised: y along N .

Original: Any subset X (cid:48)
Revised: Any subset X (cid:48)

Original: =
Revised: =

Original: x (cid:48) appearing in X = x is called a part of the explanation .
Revised: x (cid:48) appearing in X = x is called a part of the explanation .

Original: If ( M, u ) | =
Revised: If ( M, u ) | =

Original: X = x , we say that ( X = x , N ) is an actual sufﬁcient explanation of Y =
Revised: X = x , we say that ( X = x , N ) is an actual sufﬁcient explanation of Y =

Original: y in ( M, u ) .
Revised: y in ( M, u ) .

Original: If N = Y we speak of a direct sufﬁcient explanation .
Revised: If N = Y we speak of a direct sufﬁcient explanation .

Original: Of course we do not want to add redundant parts to a sufﬁcient set, as a good explanation should be as concise as possible.
Revised: Of course we do not want to add redundant parts to a sufﬁcient set, as a good explanation should be as concise as possible.

Original: Therefore a good sufﬁcient explanation ought to be minimal with respect both to X and to N .
Revised: Therefore a good sufﬁcient explanation ought to be minimal with respect both to X and to N .

Original: Deﬁnition 10 A sufﬁcient explanation ( X 1 = x 1 , N 1 ) dominates an explanation ( X 2 = x 2 , N 2 ) if both are explanations of the same Y
Revised: Deﬁnition 10 A sufﬁcient explanation ( X 1 = x 1 , N 1 ) dominates an explanation ( X 2 = x 2 , N 2 ) if both are explanations of the same Y

Original: =
Revised: =

Original: y , X 1 ⊆
Revised: y , X 1 ⊆

Original: X 2 , and N 1 ⊆
Revised: X 2 , and N 1 ⊆

Original: N 2 .
Revised: N 2 .

Original: ( X 1 = x 1 , N 1 ) strictly dominates ( X 2 = x 2 , N 2 ) if ( X 1 = x 1 , N 1 ) dominates ( X 2 = x 2 , N 2 ) and not vice versa.
Revised: ( X 1 = x 1 , N 1 ) strictly dominates ( X 2 = x 2 , N 2 ) if ( X 1 = x 1 , N 1 ) dominates ( X 2 = x 2 , N 2 ) and not vice versa.

Original: This allows us to deﬁne what makes for a good sufﬁcient explanation of anactually observed output.
Revised: This allows us to deﬁne what makes for a good sufﬁcient explanation of an observed output.

Original: Deﬁnition 

Revised: Deﬁnition

Original: An actual sufﬁcient explanation ( X 1 = x 1 , N 1 ) of Y =
Revised: An actual sufﬁcient explanation ( X 1 = x 1 , N 1 ) of Y =

Original: y in ( M, u ) is good if it is not dominated by any other actual sufﬁcient explanation in ( M, u ) .
Revised: y in ( M, u ) is good if it is not dominated by any other actual sufﬁcient explanation in ( M, u ) .

Original: To conclude the analysis of sufﬁcient explanations, we offer a ﬁrst result that shows the extreme limitation Independence poses.
Revised: To conclude the analysis of sufﬁcient explanations, I offer a ﬁrst result that shows the extreme limitation Independence poses.

Original: We have seen three very different notions of sufﬁciency, and yet under Independence they all collapse into one.
Revised: We have seen three very different notions of sufﬁciency, and yet under Independence they all collapse into one.

Original: Theorem 12 If a causal model M that agrees with h satisﬁes Independence then the following statements are all equivalent:
Revised: Theorem 12 If a causal model M that agrees with h satisﬁes Independence then the following statements are all equivalent:

Original: • X = x is weakly sufﬁcient for Y =
Revised: • X = x is weakly sufﬁcient for Y =

Original: y in M .
Revised: y in M .

Original: • X
Revised: • X

Original: = x is strongly sufﬁcient for Y =
Revised: = x is strongly sufﬁcient for Y =

Original: y in M .
Revised: y in M .

Original: • X = x is directly sufﬁcient for Y =
Revised: X = x is directly sufﬁcient for Y =

Original: y in M .
Revised: y in M .

Original: (Proofs of all Theorems are to be found in the Supplementary Material.)
Revised: (Proofs of all Theorems are to be found in the Supplementary Material.)

Original: 5. Counterfactual Explanations
Revised: 5. Counterfactual Explanations

Original: Counterfactual Explanations inform us which variables would have had to be different (and different in what way) for the outcome to be different.
Revised: Counterfactual Explanations inform us which variables would have had to be different (and different in what way) for the outcome to be different.

Original: Although the XAI literature usually only considers one interpretation, it is important to distinguish between three different interpretations.
Revised: Although the XAI literature usually only considers one interpretation, it is important to distinguish between three different interpretations.

Original: The ﬁrst interpretation is the standard one as it was introduced into the XAI literature (Wachter et al., 2017), and is in fact appropriate only when explaining the behavior of the ML-model, i.e., when we ﬁnd ourselves on the ﬁrst rung of the causal ladder.
Revised: The ﬁrst interpretation is the standard one as it was introduced into the XAI literature (Wachter et al., 2017), and is in fact appropriate only when one is looking either for model explanations or for a very restricted type of action-guiding explanations.

Original: An observation I =
Revised: An observation I =

Original: i and output Y
Revised: i and output Y

Original: =
Revised: =

Original: y is explained by offering an observation that is identical to the ﬁrst one except for some variables X that take on values x (cid:48) rather than x .
Revised: y is explained by offering an observation that is identical to the ﬁrst one except for some variables X that take on values x (cid:48) rather than x .

Original: The appropriate causal reading of this interpretation goes: if we had set the variables in X to x (cid:48) rather than x and held ﬁxed all other variables at their actual values, then Y would have been y (cid:48) rather than y .
Revised: The appropriate causal reading of this interpretation goes: if we had set the variables in X to x (cid:48) rather than x and held ﬁxed all other variables at their actual values, then Y would have been y (cid:48) rather than y .

Revised: This is a deterministic version of what is usually called the direct effect (Pearl, 2001), which is action-guiding only if one can in fact hold ﬁxed all other variables.

Original: Here I will call this relation direct counterfactual dependence .
Revised: Here I will call this relation direct counterfactual dependence .

Original: The second interpretation is the usual one found in the causation literature, which takes into account that changing X might change the values of other variables as well: if we had set the variables in X to x (cid:48) rather than x and had not intervened on any of the other variables, then Y would have been y (cid:48) rather than y .
Revised: The second interpretation is the usual one found in the causation literature, which takes into account that changing X might change the values of other variables as well: if we had set the variables in X to x (cid:48) rather than x and had not intervened on any of the other variables, then Y would have been y (cid:48) rather than y .

Original: This relation is standardly referred to as counterfactual dependence (Pearl, 2009).
Revised: This relation is standardly referred to as counterfactual dependence (Pearl, 2009).

Original: The difference between these ﬁrst two interpretations is gaining traction in recent work on algorithmic recourse (which is the term used for action-guiding given an already existing observation) 
Revised: The difference between these ﬁrst two interpretations is gaining traction in recent work on algorithmic recourse (which is the term used for action-guiding given an already existing observation) (Karimi et al., 2021a).

