1 Introduction↩︎

Causal explanations [1] are of particular interest when an observed event is unexpected under a given probabilistic model. Typical examples of rare events include natural disasters [2], stock market crashes [3], failures in technical systems [4], and extreme gene expression patterns [5]. Current approaches to root cause analysis of rare events are typically centered on identifying a small set of causal origins, called root causes [6]–[13]. While root cause analysis is informative, it does not capture the mechanisms through which these causes propagate to produce the observed outcome, which may include interacting mechanisms and modulated contextual factors. As a result, causal explanations cannot, in general, be reduced to identifying a limited number of root causes. For purposes of interpretability and scientific insight, it is therefore desirable to provide causal explanations that identify not only root causes, but also the relevant subgraph of the causal model that connects these causes to the observed event. Furthermore, the full causal model is often not known [14], therefore causal discovery can also be viewed as the problem of identifying a minimal causal structure that strongly explains a particular observed event [15]. This perspective highlights the need for a formal notion of event-specific causal explanation that remains well-defined under partial model knowledge and uncertainty about the underlying detailed causal graph.

1.0.0.1 Our contributions.

We introduce a formalization of causal pathways of events connecting root causes to a target. Based on this framework, we define several quantities that measure the quality of an explanation, motivated by the ideas of plausibility and falsifiability. Although most of our events refer to numeric variables, we emphasize that our concepts generalize to variables with arbitrary ranges, e.g., variables may describe tokens or sentences when a probability distribution is defined via semantic embedding.

1.0.0.2 Related work.

Existing work on causal modeling under rare events has largely focused on extreme-value regimes with asymptotic or tail-based limits, often subject to parametric assumptions [16], [17]. While this line of work provides insight into causal mechanisms in the tail of continuous variables, it does not address how causal structure behaves under rare but non-extreme events as rarity does not imply extremeness: events may be statistically rare while occurring at non-tail values [18], such as values that are unexpectedly close to zero or values of highly unbalanced binaries. Consequently, approaches based on asymptotic extremes are not directly applicable in this setting. In contrast, we study causal contributions for rare events without relying on asymptotic limits. Our framework does not assume continuous variables and applies to variables defined over arbitrary spaces, such as discrete domains. In addition, related work on mediation analysis has primarily addressed decomposing causal effects into path-specific components in order to quantify how effects propagate along different causal pathways [19]–[21]. While this literature enables attribution of portions of a causal effect to specific paths, it does not address a complementary question: which parts of a causal model provide a good causal explanation for a particular observed event. Therefore, our goal is not mediation analysis, but rather event-level interpretability of causal structure for rare events. Another related line of work is causal abstraction, which provides a formal framework for relating causal models that represent the same system at different levels of granularity [22]–[24]. However, existing approaches to causal abstraction operate at the level of the entire causal model, whereas our paper formalizes an event-level notion of causal pathway abstraction for the portion of a causal graph that explains a target rare event.

The paper is structured as follows. To quantify the extent to which a set of root-cause events explains a set of observed events, Section 2 discusses causal relations between binary variables. Section 3 defines causal pathways from root causes to a target event and quantifies to what extent the pathway explains the target. Section 4 describes how causal pathways of events, represented as binary variables, are obtained via abstraction of more detailed causal models. Section 5 demonstrates how our concepts can be applied to evaluate causal explanations.

2 Explaining Clusters of Events↩︎

Assume we are given \(k\) events, formalized by binary variables \(\mathbf{B}:=\{B_1,\dots,B_k\}\), where \(B_i=1\) denotes event \(i\) happening. Further assume that the variables \(\mathbf{B}\) are connected by a causal DAG \({\cal C}\) with joint probability distribution \(P_\mathbf{B}\). For an arbitrary index set \(S\subseteq K:= \{1, \dots,k\}\), let \(\mathbf{B}_S\) denote the vector of variables \((B_i)_{i\in S}\), and let \(\mathbf{B}_S=\mathbf{1}\) denote the joint event \((B_i=1)_{i\in S}\). Then, assuming the Markov condition [25], the probability of observing the event \(P_\mathbf{B}(\mathbf{B}=\mathbf{1})\) factorizes as \[P_\mathbf{B}(\mathbf{B}=\mathbf{1}) = \prod_{i=1}^k P_\mathbf{B}(B_i=1 \mid \mathbf{B}_{\mathrm{Pa}(i)} =\mathbf{1}),\] where \(\mathrm{Pa}(i)\) denotes the indices of the parents of \(B_i\) in \({\cal C}\). Now consider the scenario that an external event “corrupted” the data generating process in the sense that one or several of the causal mechanisms \(P_\mathbf{B}(B_i \mid \mathbf{B}_{\mathrm{Pa}(i)})\) are replaced with different mechanisms \(\tilde{P}_\mathbf{B}(B_i \mid \mathbf{B}_{\mathrm{Pa}(i)})\). Let \(R\subseteq K\) denote the set of indices of the affected nodes. We define the variables \(\mathbf{B}_R\) as the root-cause variables of the event \(\mathbf{B}= \mathbf{1}\), and call the tuple \(({\cal C}, \mathbf{B}, \mathbf{B}_R, P_\mathbf{B})\) a cluster of events with root causes \(R\). The “hard intervention” [25] \(do(\mathbf{B}_R=\mathbf{1})\) is a special case of the more general class of “soft interventions” [26], which yields \[\label{eq:clusterInter} P_\mathbf{B}(\mathbf{B}=\mathbf{1}| do(\mathbf{B}_R =\mathbf{1})) = \prod_{i\not \in R} P_\mathbf{B}(B_i=1| \mathbf{B}_{\mathrm{Pa}(i)} =\mathbf{1}).\tag{1}\]

For the case of soft interventions, the right-hand side of 1 is only an upper bound on the likelihood of the cluster event \(\mathbf{B}= \mathbf{1}\), because the term \(\prod_{i\in R} \tilde{P}_\mathbf{B}(B_i=1 \mid\mathbf{B}_{\mathrm{Pa}(i)}=\mathbf{1})\) is missing. Whenever the right-hand side of (1 ) is close to \(1\), the intervention \(do(\mathbf{B}_R=\mathbf{1})\) is a plausible explanation for the cluster event. To formalize this idea, we first introduce a concept from [27] with slightly different notation:

Following this terminology we define:

Good candidates for root causes are therefore given by sets \(R\) with high explanation score. Furthermore, while [27] emphasize that explanation score is–for good reasons–not generally additive over disjoint unions of features, one can easily check that \({\cal E}_{R \to K}\) does have this property:

We call \({\cal E}_{i\to K}\) the contribution of mechanism \(P_\mathbf{B}(B_i \mid\mathbf{B}_{\mathrm{Pa}(i)})\) to the cluster event. By the additivity stated in Lemma 1, for every \(R \subseteq K \setminus \{i\}\), we also have \({\cal E}_{i\to K} ={\cal E}_{R\cup \{i\} \to K} - {\cal E}_{R \to K}\). Hence, the contribution of each mechanism \(i\) to the cluster explanation does not depend on which other root causes are considered simultaneously. Consequently, there is no need to average over contexts of variables to condition on as in Shapley-value “fair attribution” [28], in contrast to causal and statistical attribution settings where marginal contributions may depend on the chosen context [29]–[32]. In addition, note that for any discrete distribution \(Q\), \(-\log Q(x)\) can be written as the KL-divergence \(D_{KL}(\delta_x\|Q)\), where \(\delta_x\) is the point mass on \(x\), which enables rewriting \({\cal E}_{R\to K}\) in terms of KL-divergences¹. Using this observation, Appendix 7.1 shows that \({\cal E}_{R\to K}\) can be seen as a degenerate instance of distribution change attribution in the sense of [33].

