Natural Counterfactuals With Necessary Backtracking

Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-10 18:33:37

Abstract: Counterfactual reasoning is pivotal in human cognition and especially important for providing explanations and making decisions. While Judea Pearl's influential approach is theoretically elegant, its generation of a counterfactual scenario often requires too much deviation from the observed scenarios to be feasible, as we show using simple examples. To mitigate this difficulty, we propose a framework of natural counterfactuals and a method for generating counterfactuals that are more feasible with respect to the actual data distribution. Our methodology incorporates a certain amount of backtracking when needed, allowing changes in causally preceding variables to minimize deviations from realistic scenarios. Specifically, we introduce a novel optimization framework that permits but also controls the extent of backtracking with a "naturalness'' criterion. Empirical experiments demonstrate the effectiveness of our method. The code is available at https://github.com/GuangyuanHao/natural_counterfactuals.

Main paper: 
 introduction: 
 Counterfactual reasoning, which aims to answer what would have been the case if certain features of the actual situation had been different, is often used in human cognition, underpinning our ability to simulate actions, perform self-reflection, provide explanations, and inform decisions. For AI systems to mirror such human-like decision-making processes, incorporating counterfactual reasoning is crucial. Judea Pearl's structural approach to counterfactual modeling and reasoning stands as a cornerstone in machine learning. Within this framework, counterfactuals are conceptualized as being generated by surgical interventions on X that sever causal links while leaving its causally upstream variables untouched (Pearl, 2009) . Such non-backtracking counterfactual reasoning (i.e., directly performing an intervention on X and leaving X's causal ancestors intact) can yield valuable insights into the consequences of hypothetical actions. Consider a scenario: Tom on a high-speed bus fell after a sudden break and as a result injured Jerry. The non-backtracking counterfactual reasoning would tell us that if Tom had stood still (despite the sudden braking), then Jerry would not have been injured. Pearl's approach supplies a principled machinery to reason about conditionals of this sort, which are usually useful for explanation, planning, and responsibility allocation.However, such surgical interventions may not be feasible in practice and then may not help too much in self-reflection. In this example, preventing Tom's fall in a sudden braking scenario contradicts mechanisms that may be extremely hard or even impossible to disrupt, like the inertia laws of physics. This incongruity poses challenges to the practical application of such counterfactual reasoning. From a legal perspective, Tom's fall to cause Jerry's injury could be given a "necessity defense", noting that the sudden braking has left him with no alternatives (Conde, 1981) . Thus, the non-backtracking counterfactual about Tom standing still despite the sudden braking may be of very limited relevance to practical concerns in such cases.We consider the scenario where a primary goal of counterfactual reasoning is constructive -to reflect upon and better situations. With this objective in mind, it is expected to provide actionable insights. In view of this challenge to non-backtracking counterfactuals, we introduce "natural counterfactuals" in this paper, which are counterfactuals derived from interventions that are natural, in the sense of being sufficiently plausible within observed data contexts. In the above example, it is too implausible for Tom not to fall given a sudden stop. A more plausible intervention might be one targeting a prior event, such as making the bus slow down earlier, which amounts to some backtracking.Natural counterfactuals are therefore expected to align better with real-world scenarios and facilitate more practical solutions or more realistic decisions. In machine learning, while non-backtracking counterfactuals can sometimes produce unlikely or unrealistic scenarios, our methodology relies on empirically supported interventions, and hence mitigate the risk of implausible outcomes and ensure empirically grounded results.In the standard Pearlian framework, counterfactual reasoning is concerned with inferences of the consequences of a change that is due to an intervention that directly brings about the change. In our approach, however, such direct interventions will be avoided if they are not sufficiently "natural", and interventions on causally preceding variables will be invoked when necessary. Our natural counterfactuals are intended to stay closer to reality than the standard Pearlian counterfactuals, by locating changes that are feasible and effective at realizing the desired counterfactual supposition. Our approach significantly enhances the real-world applicability of counterfactual reasoning, as evidenced by experiments showing an improved quality of counterfactual generation. More specifically, this paper makes the following contributions:• We develop a general framework of what we call natural counterfactuals, which are both more flexible and more realistic than the standard framework. • We propose a novel optimization framework for generating natural counterfactuals. By combining a naturalness constraint with a principle of minimal change that discourages unnecessary backtracking, we seek to strike a best balance between natural and nonbacktracking counterfactuals. • We present a detailed method in the general framework and test it empirically. The empirical results, on both simulated and real data, demonstrate the efficacy of our method. 