Original: (Karimi et al., 2021a). In fact, a slightly simpler version of Example 1 was initially used to illustrate this point precisely (Karimi et al., 2021b).
Revised: In fact, a slightly simpler version of Example 1 was initially used to illustrate this point precisely (Karimi et al., 2021b).

Original: Imagine an unsuccessful applicant with an income of 75 , 000 and 25 , 000 in savings.
Revised: Imagine an unsuccessful applicant with an income of 75 , 000 and 25 , 000 in savings.

Original: On the ﬁrst interpretation, the “cheapest” counterfactual explanation would be that if their income had been 100 , 000 rather than 75 , 000 and everything else would have remain ﬁxed, then they would have gotten the loan.
Revised: On the ﬁrst interpretation, the “cheapest” counterfactual explanation would be that if their income had been 100 , 000 rather than 75 , 000 and everything else would have remain ﬁxed, then they would have gotten the loan.

Original: The second interpretation takes into account the important fact that increases in income also result in increases in savings, and thus it would offer a cheaper counterfactual explanation: if their income had been 85 , 000 rather than 75 , 000 then they would have gotten the loan.
Revised: The second interpretation takes into account the important fact that increases in income also result in increases in savings, and thus it would offer a cheaper counterfactual explanation: if their income had been 85 , 000 rather than 75 , 000 then they would have gotten the loan.

Original: Does this mean that the second interpretation is the best one here?
Revised: Does this mean that the second interpretation is the best one here?

Original: As before, an applicant hearing this explanation could be led to believe that they can ignore their savings, and thus spend some of it, which would of course invalidate the explanation given.
Revised: As before, an applicant hearing this explanation could be led to believe that they can ignore their savings, and thus spend some of it, which would of course invalidate the explanation given.

Original: Similar to our analysis of sufﬁcient explanations, rather than choosing either one I contend instead that we should extend the explanation with information that ensures these misunderstandings cannot arise.
Revised: Similar to the analysis of sufﬁcient explanations, rather than choosing either one I contend instead that we should extend the explanation with information that ensures these misunderstandings cannot arise.

Original: This happens in two steps.
Revised: This happens in two steps.

Original: As a ﬁrst step I introduce a third interpretation of counterfactual explanations, which is the explanatory counterpart to the most recent variant of Halpern & Pearl’s inﬂuential deﬁnition of actual causation (the so-called modiﬁed deﬁnition)
Revised: As a ﬁrst step I introduce a third interpretation of counterfactual explanations, which is the explanatory counterpart to the most recent variant of Halpern & Pearl’s inﬂuential deﬁnition of actual causation (the so-called modiﬁed deﬁnition)

Original: (Halpern, 2016).
Revised: (Halpern, 2016).

Original: It generalizes the former two interpretations by including a “witness” to the explanation, meaning a set of variables W so that if we had set the variables in X to x (cid:48) rather than x , held ﬁxed the variables in W at their actual values, and had not intervened on any other variables, then Y would have been y (cid:48) rather than y .
Revised: It generalizes the former two interpretations by including a “witness” to the explanation, meaning a set of variables W so that if we had set the variables in X to x (cid:48) rather than x , held ﬁxed the variables in W at their actual values, and had not intervened on any other variables, then Y would have been y (cid:48) rather than y .

Original: The ﬁrst interpretation above implicitly assumes that the witness contains all other variables, whereas the second interpretation assumes that the witness is empty.
Revised: The ﬁrst interpretation above implicitly assumes that the witness contains all other variables, whereas the second interpretation assumes that the witness is empty.

Original: By spelling out the witness explicitly, the third interpretation includes the important information that the explanation remains valid only when holding ﬁxed certain variables at their actual values.
Revised: By spelling out the witness explicitly, the third interpretation includes the important information that the explanation remains valid only when holding ﬁxed certain variables at their actual values.

Original: The ﬁrst counterfactual explanation above, for example, depends for its validity on holding ﬁxed X 2 (and is indifferent to the values of X 3 and X 4 ).
Revised: The ﬁrst counterfactual explanation above, for example, depends for its validity on holding ﬁxed X 2 (and is indifferent to the values of X 3 and X 4 ).

Original: Formally the causal notion that this third interpretation relies on is the latest deﬁnition of actual causation by Halpern & Pearl, which is best understood as a generalization of counterfactual dependence in the following way.
Revised: Formally the causal notion that this third interpretation relies on is the latest deﬁnition of actual causation by Halpern & Pearl, which is best understood as a generalization of counterfactual dependence in the following way.

Original: (Throughout the rest of this paper, values x (cid:48) are assumed to differ from x for each variable in X .)
Revised: (Throughout the rest of this paper, values x (cid:48) are assumed to differ from x for each variable in X .)

Original: Deﬁnition 13 Y =
Revised: Deﬁnition 13 Y =

Original: y counterfactually depends on X = x rather than X
Revised: y counterfactually depends on X = x rather than X

Original: = x (cid:48) relative to W = w in ( M, u ) if X is a minimal set such that ( M, u )
Revised: = x (cid:48) relative to W = w in ( M, u ) if X is a minimal set such that ( M, u )

Original: | =
Revised: | =

Original: X = x ∧ W = w ∧ Y =
Revised: X = x ∧ W = w ∧ Y =

Original: y and ( M, u ) | =
Revised: y and ( M, u ) | =

Original: [ X ← x (cid:48) , W ← w ] Y (cid:54) =
Revised: [ X ← x (cid:48) , W ← w ] Y (cid:54) =

Original: y .
Revised: y .

Original: If W = V \ ( X ∪ { Y } ) , we speak of direct counterfactual dependence .
Revised: If W = V \ ( X ∪ { Y } ) , we speak of direct counterfactual dependence .

Original: If W = ∅ , we speak of standard counterfactual dependence .
Revised: If W = ∅ , we speak of standard counterfactual dependence .

Original: (Note that in both cases we do not need to specify W .)
Revised: (Note that in both cases we do not need to specify W .)

Original: The second step is to apply the insight we gained when analysing sufﬁcient explanations, namely that an explanation should also mention the variables that are assumed to be safeguarded from interventions.
Revised: The second step is to apply the insight we gained when analysing sufﬁcient explanations, namely that an explanation should also mention the variables that are assumed to be safeguarded from interventions.

Original: This is precisely the information we need to avoid the above misunderstanding: it should be spelled out that the explanation assumes the applicant’s savings will continue to be determined as they were.
Revised: This is precisely the information we need to avoid the above misunderstanding: it should be spelled out that the explanation assumes the applicant’s savings will continue to be determined as they were.

Original: Therefore we deﬁne counterfactual explanations as counterfactual analogs of sufﬁcient explanations.
Revised: Therefore I deﬁne counterfactual explanations as counterfactual analogs of sufﬁcient explanations.