Although we use the term "event" throughout the paper, which may suggest occurrences at specific time instants, our framework more generally applies to explaining why a specific statistical unit behaves differently from a reference population, for instance an individual patient with a monogenic disorder may be viewed as an outlier relative to healthy individuals [5].

3 Causal Pathways↩︎

We now focus on explaining a certain target event \(B_t=1\) within a cluster, and therefore focus on the subset of nodes that are crucial to understand how the target event has been triggered by the root causes via a causal pathway:

To understand the distinction between \(\mathcal{P}\) and \({\cal C}\), observe that knowledge of \(\mathcal{P}\) alone suffices to compute \(P_\mathbf{B}(\mathbf{B}|do(\mathbf{B}_R=\mathbf{b}_R))\) for all \(\mathbf{b}_R\), since the do-intervention removes all incoming edges to \(R\). Thus, \(\mathcal{P}\) is sufficient for explaining why the identified root causes render the target likely. However, computing the outcome when some root causes remain unintervened upon—as distinct from setting them to zero—requires knowledge of the incoming edges to \(R\). In this regard, \({\cal C}\) enables assessment of each individual root cause’s relevance and enhances the transparency of the explanation by allowing evaluation of whether the selected root causes are well-justified. In essence, \({\cal C}\) provides a meta-level component to the explanation by elucidating the causal pathway itself. Nevertheless, one can readily verify that the specific choice of \({\cal C}\) does not affect the scores \({\cal E}^K_{R\to t}\) and \({\cal E}_{R\to t}\), which are defined later in this section.

Note, however, that \(\mathcal{P}\) may also contain edges into nodes in \(R\). This allows us to describe, for instance, scenarios like chains of root causes, in which each one “contributes” to the target because it does not propagate the perturbation as expected and adds an additional perturbation (or additional “unexpectedness”) instead, see Example 1.

In the rest of this paper, we will refer to \(\mathcal{P}\) as the causal pathway, when the other “ingredients” are clear from the context. To quantify to what extent the root causes explain the target event we again use the concept of explanation score:

While Definition 4 can be used to measure relevance to the target for any set of nodes, we will mostly use \({\cal E}_{R\to t}\) to measure the relevance of the root causes. However, \({\cal E}_{R\to t}\) does not capture the extent to which the remaining events in \(K \setminus (R \cup\{t\})\) are crucial for explaining the target event–in contrast with common intuitive notions of a causal pathway, which entail some relevance of all events mentioned in a verbal explanation. This motivates the following concept, which is a modification of the explanation score:

Definition 5 formalizes the idea that making the root cause events happen renders all the events “relatively likely”, where likelihood is assessed relative to the “unlikeliness” of the target. Note that \({\cal E}^K_{R\to t}\) is negative if the likelihood of \(\mathbf{B}=\mathbf{1}\), given the root causes \(\mathbf{B}_R=\mathbf{1}\), is smaller than the a priori likelihood of \(B_t=1\)–in other words the explanation “explains” the target event with an even more unlikely cluster of events. The following result shows the difference between Definitions 5 and 4:

It is important to note that \(\mathbf{B}_S\) need not necessarily be mediators that propagate the information from \(\mathbf{B}_R\) to \(B_t\). We will later also have cases where they describe “context” events that control the propagation of information from \(\mathbf{B}_R\) to \(B_t\) as in Example 4. 3 measures to what extent configurations other than \(\mathbf{B}_S=\mathbf{1}\) are relevant when the intervention \(do(\mathbf{B}_R=\mathbf{1})\) changes the probability of \(B_t=1\). For a “good” pathway, we expect 3 to be small, because all its events should occur with high probability when the root cause variables are set to \(1\) and the target event is observed to happen. Lemma 2 entails in particular that \({\cal E}^K_{R\to t}>0\) implies \({\cal E}_{R\to t}>0\), and thus guarantees that root causes with positive pathway explanation score are indeed relevant for the target, and not only for the other nodes in the pathway.

Pathway explanation score and cluster explanation score are related as follows:

Lemma 3 shows an affine relation between the two types of scores, which implies the modified additivity \[\label{eq:modAddivity} {\cal E}^K_{R \to t} - {\cal E}^K_{\emptyset \to t} = \sum_{i\in R} ({\cal E}^K_{\{i\}\to t} - {\cal E}^K_{\emptyset \to t}).\tag{4}\] Lemma 3 also shows that the pathway explanation score defines a stricter assessment of an explanation compared to the cluster explanation score: an explanation is considered good only if \(P_\mathbf{B}(\mathbf{B}=\mathbf{1}\mid do(\mathbf{B}_R=\mathbf{1}))\) is much closer to \(1\) than the baseline probability \(P_\mathbf{B}(B_t=1)\), while the cluster explanation score is close to 1 if \(P_\mathbf{B}(\mathbf{B}=\mathbf{1}\mid do(\mathbf{B}_R=\mathbf{1}))\) is much closer to \(1\) than the typically smaller baseline probability \(P_\mathbf{B}(\mathbf{B}=\mathbf{1})\).

Example 5 further illustrates the distinction between pathway and cluster explanation scores.

In Appendix 7.3 we show that even when each event strongly explains the next one in a causal chain, this does not guarantee that the initial event has a high pathway explanation score. Therefore, strong local causal explanations do not necessarily imply a strong global causal explanation. The following result describes a necessary condition for the pathway explanation score to be close to \(1\):

The proof follows easily from \(P_\mathbf{B}(B_i=1 \mid \mathbf{B}_{\mathrm{Pa}(i)}=\mathbf{1})\leq P_\mathbf{B}(B_i=1)/P_\mathbf{B}(\mathbf{B}_{\mathrm{Pa}(i)}=\mathbf{1})\), or as a special case of Lemma 3.3 in [6]. Lemma 4 states that good explanations need to avoid events as mediators that are significantly less likely than their parent event. As a simple application of Lemma 4, we also get a low explanation score when a mediator event is less rare than the target since we get a significant gap for the target itself, as in Example 6.