related work: 
 Non-backtracking Counterfactual Generation. As will become clear, our theory is presented in the form of counterfactual sampling or generation. (Pawlowski et al., 2020b; Kocaoglu et al.; Dash et al., 2022; Sanchez & Tsaftaris) use the deep generative models to learn an SCM from data given a causal graph; these works strictly follow Pearl's theory of non-backtracking counterfactuals. Our case studies will examine some of these models and demonstrate their difficulties in dealing with unseen inputs.Backtracking Counterfactuals. Backtracking in counterfactual reasoning has drawn plenty of attention in philosophy (Hiddleston, 2005; Khoo, 2017) , psychology (Dehghani et al., 2012) , and cognitive science (Gerstenberg et al., 2013) . (Hiddleston, 2005) proposes a theory that is in spirit similar to ours, in which backtracking is allowed but limited by some requirement of matching as much causal upstream as possible. Gerstenberg et al. (2013) shows that people use both backtracking and non-backtracking counterfactuals in practice and tend to use backtracking counterfactuals when explicitly required to explain causes for the supposed change in a counterfactual. von Kügelgen et al. ( 2022) is a most recent paper explicitly on backtracking counterfactuals. The main differences between that work and ours are that von Kügelgen et al. (2022) requires backtracking all the way back to exogenous noises and measures closeness on noise terms, which in our view are less reasonable than limiting backtracking to what we call "necessary backtracking" and measuring closeness directly on endogenous, observable variables. Moreover, their backtracking counterfactuals sometimes allow gratuitous changes, as we explain in the Appendix.Algorithmic Recourse. Algorithmic recourse emphasizes providing individuals with actionable recommendations to achieve a desired outcome from a predictive model (Karimi et al., 2020a; Upadhyay et al., 2021; Karimi et al., 2020b; Rawal et al., 2020; Pawelczyk et al., 2022) . The concepts of (non-)backtracking counterfactuals and algorithmic recourse both revolve around the idea of understanding and potentially altering decisions made by predictive models. However, they target different aspects and objectives of the interpretability and fairness spectrum in AI. The primary focus here is on guiding individuals on what they can change to achieve a desired outcome from a model. Though natural counterfactuals might offer suggestions to individuals on potential interventions, they operate within a broader framework, focusing on the feasibility of these interventions rather than their direct applicability to individual cases.Counterfactual Explanations. A prominent approach in explainable AI is counterfactual explanation (Wachter et al., 2018; Dhurandhar et al., 2018; Mothilal et al., 2020; Barocas et al., 2020; Pawlowski et al., 2020a; Verma et al., 2020; Schut et al., 2021; Karimi et al., 2020a) , on which our work is likely to have interesting bearings. Most works on this topic define some sense of minimal changes of an input sample with a predicted class c such that adding the minimal changes into the input would make it be classified into another (more desirable) class. Although this paper does not discuss counterfactual explanations, our framework may well be used to define a novel notion of counterfactual explanation by requiring the counterfactual instances to be "natural" in our sense. 
notations and background: 
 We use a structural causal model (SCM) to represent the data generating process of a causal system. A SCM is a mathematical structure consisting of a triplet M :=< U, V, F >, with two disjoint variable sets U = {U 1 , ..., U N } (exogenous or noise variables whose values or probability distributions are given as inputs) and V = {V 1 , ..., V N } (endogenous or observed variables whose values are determined by other variables in the model), and a function set F = {f 1 , ..., f N }, one for each endogenous variable. Note that we assume there is an equal number of exogenous and endogenous variables, or in other words, there is exactly one distinct noise variable affecting each endogenous variable. This assumption can be relaxed but we will adopt it in this paper for simplicity. Each function, f i ∈ F, specifies how an endogenous variable V i is determined by its direct causes, PA i ⊆ V:EQUATIONWe assume the SCM is recursive, meaning that the transitive closure of the direct causal relations will not relate any variable to itself. A probabilistic SCM is a pair, < M, p(U) >, consisting of a (recursive) SCM and a probability distribution p(U) for the noise variables. Since the SCM is recursive, p(U) induces a joint distribution over U and V. A causal world is a pair < M, u >, with u a specific value setting of U.A counterfactual ponders what would happen in a scenario that differs from the actual one in a certain way. Following a standard notation, terms with with a * superscript refer to a counterfactual world. For example, u * k denotes U k 's value in a counterfactual world. Let A, B, E be sets or vectors of endogenous variables. Here is a general counterfactual question: given the observation of E = e, what would the value B have been if the value of A were a * (instead of the actual value a)? The Pearlian, non-backtracking reading of this question takes the counterfactual supposition of A = a * to be realized by an intervention on A (which will leave every causal ancestor of A in the model invariant). Given this understanding, counterfactual inference includes three steps. (1) Abduction: The noise distribution is updated based on the given evidence E = e. (2) Action: The causal model is modified, in which A is fixed to a * while keeping other components the same as before. (3) Prediction: The counterfactual outcome of B, denoted as B * , is inferred using the modified model and the updated noise distribution. In this setting, the counterfactual world is stipulated to share the same values of the noise variables with the actual world, and breaks by intervention some causal links that are present in the actual world. However, some interventions, e.g., having Tom stand still while keeping the sudden brake, might be infeasible in reality. 