Original: Deﬁnition 14 3 Given a causal setting ( M, u ) , if (( X = x , W = w ) , N ) is an actual sufﬁcient explanation of Y =
Revised: Deﬁnition 14 4 Given a causal setting ( M, u ) , if (( X = x , W = w ) , N ) is an actual sufﬁcient explanation of Y =

Original: y and (( X = x (cid:48) , W = w ) , N ) is a sufﬁcient explanation of Y =
Revised: y and (( X = x (cid:48) , W = w ) , N ) is a sufﬁcient explanation of Y =

Original: y (cid:48) with y (cid:48) (cid:54) =
Revised: y (cid:48) with y (cid:48) (cid:54) =

Original: y then we say that X
Revised: y then we say that X

Original: = x rather than X = x (cid:48) is a counterfactual explanation of Y =
Revised: = x rather than X = x (cid:48) is a counterfactual explanation of Y =

Original: y relative to ( W = w , N ) .
Revised: y relative to ( W = w , N ) .

Original: We also write this as ( X = ( x , x (cid:48) ) , W = w , N ) .
Revised: We also write this as ( X = ( x , x (cid:48) ) , W = w , N ) .

Original: Just as with sufﬁcient explanations, we want an explanation to be as concise as possible.
Revised: Just as with sufﬁcient explanations, we want an explanation to be as concise as possible.

Original: Deﬁnition 

Revised: Deﬁnition

Original: Given a causal setting ( M, u ) , we say that a counterfactual explanation ( X 1 = ( x 1 , x 1 (cid:48) ) , W 1 = w 1 , N 1 ) dominates an explanation ( X 2 = ( x 2 , x 2 (cid:48) ) , W 2 = w 2 , N 2 ) if both are explanations of the same Y =
Revised: Given a causal setting ( M, u ) , we say that a counterfactual explanation ( X 1 = ( x 1 , x 1 (cid:48) ) , W 1 = w 1 , N 1 ) dominates an explanation ( X 2 = ( x 2 , x 2 (cid:48) ) , W 2 = w 2 , N 2 ) if both are explanations of the same Y =

Original: y , X 1 ⊆
Revised: y , X 1 ⊆

Original: X 2 , W 1 ⊆ W 2 , and N 1 ⊆
Revised: X 2 , W 1 ⊆ W 2 , and N 1 ⊆

Original: N 2 .
Revised: N 2 .

Original: As before, strict domination means that a dominating explanation is not dominated by the explanation being dominated.
Revised: As before, strict domination means that there is domination only in one direction.

Original: A counterfactual explanation is good if it is not dominated by any other counterfactual explanation.
Revised: A counterfactual explanation is good if it is not dominated by any other counterfactual explanation.

Original: The following result shows that the previous two steps are indeed steps in the same direction:
Revised: The following result shows that the previous two steps are indeed steps in the same direction:

Original: Theorem 16 Given a causal setting ( M, u ) , the following two statements are equivalent:
Revised: Theorem 16 Given a causal setting ( M, u ) , the following two statements are equivalent:

Original: • there exist W 1 and w 1 ∈ R
Revised: • there exist W 1 and w 1 ∈ R

Original: ( W 1 )
Revised: ( W 1 )

Original: so that Y =
Revised: so that Y =

Original: y counterfactually depends on X = x rather than X
Revised: y counterfactually depends on X = x rather than X

Original: = x (cid:48) relative to W 1 = w 1 in ( M, u ) .
Revised: = x (cid:48) relative to W 1 = w 1 in ( M, u ) .

Original: • there exist W 2 , w 2 ∈ R ( W 2 ) , and N so that ( X = ( x , x (cid:48) ) , W 2 = w 2) , N ) is a good counterfactual explanation of Y =
Revised: • there exist W 2 , w 2 ∈ R ( W 2 ) , and N so that ( X = ( x , x (cid:48) ) , W 2 = w 2 , N ) is a good counterfactual explanation of Y =

Original: y .
Revised: y .

Original: Coming back to our example, we can offer the cheaper of the two explanations and yet avoid any misunderstanding by saying that X 1 = 75 , 000 rather than X 1 = 85 , 000 is a good counterfactual explanation of Y = 

Revised: Coming back to our example, we can offer the cheaper of the two explanations and yet avoid any misunderstanding by saying that X 1 = 75 , 000 rather than X 1 = 85 , 000 is a good counterfactual explanation of Y =

Original: relative to ( { X 3 = 2 , 500 } , { X 2 } ) .
Revised: relative to ( { X 3 = 2 , 500 } , { X 2 } ) .

Original: 6. Actual Causation and Explanation
Revised: 6. Actual Causation and Explanation

Original: Actual Causes are those events which explain an output in the most general sense of explanation.
Revised: Actual Causes are those events which explain an output in the most general sense of explanation.

Original: 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.
Revised: 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.

Original: I here aim to set this straight by explicitly connecting actual causation to the notions of explanation that we have come across.
Revised: I here aim to set this straight by explicitly connecting actual causation to the notions of explanation that we have come across.

Original: 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.
Revised: 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.

Original: 4 (The causal counterpart of this message is what initiated the formal causation literature some ﬁfty years ago (Lewis, 1973).)
Revised: 5 (The causal counterpart of this message is what initiated the formal causation literature some ﬁfty years ago (Lewis, 1973).)

Original: It is clear from our deﬁnitions that sufﬁcient explanations are weaker than counterfactual explanations.
Revised: It is clear from the deﬁnitions that sufﬁcient explanations are weaker than counterfactual explanations.

Original: 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.
Revised: 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.

Original: Therefore we 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.
Revised: 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.

Original: 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.
Revised: 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.

Original: Informally, X = x rather than X = x (cid:48) is an actual cause of Y =
Revised: Informally, X = x rather than X = x (cid:48) is an actual cause of Y =

Original: y if X = x is part of a good sufﬁcient explanation for Y =
Revised: y if X = x is part of a good sufﬁcient explanation for Y =

Original: y that could have not been made better by setting X to x (cid:48) .
Revised: y that could have not been made better by setting X to x (cid:48) .

Original: First we make precise how changing the values of some variables can turn a sufﬁcient explanation into a better one.
Revised: First I make precise how changing the values of some variables can turn a sufﬁcient explanation into a better one.

Original: Deﬁnition 

Revised: Deﬁnition

Original: If (( X = x , W = w ) , N ) is a sufﬁcient explanation of Y =
Revised: If (( X = x , W = w ) , N ) is a sufﬁcient explanation of Y =

Original: y , we say that X
Revised: y , we say that X

Original: = x (cid:48) can replace X = x
Revised: = x (cid:48) can replace X = x

Original: if there exists a dominating explanation that includes ( X = x (cid:48) , W = w ) .
Revised: if there exists a dominating explanation that includes ( X = x (cid:48) , W = w ) .

Original: The following result ensures that the focus on W (instead of its subsets) in Deﬁnition 17 is without loss of generality.
Revised: The following result ensures that the focus on W (instead of its subsets) in Deﬁnition 17 is without loss of generality.