3.0.0.1 Relation to probability of necessity and sufficiency.

At first glance, the notion of explaining pathway seems to miss the aspect of necessity: while it formalizes the condition that the target event and the mediators between root causes and target are likely to happen given the root causes happened, it does not explicitly state that the target event is unlikely to happen in the absence of the root causes. Likewise, one may be surprised that explanation score of [27] in general fails to formalize necessity. However, there is a sense in which necessity is implicit because the target event is rare: the intervention \(do(\mathbf{B}_R=\mathbf{1})\) is, with high probability necessary for \(B_t=1\) because without it the latter is unlikely by assumption. This is formally shown in Appendix 8, based on the notion probability of necessity [34] (where we also argue why probability of sufficiency is of minor relevance in our context). This result, however, does not imply that it is necessary to set all root causes to \(1\). The choice of a sufficiently small set of root causes will be driven by trading off simplicity of the explanation (in the sense of a small number of root causes) with pathway explanation score.

In Appendix 9 we explain why our causal explanations refer to interventional probabilities only, rather than causal counterfactuals in the sense of rung 3 in the ladder of causation [35].

3.0.0.2 Measuring quality of explanations.

We now state formal criteria that should be satisfied by a causal pathway to be a good explanation:

1. Causal relevance of all events for the target: Given some cluster \(({\cal C}, \mathbf{B}, \mathbf{B}_R, P_\mathbf{B})\), every event \(i\in K\) should satisfy \({\cal E}_{\{i\} \cup S \to t} - {\cal E}_{S \to t}> 0\) for some \(S\subset K\). Depending on one’s preferences regarding the trade-off between simplicity and detail, one may set a threshold for the desired increase in explanation score. Note that we do not consider pathway explanation scores \({\cal E}^K_{R \to t}\) for this purpose because we want to assess relevance for the target and not for other events in the pathway.

2. Sparsity of root causes: Following Occam’s Razor we a priori prefer explanations with a small number of root causes. Whenever the target event is the result of the respective data point being drawn from a different distribution this inductive bias is also aligned with the “sparse mechanism shift hypothesis” in [36]. However, rare events may also originate from selecting the statistical unit / time instant for which this rare event happened. Later we will have target events that come from thresholding continuous variables and the statistical unit we discuss is selected because it is the most extreme value within a population. Rare events that result from this selection bias are not necessarily points that are “out of distribution”. In this case, we may also expect multiple root causes. For instance, if \(X_1,\dots,X_d\) are independent identically distributed Gaussians, extreme values of \(\sum_{i=1}^d X_i\) likely originate from extreme values of multiple \(X_i\), while \(X_i\) with subexponential \(X_i\) render single root causes more likely, see Appendix 11.

3. Selecting the right root causes: The pathway explanation score \({\cal E}^K_{R\to t}\) should be reasonably close to \(1\), without unnecessarily extending \(R\). For any desired size \(|R|\), 4 implies that the best \(R\) can be obtained by a greedy approach in a scenario where the set \(\mathbf{B}\) is fixed: starting from \(R=\emptyset\), add the event \(i\) to \(R\) that maximizes the score \({\cal E}^K_{\{i\}\cup R \to t }\). After choosing \(R\), we can reduce the graph \({\cal C}\) to the relevant part, the pathway \(\mathcal{P}\), by dropping arrows into \(R\) (although we may not drop all of them, as explained earlier).

4 Pathway Abstraction↩︎

We now introduce causal pathways that constitute approximate abstractions of more fine-grained causal models. The abstraction proceeds by reducing the set of nodes and mapping the remaining variables \(X_i\) into a binary representation. In addition, edges may be added or removed to provide the best possible approximation of the causal relationships among the resulting binary variables.²

4.1 Binary Representation of Variables↩︎

Let us first focus on generating binaries from variables with arbitrary ranges. Consider, for instance, a cause-effect pair \(X,Y\) coupled by the linear structural equation \(Y=\alpha X + N\), where \(N\) is an independent noise term and all three variables are real-valued. If we induce a large perturbation \(x\) and accordingly obtain an unusually large \(y\), we would certainly accept the verbal explanation “\(y\) is large because \(x\) has been large”. To get formal criteria for whether this explanation is plausible we introduce the events \(B_1\) and \(B_2\) corresponding to the events³ \(X\geq x\) and \(Y\geq y\), and consider only accepting the explanation if \(P(Y \geq y|X\geq x)\) is “sufficiently large” in the sense specified below. In cases where \(Y\) depends on \(X\) in a non-monotonic way, for instance, when \(Y=\sin (X) +N\), we would not accept explanations like “\(y\) is large because \(x\) is large” and instead suggest the “featurized” version “\(y\) is large because \(\sin(x)\) is large” to get a “feature-monotonic” causal relation, formally introduced here⁴:

Here, featurizing variables has two goals. First, it defines outlier scores [37] via \(S(x_j):=-\log P_\mathbf{X}(\tau_j(X_j)\geq \tau_j(x_j))\), which measures the unexpectedness of events for arbitrary data modalities. Second, it enables causal statements that are more robust than statements about the impact of setting a continuous variable to a specific value, cf. [38]. We then obtain:

Note that the non-trivial part of the statement comes from conditioning on the event \(\tau_{\mathrm{Pa}(j)}(\mathbf{X}_{\mathrm{Pa}(j)}) \geq \tau_{\mathrm{Pa}(j)} (\mathbf{x}_{\mathrm{Pa}(j)})\) instead of \(\mathbf{X}_{\mathrm{Pa}(j)}=\mathbf{x}_{\mathrm{Pa}(j)}\) used in the generating process for \(x_j\). However, feature monotonicity entails \[\begin{align} &P_\mathbf{X}\big (\tau_j(X_j)\geq \tau_j(x_j) \mid \tau_{\mathrm{Pa}(j)}(\mathbf{X}_{\mathrm{Pa}(j)}) \geq \tau_{\mathrm{Pa}(j)} (\mathbf{x}_{\mathrm{Pa}(j)}) \big ) \nonumber \\ &\geq P_\mathbf{X}\big (\tau_j(X_j) \geq \tau_j(x_j)\mid \mathbf{X}_{\mathrm{Pa}(j)}=\mathbf{x}_{\mathrm{Pa}(j)} \big ), \end{align}\] which then completes the proof. To get an intuition about the implications of Lemma 5, we consider a cause-effect pair \(X\to Y\) with feature monotonic \(P(Y \mid X)\) with respect to \(\tau_X,\tau_Y\). For any \(x\)-value, a \(y\)-value drawn from \(P(Y \mid X=x)\) satisfies \(-\log P \big(\tau_Y(Y)\geq \tau_Y(y) \mid \tau_X(X)\geq \tau_X(x) \big )\geq c\) with probability at most \(e^{-c}\). Accordingly, defining the binary events \(\tau_X(X)\geq \tau_X(x)\) and \(\tau_Y(Y)\geq \tau_Y(y)\) typically results in explanation scores close to \(1\) when the generated value \(y\) is a “strong outlier”, that is, \(P(\tau_Y(Y) \geq\tau_Y(y))\) is small.