overview: 
 We aim to address this issue by relaxing the stipulation of no backtracking on the one hand and minimizing the extent of backtracking on the other. The general idea is to use a "naturalness" criterion to detect the need of backtracking and use a novel conception of minimal change to determine a minimal extent of backtracking that is needed. Multiple criteria of naturalness may be explored in this context, and we will consider some in Sec. 4.2. This discussion will be followed by a theoretical discussion of minimal change in Sec. 4.3. Sec. 4.4 will unfold a unified optimization framework to realize natural counterfactuals by integrating naturalness and minimal change.We will refer to p(V i |PA i ), i = 1, ..., N , as local (observed) mechanisms, for it is in principle estimable from the observed data and the given causal structure. Note that a local mechanism also implicitly encodes the properties of noise variables; given a fixed value for PA i , noise U i entirely dictates the probability of V i . Throughout this paper, the term "local mechanism" will be used to encompass both the conditional distribution of an endogenous variable given its causal parents and the distribution of its corresponding noise variable.In our conception of natural counterfactuals, to say that A = a * is a feasible counterfactual supposition is to say that it could have occurred in a way that conforms to the local mechanisms in the actual world. In other ways, we need to assess whether any local mechanism that contributes to the realization of A = a * has seen a significant departure from that in the actual world. Let G(A) denote the A-realization set, which includes A and all its endogenous ancestors in the given SEM. Our task is to monitor the plausibility of each local mechanism within G(A).A minimal standard in this vein is to require that each local mechanism p(N|PA(N)) taking a specific value N = n be greater than 0, where ∀N, N ∈ G(A). However, this standard still allows counterfactual scenarios that are too far-fetched, even though not outright impossible, which might still be problematic in practical applications. For instance, in normal distributions, nearly all observed data fall within three standard deviations from the mean. Data points outside this range (beyond the 3-sigma area) are deemed as improbable or outlier events. To steer clear of these improbable data points, we advocate more stringent naturalness criteria, with a user-adjustable threshold that can be tailored to suit specific problems at hand.1 Many criteria may be proposed to capture our intention in one way or another. Below are some examples that strike us as sensible:Definition 1 (Measurements of Naturalness) Given a data point X = x, U x = u X , its parents taking a value, i.e., PA(X) = pa X , and its local mechanism p(X|PA(X)), four candidate measurements of naturalness are:(1) p(x|pa X )e H(X|pa X ) , where H(X|pa X ) = E[p(X|pa X )]: Here, the average naturalness of variable X is normalized to be 1 relative to p(X|pa X ). Specifically, when log(p(x|paX )) = [-H(X|pa X )], the naturalness of such a point x is 1. (2) p(x|pa X ))E[p(X|pa X )] : This metric aligns with (1) in setting the average naturalness to 1. Interestingly, the metric servers the low bound of (1). ( 3) P (u X ): This represents the the cumulative distribution function (CDF) of the noise value u X associated with X: when the value of X's parents are fixed, P (u X ) serves as an indicator of naturalness, where P is CDF of U X , i.e., P (u x ) = ux -∞ p(U X )dU X . The closer P (u X ) is to 0.5 (meaning it is farther from the distribution tails), the higher the naturalness of the local mechanism.(4) P (x|pa X ): the CDF value of x given pa X : This definition is similar to (4), but it uses the cumulative distribution function P (X) instead. Again, the further it is away from tails, the greater the naturalness of the local mechanism is.Definition (1) provides a measure of naturalness based on the concept of entropy. It quantifies the log density of a data point within the distribution, which can be interpreted as a measure of its naturalness. The relative naturalness of a data point is determined by comparing its expectation w.r.t. its distribution. Definition (2) serves as a lower bound to Definition (1) by comparing the probability of a data point with the average probability of the distribution. Specifically, (1) can be rewritten ase log(p(x|pa X ))-[-H(X|pa X )]. The term -log(p(x|pa X )) represents the measure of surprise in the information theory or say log(p(x)) can indicate possibility that x happens given pa X and thus naturalness of a data point (Ash, 2012) . The term [-H(X|pa X )] represents the average naturalness of the distribution. Therefore, log(p(x)) -[-H(X|pa X )] is the relative naturalness of a value x given its parents compared to the average naturalness of the entire distribution p(X|pa X ). To normalize the average naturalness to 1, the final naturalness indicator is defined as p(x|pa X )e H(X|pa X ) , which is invariant to scaling transformations. For example, for a root endogenous variable following Gaus-sian distribution N ∼ (X; µ, σ 2 ), when X = x, its naturalness is e σ 2 -(x-µ) 2 2σ 2. Whatever the σ is, the mean's naturalness is always e 1 2 and naturalness is reduces as data point is away from the mean. Following the same logic as (1), p(x) could also indicate naturalness and thus, (2)'s rationality is justified.