Original: Proposition 

Revised: Proposition

Original: If (( X = x , W = w ) , N ) is a sufﬁcient explanation of Y =
Revised: If (( X = x , W = w ) , N ) is a sufﬁcient explanation of Y =

Original: y
Revised: y

Original: and there exists a dominating explanation (( X = x (cid:48) , A = a ) , B ) for some values x (cid:48) and a ⊆ w , then X
Revised: and there exists a dominating explanation (( X = x (cid:48) , A = a ) , B ) for some values x (cid:48) and a ⊆ w , then X

Original: = x (cid:48) can replace X = x .
Revised: = x (cid:48) can replace X = x .

Original: This allows us to formally deﬁne actual causation.
Revised: This allows us to formally deﬁne actual causation.

Original: Deﬁnition 19 X
Revised: Deﬁnition 19 X

Original: = x rather than X = x (cid:48) is an actual cause of Y =
Revised: = x rather than X = x (cid:48) is an actual cause of Y =

Original: y in ( M, u ) if it is part of a good sufﬁcient explanation of Y =
Revised: y in ( M, u ) if it is part of a good sufﬁcient explanation of Y =

Original: y in which X = x cannot be replaced by X = x (cid:48) .
Revised: y in which X = x cannot be replaced by X = x (cid:48) .

Original: If (( X = x , W = w ) , N ) is the relevant good explanation, we say that X = x rather than
Revised: If (( X = x , W = w ) , N ) is the relevant good explanation, we say that X = x rather than

Original: X = x (cid:48) is an actual cause of Y =
Revised: X = x (cid:48) is an actual cause of Y =

Original: y relative to ( W = w , N ) .
Revised: y relative to ( W = w , N ) .

Original: Except for a minor technical difference regarding the implementation of minimality, Deﬁnition 19 is equivalent to the deﬁnition of causation from (Beckers, 2021).
Revised: Except for a minor technical difference regarding the implementation of minimality, Deﬁnition 19 is equivalent to the deﬁnition of causation I developed in previous work (Beckers, 2021).

Original: Although we have  arrived at the same notion, we did so along a very different path, as Beckers does not draw any connection to explanations or to action-guidance, but instead argues for the deﬁnition by contrasting it to other proposals for deﬁning causation (including Deﬁnition 13).
Revised: 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).

Original: Contrary to counterfactual explanations, actual causes do not guide you towards actions that, under the same conditions, would ensure the output to be different.
Revised: Contrary to counterfactual explanations, actual causes do not guide you towards actions that, under the same conditions, would ensure the output to be different.

Original: But they do guide you towards actions that would not ensure the actual output under the same conditions as the actual action.
Revised: But they do guide you towards actions that would not ensure the actual output under the same conditions as the actual action.

Original: For example, imagine a (very fortunate) applicant for whom X 1 = 250 , 000 , X 3 = 50 , 000 , and thus
Revised: For example, imagine a (very fortunate) applicant for whom X 1 = 250 , 000 , X 3 = 50 , 000 , and thus

Original: X 2 = 125 , 000 .
Revised: X 2 = 125 , 000 .

Original: Obviously their application is successful, and their income by itself offers a good explanation of this fact:
Revised: Obviously their application is successful, and their income by itself offers a good explanation of this fact:

Original: X 1 = 250 , 000 is a good direct sufﬁcient explanation of Y = 1 .
Revised: X 1 = 250 , 000 is a good direct sufﬁcient explanation of Y = 1 .

Original: Also, their income does not counterfactually explain the output, for the application would have been approved regardless of income.
Revised: Also, their income does not counterfactually explain the output, for the application would have been approved regardless of income.

Original: 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 ).
Revised: 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 ).

Original: 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.
Revised: 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.

Original: 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.
Revised: 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.

Original: 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.
Revised: 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.

Original: Consider for example a situation in which a short-circuit starts a ﬁre ( F = 1 ) in an ofﬁce building.
Revised: Consider for example a situation in which a short-circuit starts a ﬁre ( F = 1 ) in an ofﬁce building.

Original: 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.
Revised: 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.

Original: Matching equations for this story are: B = F ∧ ¬ S , S = F .
Revised: Matching equations for this story are: B = F ∧ ¬ S , S = F .

Original: In this scenario, the ﬁre offers a good sufﬁcient explanation of the building not burning down relative to the sprinklers functioning as they should.
Revised: In this scenario, the ﬁre offers a good sufﬁcient explanation of the building not burning down relative to the sprinklers functioning as they should.

Original: But obviously it would be unwise to conclude from this that we should start ﬁres in order to prevent buildings from burning down!
Revised: But obviously it would be unwise to conclude from this that we should start ﬁres in order to prevent buildings from burning down!

Original: This point can be brought out by noting that the ﬁre is not a 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.
Revised: 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.

Revised: (Concretely: F =can replace F = 1 in the good sufﬁcient explanation ( F = 1 , S ) of B = 0 .)

Original: 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.
Revised: 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.

Original: 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).
Revised: 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).

Original: Theorem 

Revised: Theorem

Original: If X 1 = x 1 rather than X 1 = x 1 (cid:48) is a counterfactual explanation of Y =
Revised: If X 1 = x 1 rather than X 1 = x 1 (cid:48) is a counterfactual explanation of Y =

Original: y in ( M, u ) (relative to some ( W = w , N ) )
Revised: y in ( M, u ) (relative to some ( W = w , N ) )

Original: then for some X 2 ⊆ X 1 , X 2 = x 2 rather than X 2 = x 2 (cid:48) is an actual cause of Y =
Revised: then for some X 2 ⊆ X 1 , X 2 = x 2 rather than X 2 = x 2 (cid:48) is an actual cause of Y =

Original: y in ( M, u ) (where x 2 and x 2 (cid:48) are the relevant restrictions to X 2 ).
Revised: y in ( M, u ) (where x 2 and x 2 (cid:48) are the relevant restrictions to X 2 ).

Original: Finally, replacing strong sufﬁciency with direct sufﬁciency offers a notion of direct causation .
Revised: Finally, replacing strong sufﬁciency with direct sufﬁciency offers a notion of direct causation .

Original: Deﬁnition 21 X
Revised: Deﬁnition 21 X

Original: = x is a direct cause of Y =
Revised: = x is a direct cause of Y =

Original: y in ( M, u ) if it is part of an actual direct good sufﬁcient explanation of Y =
Revised: y in ( M, u ) if it is part of an actual direct good sufﬁcient explanation of Y =

Original: y in ( M, u ) .
Revised: y in ( M, u ) .

Original: Proposition 22 If X = x is a direct cause of Y =
Revised: Proposition 22 If X = x is a direct cause of Y =

Original: y in ( M, u ) then there exist values x (cid:48) such that
Revised: y in ( M, u ) then there exist values x (cid:48) such that

Original: X = x rather than X = x (cid:48) is an actual cause of Y =
Revised: X = x rather than X = x (cid:48) is an actual cause of Y =

Original: y .
Revised: y .

Original: We can now present two more results regarding the limitation of Independence .
Revised: We can now present two more results regarding the limitation of Independence .

Original: Similar to the ﬁrst result, many of the distinctions we have just made disappear.
Revised: Similar to the ﬁrst result, many of the distinctions we have just made disappear.