The following result generalizes Lemma 5 to general DAGs:

The proof is provided in Appendix 12. To understand the significance of Theorem 1, consider the joint distribution⁵ \(P_\mathbf{B}(\mathbf{B}):= \prod_{j\in N} P_\mathbf{X}( B_j \mid \mathbf{B}_{\mathrm{Pa}(j)})\). Note that \(P_\mathbf{B}(\mathbf{B})\) is, by construction, Markov relative to \(\mathcal{G}\), whereas the true distribution \(P_\mathbf{X}(\mathbf{B})\) need not be; even for \(X_1\to X_2\to X_3\), binarization can destroy the conditional independence \(X_1\mathpalette{\independenT}{\perp}X_3\,|X_2\). Let us, nevertheless, consider a scenario where \(P_\mathbf{B}(\mathbf{B})\) is a reasonable approximation for \(P_\mathbf{X}(\mathbf{B})\), because we can easily compute interventional probabilities for the former. We then observe that \(L\) is the negative log-likelihood of \(P_\mathbf{B}(\mathbf{B}=\mathbf{1}\mid do(\mathbf{B}_R=\mathbf{1}))\), and thus it follows from Theorem 1 that the pathway explanation score is close to \(1\) with high probability, whenever \(\mathbf{x}_R\) induces a value \(x_t\) of the target node such that \(P_\mathbf{B}(B_t=1)\) is small. Although feature monotonicity may not hold in general, we assume that good explanations should use features that do not deviate too much from monotonicity and use the p-value in 6 as guidance on which deviations from explanation score \(1\) we tolerate for the given number \(n\) of nodes, regardless of whether feature monotonicity holds exactly for a given case.

4.2 Causal Pathways as Approximate Abstractions↩︎

To enable simple causal explanations, we introduce pathways (potentially with fewer links and variables than the original graph) with distributions that do not perfectly align with the true one, but do not deviate too much in terms of interventional and observational probabilities. This raises the issue that the interventions \(do(B_i=b_i)\), with \(b_i=0,1\), are only well-defined in the model \(P_\mathbf{B}\), but are ill-defined in \(P_\mathbf{X}\) since there are generally many distinct interventions \(do(X_j=x_j)\) that correspond to the same value of \(B_j\). This aligns with the known ill-definedness of interventions on coarse-grained variables [39]. Consider, for instance, the event \(X_j\geq x_j\) for some unusually large \(x_j\). Then \(do(X_j=x_j-\epsilon)\), which sets \(B_j\) to \(0\), will typically result in distributional changes that are close to \(do(X_j=x_j)\), which sets \(B_j\) to \(1\), and thus far from the effect of “normal” \(x_j\). We solve this problem by interpreting \(do(\mathbf{B}_j=\mathbf{b}_j)\) in the original model as a randomized intervention that draws \(X_j\) from the distribution \(P_{\mathbf{X}}(X_j \mid B_j=b_j)\), following the idea of “natural micro-realization” given by Definition 8 in [40]. For multi-node interventions we draw independently from the product distribution \(\prod_{i\in S} P_{\mathbf{X}}(X_i \mid B_i=b_i)\) and thus define:

For an unconfounded cause-effect pair \(X\to Y\), for instance, this definition ensures that \(P(Y \mid do(X\geq x)) = P(Y \mid X\geq x)\). We now introduce pathway abstraction formally:

Note that the pathway explanation score can also be rephrased in terms of KL-divergences as \[\label{eq:asKL} {\cal E}^K_{R\to t} = 1 - \frac{D_{KL}(\delta_\mathbf{1}(\mathbf{B}) \| P_\mathbf{B}(\mathbf{B}\mid do(\mathbf{B}_R=\mathbf{1}))}{ -\log P_\mathbf{B}(B_t=1)},\tag{8}\] where \(\delta_\mathbf{1}\) denotes the point mass on \(\mathbf{B}=\mathbf{1}\) (see remark after Definition 2). Hence, the explanation score measures the distance of \(\delta_\mathbf{1}\) from the interventional distribution in \(P_\mathbf{B}\) with respect to \(\mathcal{P}\), where all root causes are set to \(1\), and accuracy measures the distance between the distribution of the abstraction and the true distribution projected to the binaries. In this sense, both quantities measure the quality of the explanation in terms of KL-divergences. Note that minimizing the KL-divergence 7 is equivalent to maximizing the expected log-likelihood \(\mathbb{E}_{P_\mathbf{X}(\mathbf{B}\mid do(\mathbf{b}_S))} [\log P_\mathbf{B}(\mathbf{B}\mid do(\mathbf{b}_S))]\). Therefore trading off explanation score with accuracy amounts to trading off two likelihood terms, namely the one of the event \(\mathbf{B}=\mathbf{1}\) under the model \(P_\mathbf{B}(\mathbf{B}\mid do(\mathbf{B}_R=\mathbf{1}))\) and the likelihood of observations from \(P_\mathbf{X}(\mathbf{B}\mid do(\mathbf{b}_S))\) under the models \(P_\mathbf{B}(\mathbf{B}\mid do(\mathbf{b}_S))\).

4.3 Examples of Pathway Abstractions↩︎

We first start with a simple example where the abstraction consists only in binarization of two continuous variables:

Example 2 shows that the pathway explanation score conceptualizes a certain kind of robust causal statements: It does not measure whether the specific value \(x\) increases the probability of the event \(Y\geq y\). Instead, it measures whether ‘typical values’ of \(X\) from the interval \([x,\infty)\) have this effect, which is a more valuable insight since it is more general and falsifiable. The following example shows why root nodes in a pathway abstraction are not necessarily root causes, but potentially non-rare events that are nevertheless crucial as context.

The following example shows a similar case obtained by binarization of continuous variables. We consider the causal DAG in Figure [fig:3B], left, where an extreme value \(y\) is induced at \(Y\). For understanding the impact of this perturbation on \(Z\), it is important to know that \(X\) attained normal values to exclude strong influence of the confounder in the perturbed regime. Hence, \(X\) being normal is required as context information. The example also shows that binarization can result in accuracy smaller than \(1\) because the natural micro-realization of the intervention does not capture the dependencies of different parents entirely. However, for our purpose of causal explainability, we do not need accuracy 1 in order to get simple and verbalizable explanations.

In Appendix 13.2 we consider additional pathway abstractions for the three-variable binary causal DAG and give quantitative conditions under which the direct link or the confounder can be removed while maintaining high abstraction accuracy.