(1) and ( 2)'s ideas directly depend on endogenous conditional distribution, while (3) and ( 4) go back to exogenous level. ( 3) and ( 4) is useful in practical implementations. For (3), to achieve SCMs in machine learning system, a common assumption is that the support of the distribution p(X) does not contain disjoint sets, and a nonlinear, nonparametric generative model X = f (pa X , U X ) is implemented, where f is monotonically increasing w.r.t. U X , which is usually assumed to be a standard Gaussian (Lu et al., 2020) . In this case, under each local mechanism, noise distribution follows a standard Gaussian. Data points from tails can be thought of less impossible events. Similar to (4), following the same assumption, the CDF P (x|pa X ) and the CDF P (U X ) is one-to-one mapping, i.e., specifically, P (x|pa X ) = P (u X ), where x = f (pa X , u X ). For example, if f is a linear SEM, assuming U X follows standard Gaussian distribution, p(X|pa X ) = N(X; pa X , 1) and thus P (X = x|pa X ) = P (u X ), where x = f (pa X , u X ). Hence, P (X) can also represent naturalness.These four definitions of naturalness provide different perspectives on measuring the naturalness of a data point based on its generative model. They capture various aspects such as the probability, entropy, relative naturalness, and the distribution of exogenous or endogenous variables.Accordingly, we define local ϵ-natural generation as follows and introduces an epsilon threshold to determine whether a local mechanism is considered natural.Definition 2 (Local ϵ-Natural Generation) Given a data point X = x, its parents taking a value, i.e., PA(X) = pa X , and its local mechansim p(X|PA(X)), according to Definition 2, the local mechanism p(x|pa x ) satisfies local ϵ-natural generation, where ϵ is a small constant, if one of the following conditions is satisfied:(1) p(x|pa X )e H(X|pa X ) > ϵ: The naturalness of the data point, measured by p(x) multiplied by the exponential of the entropy of the distribution, is greater than ϵ.(2) p(x|pa X )) E[p(X|pa X )] > ϵ: The naturalness of the data point, measured by the ratio of its probability to the average probability of the distribution, is greater than ϵ.(3) ϵ 2 < P (u X ) < 1 -ϵ 2 :The CDF of the exogenous variable U X 's value u x of x falls within the range( ϵ 2 , 1 -ϵ 2 ). (4) ϵ 2 < P (x|pa X ) < 1-ϵ 2 :The CDF of x given its parents' value pa X falls within the range( ϵ 2 , 1 -ϵ 2 ).These criteria provide different perspectives on naturalness and allow for flexibility in assessing the naturalness of the generative process. The first two standards focus on the relative naturalness of the data point within the distribution, while the last two standards consider the positioning of the data point within the cumulative distribution function.Note that definitions (3) and (4) focus on the tails of the distribution and aim to identify data points that deviate from the expected or typical values. In the case of a standard Gaussian distribution, the tails are considered less natural. These definitions provide a practical approach for detecting unnatural data points by leveraging the properties of the Gaussian distribution. Definition (3) is the most feasible and practical way to implement naturalness assessment, where we only need to deal with standard Gaussian distributions. It allows for the identification of data points in the tails, which are often considered less natural.Finally, we define ϵ-Natural Generation for A-realization set G(A) to judge whether all local mechanisms are natural. Here is the general definition:Definition 3 (ϵ-Natural Generation) Given a SCM containing a set X taking value x and a set G(X) containing X and its all endogenous ancestors, taking value g(X), mechanisms of G(X) satisfy ϵ-natural generation, if ∀N ∈ G(X), each local mechanism p(N = n|pa N ) satisfies local ϵ-natural generation, where n and pa N are values of N and its parents respectively, and ϵ is a small constant. (x 1 , x 2 ) = ||P (x 1 ) -P (x 2 )|| 0 or d(x 1 , x 2 ) = ||P (x 1 ) -P (x 2 )|| 1 , where P (x * ) = x * -∞ p(x)dx.The two