Original: Theorem 

Revised: Theorem

Original: If a causal model M satisﬁes Independence then for any W 1 , W 2 , N 1 , N 2 , it holds that there exist values w 1 ∈ R
Revised: If a causal model M satisﬁes Independence then for any W 1 , W 2 , N 1 , N 2 , it holds that there exist values w 1 ∈ R

Original: ( W 1 )
Revised: ( W 1 )

Original: so that ( X = ( x , x (cid:48) ) , W 1 = w 1 , N 1 ) is a good counterfactual explanation of Y =
Revised: so that ( X = ( x , x (cid:48) ) , W 1 = w 1 , N 1 ) is a good counterfactual explanation of Y =

Original: y iff there exist values w 2 ∈ R
Revised: y iff there exist values w 2 ∈ R

Original: ( W 2 )
Revised: ( W 2 )

Original: so that ( X = ( x , x (cid:48) ) , W 2 = w 2 , N 2 ) is a good counterfactual explanation of Y =
Revised: so that ( X = ( x , x (cid:48) ) , W 2 = w 2 , N 2 ) is a good counterfactual explanation of Y =

Original: y .
Revised: y .

Original: The next result shows that the different notions of causationwe considered also reduce to each other.
Revised: The next result shows that the different notions of causation also reduce to each other.

Original: Theorem 24 If a causal model M satisﬁes Independence then the following statements are all equivalent:
Revised: Theorem 24 If a causal model M satisﬁes Independence then the following statements are all equivalent:

Original: • X = x is a direct cause of Y =
Revised: • X = x is a direct cause of Y =

Original: y in ( M, u ) .
Revised: y in ( M, u ) .

Original: • there exist values x (cid:48)
Revised: • there exist values x (cid:48)

Original: so that X = x rather than X = x (cid:48) is an actual cause of Y =
Revised: so that X = x rather than X = x (cid:48) is an actual cause of Y =

Original: y in ( M, u ) .
Revised: y in ( M, u ) .

Original: • X
Revised: • X

Original: = x is part of a good sufﬁcient explanation of Y =
Revised: = x is part of a good sufﬁcient explanation of Y =

Original: y in ( M, u ) .
Revised: y in ( M, u ) .

Original: 6.1. Actual Causation and Fairness
Revised: 6.1. Actual Causation and Fairness

Original: So far this paper has focused on explanations as they relate to forward-looking action-guiding, but explanations (and actual causes in particular) are also of fundamental importance in backwardlooking contexts, where the primary aim is to evaluate what has already happened.
Revised: So far this paper has focused on explanations as they relate to forward-looking action-guiding, but explanations (and actual causes in particular) are also of fundamental importance in backwardlooking contexts, where the primary aim is to evaluate what has already happened.

Original: One such area that is of particular concern to XAI is that of fairness, which aims to evaluate whether a protected variable “contributed” in some way or other to produce an output.
Revised: One such area that is of particular concern to XAI is that of fairness, which aims to evaluate whether a protected variable “contributed” in some way or other to produce an output.

Original: Here as well do we ﬁnd a growing inﬂuence of causal models, most prominently in the work on counterfactual fairness, which in its basic form can be characterized as the condition that an output should not standardly counterfactually depend – see Def. 13 – on a protected variable (Kusner et al., 2017; Loftus et al., 2018).
Revised: Here as well do we ﬁnd a growing inﬂuence of causal models, most prominently in the work on counterfactual fairness, which in its basic form can be characterized as the condition that an output should not standardly counterfactually depend – see Def. 13 – on a protected variable (Kusner et al., 2017; Loftus et al., 2018).

Original: As we have seen, actual causes offer better explanations than counterfactual explanations, and thus I propose that fairness should be evaluated using actual causation instead.
Revised: As we have seen, actual causes offer better explanations than counterfactual explanations, and thus I propose that fairness should be evaluated using actual causation instead.

Original: Interestingly though, there already is a growing consensus that counterfactual dependence is too strong a condition and should be replaced with counterfactual dependence along unfair paths (again relying for its foundations on Pearl’s work (Pearl, 2001))
Revised: Interestingly though, there already is a growing consensus that counterfactual dependence is too strong a condition and should be replaced with counterfactual dependence along unfair paths (again relying for its foundations on Pearl’s work (Pearl, 2001))

Original: (Loftus et al., 2018; Nabi and Shpitser, 2018; Chiappa, 2019; Wu et al., 2019).
Revised: (Loftus et al., 2018; Nabi and Shpitser, 2018; Chiappa, 2019; Wu et al., 2019).

Original: Our deﬁnition of actual causation naturally accommodates this replacement as well, for actual causation occurs along a network N .
Revised: My deﬁnition of actual causation naturally accommodates this replacement as well, for actual causation occurs along a network N .

Original: Therefore we can deﬁne fairness by demanding that protected variables do not cause the outcome along an unfair network, i.e., a network that consists entirely of unfair paths.
Revised: Therefore we can deﬁne fairness by demanding that protected variables do not cause the outcome along an unfair network, i.e., a network that consists entirely of unfair paths.

Original: Concretely, I propose the following deterministic deﬁnition of fairness, which can easily be extended into a probabilistic deﬁnition by moving to probabilistic causal models.
Revised: Concretely, I propose the following deterministic deﬁnition of fairness, which can easily be extended into a probabilistic deﬁnition by moving to probabilistic causal models.

Original: Deﬁnition 

Revised: Deﬁnition

Original: Given a causal model M , a protected variable A , and a set of unfair paths P a M , we say that the model is fair for A relative to P a M if there does not exist a context u and values y, a, a (cid:48) so that A = a rather than A = a (cid:48) causes Y =
Revised: Given a causal model M , a protected variable A , and a set of unfair paths P a M , we say that the model is fair for A relative to P a M if there does not exist a context u and values y, a, a (cid:48) so that A = a rather than A = a (cid:48) is an actual cause of Y =

Original: y in ( M, u ) relative to some ( W = w , N ) such that P a N ⊆ P a M , where P a N are the paths in N .
Revised: y in ( M, u ) relative to some ( W = w , N ) such that P a N ⊆ P a M , where P a N are the paths in N .

Original: We consider a very simple example as an illustration.
Revised: We consider a very simple example as an illustration.

Original: The equations are Y =
Revised: The equations are Y =

Original: ¬ B ∧¬ C , B
Revised: ¬ B ∧¬ C , B

Original: =
Revised: =

Original: A ,
Revised: A ,

Original: C =
Revised: C =

Original: ¬ A , all variables are binary, and all paths from A to Y are unfair (meaning path-speciﬁc counterfactual fairness reduces to counterfactual fairness).
Revised: ¬ A , all variables are binary, and all paths from A to Y are unfair (meaning path-speciﬁc counterfactual fairness reduces to counterfactual fairness).

Original: Y never standardly counterfactually depends on A , and thus existing notions of path-speciﬁc counterfactual fairness would always consider this model fair for A .
Revised: Y never standardly counterfactually depends on A , and thus existing notions of path-speciﬁc counterfactual fairness would always consider this model fair for A .