5 Evaluating Causal Explanations↩︎

The assumption that all mechanisms of non-root causes operated as expected for the statistical observation to be explained entails testable implications. Phrasing causal explanations in terms of pathways paves the way for several types of consistency checks. First, Lemma 5 and Theorem 1 show how to test consistency with data from the “normal regime” because this enables the estimation of the respective probabilities \(P\). Second, for any given agent generating causal explanations (human experts or GenAI tools like LLMs), we can check internal consistency, both in a qualitative and in a quantitative sense: For the qualitative test of a pathway \(\mathcal{P}\), we can ask the agent about substructures. If \(\mathcal{P}\) contains the edge \(B_i\to B_j\), we can ask “is \(B_i\) a cause of \(B_j\) in the context of \(\mathbf{B}_{\mathrm{Pa}(j) \setminus \{i\}}\)?”. For a quantitative test, we can ask about the probability that \(B_i=1\) results in \(B_j=1\), given \(\mathbf{B}_{\mathrm{Pa}(j) \setminus \{i\}}=\mathbf{1}\). In other words, we check whether the claimed pathway is consistent with believed probabilities, regardless of whether these are Bayesian beliefs or whether they are inferred from data. We asked LLMs to describe realistic hypothetical causal pathways that explain why an imaginary person in the USA became homeless. We describe more details in Appendix 14.

We obtained, among others, a pathway \(A\to B \to C \to D \to E\), with the following events:
A. 35-year-old male developed schizophrenia at age 19; lacks consistent access to psychiatric medication due to gaps in Medicaid coverage
B. Fired from three consecutive jobs over 2 years due to paranoid behavior and absences during acute episodes
C. With no income and $2,000 in savings depleted on food/medication, evicted from rental apartment
D. No family support available—parents died; only sibling cut contact after threatening incident during psychotic break
E. Living on streets for 18+ months; chronic homelessness now established

We then asked the same LLM, in separate conversations, about the probability that each child node was caused by its parent nodes and obtained the following⁶ results: \(P(B \mid A)\approx 0.55\), \(P(C \mid B)\approx 0.8\), \(P(D \mid C)\approx 0.05\), \(P(E \mid D)\approx 0.2\). Further, the a-priori probability of being homeless long-term was estimated at \(P(E)\approx 0.0005\). Taking \(A\) as root cause results in a pathway explanation score of only \[{\cal E}^K_{R\to t} = 1 -\frac{\log (0.55 \cdot 0.8 \cdot 0.05 \cdot 0.2)}{\log 0.0005} \approx 1- \frac{5.43}{7.60} \approx 0.29,\] since the explanation is mainly weak at the link \(C\to D\). After all, \(C\) does not refer to any mental illness that would explain event \(D\). The evaluation suggests to revise the pathway or events in a way that parent events contain the essential information needed to explain the child event, e.g., by drawing an additional direct link \(A\to D\) or rephrasing \(C\). Looking at the LLM output in Appendix 14.1, the reader may notice that it is inconsistent regarding the hypothesized pathway: the displayed chain suggests the pathway \(A \to B \to C \to D \to E\), whereas at the end it specifies edges which contain no arrow \(C\to D\), but include an additional link \(D\to E\). We also checked that this alternate pathway yields a higher explanation score, aligning with the fact that it avoids the weak link \(C\to D\).

As a disclaimer for this example, we would like to emphasize that high explanation scores are not sufficient to prove that a causal explanation is valid. The score does not verify that the underlying causal DAG is true, or that the hypothetical interventional probabilities computed from the DAG reflect the true ones.

6 Discussion↩︎

Verbal causal explanations of unexpected events are commonly based on chains of events causing each one after another, although the actual causal relation can be a complex graph. We have introduced a formal definition of causal pathway that allows us to quantify the extent to which such simple explanations are consistent with data. The reason why our framework renders explanations falsifiable is that it requires an explicit commitment on which of the mechanisms in a causal network behaved as usual and which ones did not. We do not claim that our framework already enables automated testing of the plausibility of causal explanations. Instead, the goal is to introduce concepts that enable the discussion of plausibility in a quantitative and data-driven way.

Impact Statement↩︎

There is broad consensus that producing reliable and transparent causal explanations would represent a significant advancement in generative AI. Moreover, a key research challenge is to develop methods to assess whether causal statements generated by AI systems are truthful [41]. The present work advances this objective by formalizing causal explanations. We present concepts that facilitate empirical falsification and transparency by representing causal explanations as explicit pathways. Through this approach, we aim to contribute toward improved automated verification methods for causal explanations, with a specific focus on identifying flawed causal accounts of socially significant events. While the inherent ambiguity of verbal explanations makes formal plausibility assessments difficult, LLMs can be guided to generate explanations conforming to the causal pathway format proposed here. Our example in Section 5 demonstrates as a proof of concept illustrating how probabilities derived from the same LLM can serve to evaluate the plausibility of its own explanations.

Acknowledgments↩︎

We thank William Roy Orchard for carefully reading the paper and providing suggestions for improvement.

7 More Details on Explanation Scores↩︎

7.1 Cluster Explanation Score and Distribution Change↩︎

The proof is a straightforward calculation using \[\begin{align} & D_{KL}\left[ \delta_\mathbf{1}(B_i \mid\mathbf{B}_{\mathrm{Pa}(i)})\| P_\mathbf{B}(B_i \mid\mathbf{B}_{\mathrm{Pa}(i)})\right] = \sum_{\mathbf{b}_{\mathrm{Pa}(i)}} D_{KL}\left[ \delta_\mathbf{1}(B_i \mid\mathbf{b}_{\mathrm{Pa}(i)})\| P_\mathbf{B}(B_i \mid\mathbf{b}_{\mathrm{Pa}(i)})\right] \delta_\mathbf{1}(\mathbf{b}_{\mathrm{Pa}(i)})\\ & = - \log P_\mathbf{B}(B_i=1 \mid \mathbf{B}_{\mathrm{Pa}(i)}=\mathbf{1}). \end{align}\]

7.2 Explaining Clusters versus Pathways↩︎

7.3 High Pairwise Explanation Score are Not Sufficient↩︎

8 Relation to Probabilities of Necessity and Sufficiency↩︎

As shown by [34], see Eq. (45), under monotonicity and unconfoundedness these quantities can be expressed as follows:

Despite the impression that explanation score is rather related to sufficiency than necessity, we observe that PS can be rather small even when the explanation score is close to \(1\). Assume, for instance, that \(p(y^1)=10^{-12}\) and \(p(y^1 \mid x^1)=10^{-2}\). Since the denominator in 11 is close to \(1\), PS is roughly \(1/100\), while explanation score is \(5/6\). In contrast, when the target is rare and the explanation score is positive enough, the following lemma provides a lower bound showing that PN is close to \(1\):

To illustrate this result with numbers, assume that we observe an event with outlier score \(S(y^1):=- \log p(y^1)=5\), which corresponds to an observation that is unique among \(e^5 \approx 148\) observations. For \({\cal E}(x^1\to y^1)=1/5\) we obtain \(PN\geq 1- e^{-1} \approx 0.63\), meaning that it is more likely for \(X=x^1\) to be necessary than not necessary.