Original: In contrast, in a context where A = 1 , A = 1 rather than A = 0 is an actual cause of Y = 0 (along { B } ), and thus Deﬁnition 25 would consider this model unfair.
Revised: In contrast, in a context where A = 1 , A = 1 rather than A = 0 is an actual cause of Y = 0 (along { B } ), and thus Deﬁnition 25 would consider this model unfair.

Original: Imagine Y represents the outcome of a hiring process such that a candidate is hired if neither Billy ( B ) nor Cindy ( C ) rejects the applicant.
Revised: Imagine Y represents the outcome of a hiring process such that a candidate is hired if neither Billy ( B ) nor Cindy ( C ) rejects the applicant.

Original: Billy doesn’t like religious people ( A = 1 ), and therefore rejects their applications.
Revised: Billy doesn’t like religious people ( A = 1 ), and therefore rejects their applications.

Original: Cindy doesn’t like areligious people ( A = 0 ), and therefore rejects their applications.
Revised: Cindy doesn’t like areligious people ( A = 0 ), and therefore rejects their applications.

Original: So we get that a candidate’s religiosity caused their application to be denied, and this matches the intuition that the outcome is unfair in a context where religiosity is a protected variable.
Revised: So we get that a candidate’s religiosity caused their application to be denied, and this matches the intuition that the outcome is unfair in a context where religiosity is a protected variable.

Original: 7. Conclusion
Revised: 7. Conclusion

Original: This paper has integrated work on causation into the ﬁeld of action-guiding explainable AI by formally deﬁning several causal notions of explanation as well a novel account of causal fairness.
Revised: This paper has integrated work on causation into the ﬁeld of action-guiding explainable AI by formally deﬁning several causal notions of explanation as well a novel account of causal fairness.

Original: We explored the connections between these notions as well as the consequences that come with ignoring the causal structure.
Revised: I explored the connections between these notions as well as the consequences that come with ignoring the causal structure.

Original: Moreover, the proposed integration of actual causation with explanation extends beyond the practical needs of XAI to philosophy and science in general.Appendix A. Proposition 8 If X = x is directly sufﬁcient for Y = y then X
Revised: Moreover, the proposed integration of actual causation with explanation extends beyond the practical needs of XAI to philosophy and science in general.

Original: = x is strongly sufﬁcient for

Original: Y =

Original: y along some N , and if X = x is strongly sufﬁcient for Y =

Original: y along some N then X

Original: = x is weakly sufﬁcient for Y =

Original: y .

Original: Proof: Follows directly from the deﬁnitions.

Original: Theorem 12 If a causal model M that agrees with h satisﬁes Independence then the following statements are all equivalent:

Original: • X = x is weakly sufﬁcient for Y =

Original: y in M .

Original: • X

Original: = x is strongly sufﬁcient for Y =

Original: y in M .

Original: • X

Original: = x is directly sufﬁcient for Y =

Original: y in M .

Original: Proof: The implications from bottom to top are a direct consequence of Proposition 8.

Original: Assume X = x is weakly sufﬁcient for Y =

Original: y in M .

Original: This means that for all u ∈ R ( U ) we have that ( M, u ) | =

Original: [ X ← x ] Y =

Original: y .

Original: Let C

Original: =

Original: R ( V − ( X ∪ { Y } )) .

Original: Given that M agrees with h , either the equation for Y is of the form Y = U for some U ∈ U , or Y only has parents in V \ { Y } . Since the former contradicts our assumption that in all contexts ( M, u ) | = [ X ← x ] Y =
Revised: Acknowledgments

Original: y , it has to be the latter.

Original: As a consequence, interventions on all endogenous variables make the particular context irrelevant, i.e., for all c ∈ R ( C ) and

Original: all u 1 , u 2 ∈ R

Original: ( U ) , we have that ( M, u 1 ) |

Original: = [ X ← x , C ← c ] Y = y iff ( M, u 2 ) | = [ X ← x , C ← c ] Y = y . Further, for each V i ∈ V \ { Y } the equation is of the form V i
Revised: I would like to thank the reviewers, as well as Timo Freiesleben, Konstantin Genin, Amir-Hossein

Original: = U i .

Original: Although technically one could choose to deﬁne R ( V i ) such that R ( V i ) (

Original: cid:54)⊆ R

Original: ( U i ) , this comes down to deﬁning a variable with values that it cannot obtain, which serves no purpose.

Original: Therefore we can assume that for each c ∈ C there exists a context u (cid:48) such that ( M, u (cid:48) )

Original: | =

Original: C = c .

Original: Given that no members of X are parents of members of C , it also holds that ( M, u (cid:48) ) | =

Original: [ X ← x ] C = c .

Original: Since X = x is weakly sufﬁcient for Y =

Original: y , we also have that ( M, u (cid:48) )

Original: | =

Original: [ X ← x ] C = c

Original: ∧ Y =

Original: y , from which it follows

Original: that ( M, u (cid:48) ) | =

Original: [ X ← x , C ← c ] Y =

Original: y .

Original: Taken together with the previous observation that the particular context is irrelevant, we get that for all c ∈ R ( V − ( X ∪ Y )) and all u ∈ R ( U ) we have that ( M, u ) | =

Original: [ X ← x , C ← c ] Y = y , which is what we had to prove.

Original: Theorem 16 Given a causal setting ( M, u ) , the following two statements are equivalent:

Original: • there exist W 1 and w 1 ∈ R

Original: ( W 1 )

Original: so that Y =

Original: y counterfactually depends on X = x rather than X

Original: = x (cid:48) relative to W 1 = w 1 in ( M, u ) .

Original: • there exist W 2 , w 2 ∈ R ( W 2 ) , and N so that ( X = ( x , x (cid:48) ) , W 2 = w 2 ) , N ) is a good counterfactual explanation of Y =

Original: y

Original: Proof:

Original: Observation 1 Recall from Deﬁnition 3 that exogenous variables only appear in equations of the form V = U .

Original: Say R ⊆

Original: V are all variables which have such an equation, and call these the root variables.

Original: It is clear that if we intervene on all of the root variables, they take over the role of the exogenous variables.

Original: Concretely, given strong recursivity, for any setting r ∈ R ( R ) there exists a unique setting v ∈ R ( V ) so that for all contexts u ∈ R ( U ) we have that ( M, u ) | =

Original: [ R ← r ] V = v .

Original: Assume that Y =

Original: y counterfactually depends on X = x rather than X

Original: = x (cid:48) relative to

Original: W 1 = w 1 in ( M, u ) .

Original: This means that ( M, u ) | =

Original: X = x ∧ W 1 = w 1 ∧ Y =

Original: y , and ( M, u ) | =

Original: [ X ← x (cid:48) , W 1 ← w 1 ] Y (cid:54) =

Original: y .

Original: Let S

Original: =

Original: R \ ( W 1 ∪ X ) , and let s ∈ R ( S ) be the unique values so that ( M, u ) | =

Original: S = s .