Let us now argue why PS is of minor relevance for our explanations. We are interested in explaining an observed event in which both \(x^1,y^1\) occurred, while PS concerns cases in which \(X=x^0\) and \(Y=y^0\) occurred, and asks whether setting \(X\) to \(x^1\) would have produced \(Y=y^1\). Thus, PS addresses a different question from the event-specific explanations considered here. This aligns with an argument of [42] on page 7, that "... an explicit sufficiency condition is not required" when focusing on cases where both cause and effect happened. To understand why PN is so close to \(1\) even when explanation scores are only slightly above \(0\), note that \(Y=y^1\) is unlikely and thus every sufficient cause is unlikely. Thus, although other sufficient causes may exist, they are unlikely to have occurred in the particular occurrence of \(Y=y^1\) where \(X=x^1\) is also observed. This implies that, in the rare-event setting, a reasonably strong explanatory cause provides evidence for necessity.

It is therefore reasonable to search for causes with high explanation score, because such causes are likely to be necessary for the target event and therefore suitable intervention points for preventing the event, especially when the target is undesirable, such as a performance issue in a technical system. For a pathway with nodes \(B_1,\dots,B_k\) where \(B_t\) is the target, we can make this intervention interpretation precise by considering an augmented graph \(\mathcal{P}^I\) containing an intervention node \(I\) with states \(i^0,i^1\). The state \(I=i^1\) triggers the intervention \(do(\mathbf{B}_R=\mathbf{1})\) for \(I=i^1\) with \(P(i^1)=:\epsilon\). Let \(P^\epsilon(I,\mathbf{B})\) denote the joint distribution in the augmented DAG. Using Lemma 8 with \(y^1\) being the event \(B_t=1\) and \(x^1\) the intervention \(i^1\), we conclude that the probability of necessity of \(I=i^1\) for \(B_t=1\) satisfies \[\label{eq:PN} PN \geq 1- P^\epsilon(B_t=1)^{{\cal E}_{R \to t}}.\tag{13}\] For \(\epsilon\to 0\), \(P^\epsilon (\mathbf{B})\) converges to \(P(\mathbf{B})\) and \(P^\epsilon(B_t=1)\) can be replaced with \(P(B_t=1)\).

9 Why We Use Interventional Probabilities Only↩︎

One may ask why our explanations use interventional probabilities only –rung 2 in the ladder of causation [35]– while reasoning about single events may suggest using counterfactuals (rung 3). To this end, we consider the toy scenario with two binary variables encoding rare events \(X,Y\) with \(Y= X \wedge N\), where \(N\) is binary noise with \(P(N=1)\ll P(X=1)\). While the counterfactual statement “\(Y=1\) would not have occurred if \(X\) had been set to \(0\) (for that particular statistical unit)” suggests \(X\) as crucial for explaining \(Y=1\), it is of minor relevance according to the quantitative measures used here because the counterfactual relevance generalizes to only a tiny fraction of statistical units. However, for the vast majority of units, \(X\) has no effect on \(Y\), since \(N=0\) is overwhelmingly more likely than \(N=1\). While we do not in general deny the importance of counterfactuals for understanding singular events, our goal of stating explanations that are easily falsifiable and generalize to a significant fraction of statistical units lets us prefer referring to interventional probabilities only.

10 Comparison to Other Methods for Explaining Rare Events↩︎

To provide a concrete illustration with mean-based attribution, consider the logical AND gate \(Y=X \wedge N\) with \(P(X=1)=10^{-9}\) and \(P(N=1)=10^{-1}\), so that \(P(Y=1)=10^{-10}\). With our logarithmic scaling (following explanation scores), the contribution of \(X\) reads \(1 - \frac{1}{10}=90\%\), in other words its contribution is strong because it increases the probability of the output event by a large factor. By contrast, a mean-based contribution such as \(\frac{\mathbb{E}[Y \mid do(X=1)]-\mathbb{E}[Y]}{1-\mathbb{E}[Y]}\) is approximately \(10^{-1}\), which is much smaller. If we instead consider Shapley-based contributions, the contribution of \(X\) is

\[\frac{\mathbb{E}[Y \mid X=1] - \mathbb{E}[Y] + \mathbb{E}[Y \mid X=1,N=1]- \mathbb{E}[Y \mid N=1]}{2(1-\mathbb{E}[Y])} \approx 0.55\]

This gives a stronger contribution than the plain mean-based score, but it still remains substantially below the logarithmic score of \(0.9\), since of the two marginal terms are still tied to the absolute change in the mean. Since we follow arguments in [10], [27] stating that a substantially more rare factor should get significantly more attention, we prefer the logarithmic contribution analysis. Another issue with Shapley-based contributions is that they require knowing the structural causal model and are not invariant under redefining the noise: if we replace \(N\) by two variables \(N_1, N_2\), with \(Y = X \wedge N_1 \wedge N_2\) and \(P(N_1=1, N_2=1)=P(N=1)=10^{-1}\), then the Shapley contribution of \(X\) changes.

11 When to Expect Multiple Root Causes↩︎

Let us consider a simple causal DAG where \(Y\) is an additive effect of multiple independent causes: \(Y:=\sum_{i=1}^d X_i\), with \(X_i\) independently identically distributed with density \(p_X\). It is known from a large number of asymptotical results, see e.g. [43], that the shape of \(p_X\) determines whether extremes of \(Y\) tend to originate from extreme values of one of the \(X_i\) or multiples of them. It is easy to see why Gaussian \(X_i\) asymptotically result in all \(X_i\) being extreme: since the joint density is rotation invariant in \({\mathbb{R}}^d\), we can choose \(Y\) as one axis, and all directions orthogonal to \(Y\) are independent. Accordingly, conditioning on \(Y\) being large does not affect the joint distribution of the subspace orthogonal to \(Y\). Hence, there are most likely no extreme values in any normalized linear combination of \(X_i\) that is orthogonal to \(Y\). Therefore all \(X_i\) must attain large values. On the other hand, [43] show that for \(p_X\) with subexponential density, extreme values of \(Y\) originate from extrema of a single \(X_i\) in the limit of large \(y\). All the above arguments are based on the assumption that the outlier is a result of post-selection, rather than distribution shift.

If we, instead, assume that our event is a result of distribution shift, which is caused by shifts of conditional distributions at multiple nodes, we tend to ask for a common root cause behind distributional shifts happening simultaneously at actually “independent mechanisms”, cf [44], Sections 2.1 and 2.2.