Original: As R ⊆ ( S ∪ W 1 ∪ X ) , we have that ( X = x , S = s , W 1 = w 1 ) is strongly sufﬁcient for

Original: Y =

Original: y along N = V \ ( X ∪ W 1 ∪ S ) , and thus (( X = x , S = s , W 1 = w 1 ) , N ) is an actual sufﬁcient explanation of Y =

Original: y .

Original: Furthermore, changing X from x to x (cid:48) obviously has no effect on any of the other values in R .

Original: Therefore ( M, u ) | =

Original: [ X ← x (cid:48) , W 1 ← w 1 ] S = s , and thus we get that ( M, u ) |

Original: =

Original: [ X ← x (cid:48) , W 1 ← w 1 , S ← s ] Y = y (cid:48) for some y (cid:48) (cid:54) =

Original: y .

Original: As before, we can conclude that ( X = x (cid:48) , S = s , W 1 = w 1 ) is strongly sufﬁcient for Y = y (cid:48) along N , and

Original: thus (( X = x (cid:48) , S = s , W 1 = w 1 ) , N ) is a sufﬁcient explanation of Y =

Original: y (cid:48) .

Original: Combining the two previous paragraphs, we get that ( X = ( x , x (cid:48) ) , ( S = s , W 1 = w 1 )) , N ) is a counterfactual explanation of Y =

Original: y .

Original: Let ( X = ( x , x (cid:48) ) , W 2 = w 2 ) , N 2 ) be a dominating counterfactual explanation that is not strictly dominated by any other explanation that contains X = ( x , x (cid:48) ) .

Original: (It is easy to see that such an explanation must exist: one can simply keep removing elements from W 1 , S , and N until no further element can be removed while still remaining a counterfactual explanation of Y = y .)

Original: Now assume that there exist W 2 , w 2 ∈ R ( W 2 ) , and N so that (( X = x , W 2 = w 2 ) , N ) is an actual sufﬁcient explanation of Y =

Original: y and (( X = x (cid:48) , W 2 = w 2 ) , N ) is a sufﬁcient explanation of some Y =

Original: y (cid:48) with y (cid:48) (cid:54) =

Original: y . Since the ﬁrst explanation is actual, it follows immediately that ( M, u ) | = X = x ∧ W 2 = w 2 ∧ Y = y .
Revised: Karimi and Gunnar K¨onig for helpful comments on a preliminary version of this paper.

Original: Combining the second explanation with Propositionwe get that ( M, u ) | =

Original: [ X ← x (cid:48) , W 2 = w 2 ] Y = y (cid:48) .

Original: Note that we did not require X to be minimal in either direction, and [-thus the conditions as stated without minimality of X are equivalent. Therefore the conditions that include the minimality of X are also equivalent, which is what we had to prove. Proposition 
 If (( X = x , W = w ) , N ) is a sufﬁcient explanation of Y =-]
Revised: This research was initially funded by the NIAS-Lorentz Theme-Group Fellowship on “Accountability in Medical Autonomous Expert Systems: Ethical and [-thus the conditions as stated without minimality of X are equivalent. Therefore the conditions that include the minimality of X are also equivalent, which is what we had to prove. Proposition 
 If (( X = x , W = w ) , N ) is a sufﬁcient explanation of Y =-]Epistemological Challenges for Explainable

Original: y

Original: and there exists a dominating explanation (( X = x (cid:48) , A = a ) , B ) for some values x (cid:48) and a ⊆ w , then X

Original: = x (cid:48) can replace X = x .

Original: Proof: Assume (( X = x , W = w ) , N ) is a sufﬁcient explanation of Y =

Original: y and (( X = x (cid:48) , A = a ) , B ) is a dominating explanation of Y =

Original: y , with a ⊆ w .

Original: We show that (( X = x (cid:48) , W = w ) , B ) is a sufﬁcient explanation of Y =

Original: y , from which the result follows.

Original: Let C

Original: =

Original: V \ ( X ∪ A ∪ B ) .

Original: From the deﬁnition of sufﬁcient explanations, we know that for all u ∈ R ( U ) and all c ∈ R ( C ) , we have that ( M, u ) | =

Original: [ X ← x (cid:48) , A ← a , C ← c ] B = b for some b ∈ R ( B ) that includes y .

Original: Let D = V \

Original: ( X ∪ W ∪ B ) , F = W \ A , and let f be the restriction of w to F .

Original: Note that

Original: C = F ∪ D .

Original: From the previous paragraph it follows that for all u ∈ R ( U ) and all d ∈ R ( D , we have that ( M, u ) | =

Original: [ X ← x (cid:48) , A ← a , F ← f , D ← d ] B = b , and thus ( M, u ) | =

Original: [ X ← x (cid:48) , W ← w , D ← d ] B = b , which is what had to be shown.

Original: [-Theorem 
-]
Revised: [-Theorem 
-]

Original: If X 1 = x 1 rather than X 1 = x 1 (cid:48) is a counterfactual explanation of Y =

Original: y in ( M, u ) (relative to some ( W = w , N ) )

Original: then for some X 2 ⊆ X 1 , X 2 = x 2 rather than X 2 = x 2 (cid:48) is an actual cause of Y =

Original: y in ( M, u ) (where x 2 and x 2 (cid:48) are the relevant restrictions to X 2 ).

Original: Proof: Assume X 1 = x 1 rather than X 1 = x 1 (cid:48) is a counterfactual explanation of Y =

Original: y in ( M, u ) relative to ( W = w , N ) .

Original: This means that (( X 1 = x 1 , W = w ) , N ) is an actual sufﬁcient explanation of Y =

Original: y and (( X 1 = x 1 (cid:48) , W = w ) , N ) is a sufﬁcient explanation of Y =

Original: y (cid:48) with y (cid:48) (cid:54) =

Original: y .

Original: Let ( T = t , S ) be a good sufﬁcient explanation of Y =

Original: y , i.e., an actual sufﬁcient explanation of Y = y that dominates (( X 1 = x 1 , W = w ) , N ) and cannot itself be dominated by another actual sufﬁcient explanation of Y =

Original: y .

Original: (It is easy to see that such an explanation must exist: one can simply keep removing elements from (( X 1 = x 1 , W = w ) , N ) until no further element can be removed while still remaining a sufﬁcient explanation of Y = y .)

Original: Let X 2 = X 1 ∩ T .

Original: We now show that X 2 (cid:54) = ∅ by a reductio.

Original: Assume X 2 = ∅ .

Original: This means that T ⊆ W .

Original: Also, S ⊆ N . Let s ∈ R ( S ) and n ∈ R ( N ) be the actual values of S and N in ( M, u ) . Let C =
Revised: AI”, and later by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under

Original: V \ ( X 1 ∪ W ∪ N ) .

Original: Given that (( X 1 = x 1 (cid:48) , W = w ) , N ) is a sufﬁcient explanation of Y =

Original: y (cid:48) , we have that for all c ∈ R ( C ) , ( M, u ) |

Original: =

Original: [ X 1 ← x 1 (cid:48) , W ← w , C ← c ] Y = y (cid:48) .

Original: Let D = V \

Original: ( T ∪ S ) .

Original: Given that ( T = t , S ) is a sufﬁcient explanation of Y =

Original: y , we have that for all d ∈ R ( D ) , ( M, u ) |

Original: =

Original: [ T ← t , D ← d ] Y = y .

Original: By our assumption, X 1 ⊆ ( W \ T ) .

Original: Thus D

Original: =

Original: X 1 ∪ C ∪ ( W \ ( T ∪ X 1 )) ∪ ( N \ S ) .

Original: Therefore from the previous paragraph we get that for all c ∈ R ( C ) , ( M, u ) | =

Original: [ X 1 ← x 1 (cid:48) , W ← w , C ← c ] Y =

Original: y .

Original: This contradicts the paragraph before the previous, and therefore X 2 (cid:54) = ∅ .

Original: Let W 2 =

Original: T \ X 2 .

Original: [-Then we can conclude that ( X 
-]
Revised: [-Then we can conclude that ( X 
-]

Original: = x 2 , W 2 =

Original: w 2 , S ) is a good sufﬁcient explanation of Y =

Original: y , where x 2 is the restriction of x 1 to X 2 , and w 2 is the restriction of W to

Original: W 2 .

Original: Remains to be shown that x 2 (cid:48) cannot replace x 2 in this explanation, where x 2 (cid:48) is the restriction of x 1 (cid:48) to X 2 .

Original: The former just means that (( X 2 = x 2 (cid:48) , W 2 = w 2 ) , S 2 ) is not a sufﬁcient explanation of Y =

Original: y for any S 2 ⊆ S .

Original: Again we proceed by a reductio: assume that (( X 2 = x 2 (cid:48) , W 2 = w 2 ) , S 2 ) is a sufﬁcient explanation of Y =

Original: y .

Original: Let F = V \ (

Original: X 2 ∪ W 2 ∪ S 2 ) .

Original: We have that for all f ∈ R ( F ) , ( M, u ) | =

Original: [ X 2 ← x 2 (cid:48) , W 2 ← w 2 , F ← f ] Y = y .

Original: In particular, we have that ( M, u ) |

Original: =

Original: [ X 1 ← x 1 (cid:48) , W ← w ] Y = y .

Original: Recall that (( X 1 = x 1 (cid:48) , W = w ) , N ) is a sufﬁcient explanation of Y =

Original: y (cid:48) with y (cid:48) (cid:54) =

Original: y .

Original: Using

Original: Proposition 8, we get that ( M, u ) | =

Original: [ X 1 ← x 1 (cid:48) , W ← w ] Y = y (cid:48) .

Original: This contradicts the result in the previous paragraph, which concludes the proof.

Original: Proposition 22 If X = x is a direct cause of Y =

Original: y in ( M, u ) then there exist values x (cid:48) such that

Original: X = x rather than X = x (cid:48) is an actual cause of Y =

Original: y .

Original: Proof: Assume X = x is a direct cause of Y =

Original: y in ( M, u ) , i.e., it is part of a direct good sufﬁcient explanation ( X = x , W = w ) of Y =

Original: y in ( M, u ) .

Original: [-This means that all that remains to be shown, is that there exist values x (cid:
-]
Revised: [-This means that all that remains to be shown, is that there exist values x (cid:
-]

Original: ) ∈ R ( X ) such that

Original: ( X = x (cid:48) , W = w ) is not a direct sufﬁcient explanation of Y =

Original: y .

Original: We know that ( X = x , W = w ) is directly sufﬁcient for Y =

Original: y , and this does not hold if we remove any subset from either X or W .

Original: Let C

Original: =

Original: V \ ( X ∪ W ∪ { Y } ) .

Original: Then we have that for all c ∈ R ( C ) , ( M, u ) | =

Original: [ X ← x , W ← w , C ← c ] Y = y .

Original: From the minimality of X , it follows that there exists a c ∈ R ( C ) and

Original: x (cid:48) ∈ R ( X ) so that ( M, u ) | =

Original: [ X ← x (cid:48) , W ← w , C ← c ] Y (cid:54) =

Original: y .

Original: Therefore ( X = x (cid:48) , W = w ) is not a direct sufﬁcient explanation of Y =

Original: y .

Original: [-Theorem 
-]
Revised: [-Theorem 
-]

Original: If a causal model M satisﬁes Independence then for any W 1 , W 2 , N 1 , N 2 , it holds that there exist values w 1 ∈ R

Original: ( W 1 )

Original: so that ( X = ( x , x (cid:48) ) , W 1 = w 1 , N 1 ) is a good counterfactual explanation of Y =

Original: y iff there exist values w 2 ∈ R

Original: ( W 2 )

Original: so that ( X = ( x , x (cid:48) ) , W 2 = w 2 , N 2 ) is a good counterfactual explanation of Y =

Original: y .

Original: Proof: Given the symmetry, it sufﬁces to prove the implication in one direction.

Original: We invoke Theorem 16 so that we can use counterfactual dependence instead of good counterfactual explanations.

Original: Assume Y =

Original: y counterfactually depends on X = x rather than X

Original: = x (cid:48) relative to W 1 = w 1 .

Original: This means that X is a minimal set such that ( M, u ) | =

Original: X = x ∧ W 1 = w 1 ∧ Y = y and ( M, u ) | = [ X ← x (cid:48) , W 1 ← w 1 ] Y (cid:54) = y .
Revised: Germany’s Excellence Strategy – EXC number 2064/1 – Project number

Original: Take any set W 2 ⊆

Original: ( V \ ( X ∪ { Y } )) , and let w 2 be the values of W 2 in ( M, u ) .

Original: First, note that we have ( M, u ) | =

Original: X = x ∧ W 2 = w 2 ∧ Y =

Original: y .

Original: Second, since none of the members of X are parents of any of the members of W 2 , we also have that ( M, u ) | =

Original: [ X ← x (cid:48) ] W 2 = w 2 ∧ Y =

Original: y (cid:48) .

Original: Thus we also have that ( M, u ) | =

Original: [ X ← x (cid:48) , W 2 ← w 2 ]

Original: Y = y (cid:48) .

Original: As we did not use the fact that X is a minimal set, and thus both statements are equivalent when ignoring minimality on both sides, they are also equivalent when including minimality on both sides, which is what we have to prove.

Original: Theorem 24 If a causal model M satisﬁes Independence then the following statements are all equivalent:

Original: • X = x is a direct cause of Y =

Original: y in ( M, u ) .

Original: • there exist values x (cid:48)

Original: so that X = x rather than X = x (cid:48) is an actual cause of Y =

Original: y in ( M, u ) .

Original: • X

Original: = x is part of a good sufﬁcient explanation of Y =

Original: y in ( M, u ) .

Original: Proof: The implication from the ﬁrst statement to the second is a direct consequence of Proposition 22.

Original: The implication from the second statement to the third follows from the deﬁnition of actual cause.

Original: The implication from the third statement to the ﬁrst follows from Theorem 12, which shows that under Independence we may replace sufﬁciency with direct sufﬁciency, and thus the result follows from the deﬁnition of direct cause.