12 Proof of Theorem 1↩︎

Let \(1,\dots,n\) be w.l.o.g. causally ordered according to \(\mathcal{G}\). Define the random variable \[(x_j, \mathbf{x}_{\mathrm{Pa}(j)}) \mapsto L_j(x_j,\mathbf{x}_{\mathrm{Pa}(j)}):=-\log P_\mathbf{X}\big(\tau_j(X_j)\geq \tau_j(x_j) \mid \tau_{\mathrm{Pa}(j)}(\mathbf{X}_{\mathrm{Pa}(j)}) \geq \tau_{\mathrm{Pa}(j)}(\mathbf{x}_{\mathrm{Pa}(j)})\big).\] Lemma 5 entails for all \(j\not \in R\), given an arbitrary observation \(\mathbf{x}_{\{1,\dots,j-1\}}\), the random variable \(L_j\) attains values above \(\alpha\) with probability at most \(e^{-\alpha}\). Let \((T_j)_{j\in N \setminus R}\) be independent exponentially distributed random variables with density \(p(t)=e^{-t}\). Then for any \(\mathbf{x}_{\{1,\dots,j-1\}}\), the conditional distribution of \(L_j\) given \(\mathbf{x}_{\{1,\dots,j-1\}}\), is stochastically dominated by \(T_j\). Since \((L_i)_{i\in N\setminus R , i\leq j-1}\) is a function of \(\mathbf{x}_{\{1,\dots,j-1\}}\), we also conclude that \(L_j\) is stochastically dominated by \(T_j\), given any values of \((L_i)_{i\in N\setminus R , i\leq j-1}\). Hence, \(L=\sum_{j\in N\setminus R} L_j\) is stochastically dominated by \(\sum_{j\in N\setminus R} T_j\).

The sum of \(n'\) independent identically distributed exponential distributions with parameter \(\lambda\) is the Erlang distribution \({\rm Erl}(\lambda,n')\) [45], with CDF \[F(x)= 1- e^{-\lambda x} \sum_{i=0}^{n'-1} \frac{(\lambda x)^i}{i!}.\] Since \(\lambda=1\) because we have density \(p(t)=e^{-t}\) and \(n'=n-|R|\), we conclude that \(T\) attains values greater than or equal to some \(c>0\) with probability at most \[\label{eq:convStat} p= \sum_{i=0}^{n - |R| -1} \frac{c^i}{i!} e^{-c},\tag{14}\] which applies to \(L\) too because it is stochastically dominated by \(T\).

13 Additional Results on Pathway Abstractions↩︎

13.1 Further Analysis of Example 4↩︎

We first consider the simpler pathway abstraction \(B_2 \to B_3\), where both the pathway and the cluster are given by the DAG \(B_2 \to B_3\), and we set \(P_{\mathbf{B}}(B_2,B_3):=P(B_2,B_3)\). The accuracy of the bivariate abstraction is

\[r_\mathrm{bi}=1 - \frac{\max \{D_{KL}(\mathrm{Bern}(p_0) \,\|\,\mathrm{Bern}(q_0)), D_{KL}(\mathrm{Bern(p_1) \,\|\,\mathrm{Bern(q_1))}}\}}{-\log P(Z \geq z)},\]

where \(q_0\) and \(q_1\) are computed by

\[\begin{align} \label{eq:q950} q_0:=P_\mathbf{B}(B_3=1 \mid do(B_2=0))= P_\mathbf{B}(B_3=1 \mid B_2=0)= P(Z\geq z \mid Y < y), \end{align}\tag{15}\]

\[\begin{align} \label{eq:q951} q_1:=P_\mathbf{B}(B_3=1 \mid do(B_2=1))= P_\mathbf{B}(B_3=1 \mid B_2=1)= P(Z\geq z \mid Y\geq y), \end{align}\tag{16}\]

while \(p_0\) and \(p_1\) are obtained from the fine-grained interventional probability distributions under natural micro-realization given by

\[\begin{align} \label{eq:p950} p_0:=P(B_3=1 \mid do(B_2=0))= \int^y_{-\infty} P(Z\geq z \mid do(Y=y')) p(y' \mid Y < y)dy', \end{align}\tag{17}\]

\[\begin{align} \label{eq:p951} p_1:= P(B_3=1 \mid do(B_2=1))= \int_y^\infty P(Z\geq z \mid do(Y=y')) p(y' \mid Y\geq y)dy'. \end{align}\tag{18}\]

Since \(Y\) and \(Z\) are linear functions of the jointly Gaussian variables \(X, N_Y, N_Z\), the pair \((Y,Z)\) is bivariate Gaussian with covariance

\[\Sigma_{Y,Z}=\begin{pmatrix} 1 & \gamma + \alpha \beta\\ \gamma + \alpha \beta & 1 \end{pmatrix},\]

and therefore the probabilities in (15 ) and (16 ) can be computed from the bivariate Gaussian CDF of \((Y, Z)\). To compute the interventional probability \(P(Z \geq z \mid do(Y=y'))\), consider the structural equation model

\[\begin{align} \label{eq:scm} Z^*= \beta X + \gamma Y^* + N_Z, \end{align}\tag{19}\]

where \(X\) and \(N_z\) are as in the original structural model in (9 ), and \(Y^* \sim {\cal N}(0,1)\) with \(Y^* \mathpalette{\independenT}{\perp}X, N_z\).

Then, conditional on \(Y^*=y'\), we have

\[Z^*=\beta X + \gamma y' + N_Z.\]

Hence, \[P(Z \geq z \mid do(Y=y'))=P(Z^* \geq z \mid Y^*=y').\]

Therefore,

\[\begin{align} \label{eq:b952951} P(Z \geq z \mid do(B_2=1)) = \int_y^\infty P(Z^* \geq z \mid Y^*= y')p(y' \mid Y^* \geq y') dy= P(Z^* \geq z \mid Y^* \geq y'). \end{align}\tag{20}\]

Similarly we obtain

\[\begin{align} \label{eq:b952950} P(Z \geq z \mid do(B_2=0))= P(Z^* \geq z \mid Y^* < y'). \end{align}\tag{21}\]

Note that \((Y^*, Z^*)\) is a bivariate Gaussian variable with covariance

\[\Sigma_{Y^*,Z^*}=\begin{pmatrix} 1 & \gamma\\ \gamma & \beta^2 + \gamma^2 + \sigma_{N_z}^2 \end{pmatrix},\]

since \(\mathrm{Cov} (Y^*,Z^*)= \gamma\) and \(\mathrm{Var}(Z^*)=\beta^2 + \gamma^2 + \sigma_{_z}^2\). Thus, the right-hand sides of (20 ) and (21 ) can be calculated using the bivariate Gaussian CDF of \((Y^*, Z^*)\).

The pathway explanation score of the bivariate pathway abstraction is given by

\[{\cal E}_{2 \to 3}^{\{2,3\}}=1-\frac{\log P(Z \geq z \mid Y \geq y)}{\log P(Z \geq z)}.\]

Now we consider the pathway abstraction \(B_1 \to B_3 \leftarrow B_2\), with the full DAG as the cluster and \(P_\mathbf{B}(\mathbf{B}):=P(\mathbf{B})\). For the trivariate abstraction, the accuracy is

\[\begin{align} r_{tri}= 1 - \frac{\max_{S \subseteq \{1,2,3\}}\max_{\mathbf{b}_S}D_{KL}(P(\mathbf{B}\mid do(\mathbf{B}_S=\mathbf{b}_S)) \,\|\, P_\mathbf{B}(\mathbf{B}\mid do(\mathbf{B}_S=\mathbf{b}_S)))}{-\log P(Z \geq z)}. \end{align}\]

Since \(P_\mathbf{B}\) is Markov relative to the cluster, we have the factorization

\[\begin{align} p_\mathbf{B}(b_1, b_2, b_3)=p_\mathbf{B}(b_1)p_\mathbf{B}(b_2 \mid b_1)p_\mathbf{B}(b_3 \mid b_1,b_2). \end{align}\]

Thus, \(P_\mathbf{B}(\mathbf{B}\mid do(\mathbf{B}_s=\mathbf{b}_s))\) is obtained by fixing the variables in \(S\) and multipying only the mechanisms of the non-intervened variables.

We now describe the fine-grained interventional distributions \(P(\mathbf{B}\mid do(\mathbf{B}_s=\mathbf{b}_s))\). Interventions that do not involve \(B_2\) can be simply computed using the observational Gaussian distribution of \((X, Y, Z)\). For example, under \(do(B_2=1)\), the fine-grained value of \(X\) is drawn from \(P(X \mid \lvert X \rvert \leq x)\), and \(Y\) and \(Z\) are generated by the original structural equation model. Hence,

\[\begin{align} P(B_1=1, B_2=1, B_3=1 \mid do(B_1=1)) = P(Y \geq y, Z \geq z \mid \lvert X \rvert \leq x). \end{align}\]

When \(B_2\) is intervened on, we again use structural equation model in (19 ). Then the covariance matrix of the multivariate Gaussian variable \((X, Y^*, Z^*)\) is given by

\[\Sigma_{X, Y^*, Z^*}= \begin{pmatrix} 1 & 0 & \beta\\ 0 & 1 & \gamma\\ \beta & \gamma & \beta^2+\gamma^2+\sigma_{N_Z}^2 \end{pmatrix},\]

The fine-grained interventional probabilities can therefore be calculated using Gaussian probabilities of the multivariate Gaussian variable \((X, Y^*, Z^*)\). For example, under \(do(B_2=1)\),

\[P(B_1=1, B_2=1, B_3=1 \mid do(B_2=1))=\frac{P{( \lvert X \rvert \leq x, Y^* \geq y, Z^* \geq z)}}{P(Y^* \geq y)},\]

and under \(do(B_1=1, B_2=1)\) we have

\[\begin{align} P(B_1=1, B_2=1, B_3=1 \mid do(B_1=1, B_2=1))=\frac{P( \lvert X \rvert \leq x, Y^* \geq y, Z^* \geq z)}{P(\lvert X \rvert \leq x) P(Y^* \geq y)}. \end{align}\]

Figure 4 compares the accuracy and explanation score of the bivariate abstraction \(B_2 \to B_3\) with the trivariate abstraction \(B_1 \to B_3 \leftarrow B_2\) in Example 4 for \(x=1\) and different values of \(y,z\). The parameter choice creates negative confounding: because \(\alpha < 0\), large \(Y\) tends to occur with smaller values of \(X\), while because \(\beta >0\), smaller values of \(X\) tend to decrease \(Z\). The trivariate abstraction includes \(B_1\), which captures this context, and therefore achieves higher explanation scores and accuracies in the outlier region \(y\geq 2\), \(z \geq 2\).

The code used for the numerical evaluation in Example 4 and for generating the plots in Figure 4 is available at https://github.com/AH-20/CausalPathwayExplanation.

13.2 Pathway Abstraction of DAGs With Binary Variables↩︎

Consider the causal DAG given by Figure 5, where \(X,Y,Z\) are binary variables, and the events \(X=i\), \(Y=j\), \(Z=k\) are denoted by \(x^i,y^j,z^k\), respectively, for \(i,j,k \in \{0,1\}\). Note that the DAG in Figure [fig:xyz95b] is always a valid pathway abstraction with accuracy equal to \(1\) by merging the direct and indirect links into one arrow. In this section we show that the accuracy of the pathway abstraction in Figure [fig:xyz95c] depends on the relevance of the direct link \(X\to Z\) and we derive a quantitative condition for safely ignoring the confounder \(X\) to obtain the pathway abstraction in Figure [fig:xyz95d].

First, in close analogy to [10], we define a notion of “causal contribution” for a cause effect-pair \(X \to Y\):

Note that it coincides with explanation score up to a normalization factor: \[{\cal E}\left(x^i \to y^j \right) = - \frac{C \left(x^i\to y^j \right)}{\log p(y^j)}.\]

When there are mediator nodes between the cause-effect pair in addition to the direct link, the above definition does not correctly capture the “direct” contribution of the cause node to the effect node. Therefore, we introduce a notion of direct contribution that follows our principle of logarithmic contribution analysis, although it is motivated by the average controlled direct effect in Definition 3 in [46].

We are now able to describe a condition for which we can safely ignore the confounder:

Lemma 9 shows that \(X\) can be ignored when the confounding strength is small. This happens, for example, if \(X\) has little contribution to \(Y\), or if the controlled direct contribution of \(X\) to \(Z\) changes little across the values of \(X\) once \(Y\) is fixed.

Finally, we are able to describe a condition for which we can safely ignore the direct link:

Lemma 10 shows that the direct edge \(X \to Y\) can be ignored when the controlled direct contribution of \(X\) to \(Y\) is uniformly small across all values of \(X, Y, Z\).

14 Example of a Causal Pathway Generated by LLM↩︎

Generated by Anthropic Claude Opus 4.5, with 32,000 output tokens, accessed via Amazon Bedrock. The boxes contain answers in a format that is close to the original output.

14.1 Generation of an Example with Pathway↩︎

14.2 Inferring Probability \(P(B|A)\)↩︎

14.3 Inferring probability \(P(C|B)\)↩︎

14.4 Inferring Probability \(P(D|C)\)↩︎

14.5 Inferring Probability \(P(E|D)\)↩︎

References↩︎

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1. Note that [27] already introduced the explanation score as a KL divergence, as well as generalizations in terms of other distances between distributions.↩︎

2. We require approximate equality of interventional probabilities, cf. related notions of approximate abstraction in the literature [24]. Here, however, “approximate” is defined by our logarithmic scaling, which is particularly tailored to rare events.↩︎

3. For notational convenience, we avoid superscripts like in \(B^x_1,B^y_2\), but keep the dependence on \(x\) and \(y\) in mind.↩︎

4. cf. monotonicity condition in Lemma 3.5 in [6].↩︎

5. For any \(S\), we denote by \(P_\mathbf{X}(\mathbf{B}_S)\) the pushforward measure under our binarization map for fixed \(\mathbf{x}_S\).↩︎

6. Evaluating the accuracy of these probabilities is beyond the scope of this paper; in future agentic applications, they could be inferred directly from data.↩︎