local mechanism distances are invariant to distribution, meaning that, when p(x) is transformed to another continuous distribution p(y) by reversible transformation, the distance of transformed points x 1 , x 2 of y 1 , y 2 is the same as that of y 1 , y 2 before transformed.For example, for one endogenous nodeV k ∈ G(A), its local mechanism distance is d(u k , u * k ), where u k is the value of U k , the noise variable of V k . since two proposed distances are invariant to different distributions, the distances of different noise variables are transformed into one space to be compared. If directly using the values of noise variables, the differences are hard to compared. For example, in linear SEMs, assuming V 1 and V 2 follows Gaussian distribution and uniform distribution respectively. The same value differences on V 1 and V 2 respectively is not comparable.It is worth noting that when the d takes L 0 or L 1 norm, it expresses different preferences on changes of mechanisms. Using L 0 norm tends to change the fewer number of nodes. Using L 0 norm prefers smaller total changes of mechanisms.Necessary Backtracking. Causally prior variables should be changed as little as possible, in order to minimize the extent of backtracking. Moreover, some nodes may have more downstream nodes than others and should be changed less, as they will propagate changes to their descendants. Hence, we propose a notion of necessary backtracking. In the global mechanism distance, changes of each noise variables are independent and the distance on each single node has the same weight, which may sometimes lead to the bigger changes of causal earlier nodes. Hence, different weights should be set for different variables. Take global mechanism distance as an example. The modified distance is as follows:D M (g(A), g(A) * ) = u k ∈u g(A) ,u * k ∈u g(A) * w k d(u k , u * k ) (2)w k is the weight of the node U k , defined as the number of decedents of V k denoted as N D(V k ).The number of variables influenced by a variable can be used as a weight in the distance measure. For example, in a causal graph where where A cause B and C is the confounder of A and B. If A = a * , A's and B's weight is 1 and 2 respectively.We can now formulate the problem of generating natural counterfactuals as an optimization problem.The objective is to minimize the distance of actual world and counterfactual world, i.e., the distance between G(A) * and G(A), while satisfying the constraint of A = a * and ϵ-natural generation of G(A) * , where A * is the counterfactual counterpart of A. The optimization framework denoted as Natural Counterfactual Optimization is defined as follows:min g(A) * D(g(A), g(A) * ) s.t. A = a * s.t. g(A) * satisfying ϵ-natural generation (3)Where g(A) is value of set G(A) in the actual world, g(A) * is the counterfactual value to be optimized, ϵ is a small constant, and D(•) is a distance measure discussed above.g(A) * may admit multiple optimal solutions and its solution set is denoted as S g(A) . The distribution over S g(A) can be stipulated as:EQUATIONThe  
case studies: 
 In this section, we put our theoretical framework into practice, examining its validity, utility, and performance through a series of empirical experiments on four simulated datasets and two publicly available datasets, MorphoMNIST and 3DIdentBOX. Please refer to the more detailed discussions in Sec. A.On each experiment, we computed the average error between generated outcomes and ground-truth outcomes, which revealed a significant reduction in error when using our natural counterfactuals. This improvement stems from our approach's ability to perform necessary backtracking to determine plausible interventions when direct interventions are infeasible, while non-backtracking counterfactuals consistently employ direct interventions, even when they are implausible. (2019) . Given the learned SCMs and a data point from the test set as evidence, we do interventionswith random values in non-backtracking counterfactuals or do changes in our natural counterfactuals either on n 1 or on n 2 within their support. For our natural counterfactuals, given a small constant ϵ = 10 -4 , we use Eqn. 7 (refer to the concrete method in D of the Appendix) to learn feasible interventions, with w ϵ = 10 4 and the setting of other weights following Eqn. 7. We report the Mean Absolute Error (MAE) between our learned counterfactual outcomes and groundtruth outcomes on n 2 or/and n 3 on the test dataset repeated multiple times with multiple random seeds. Notice there may be no feasible interventions for some changes, as we have claimed, and thus we only report outcomes with feasible interventions, which is within the scope of our natural counterfactuals.Findings on Toy 1. We first do changes or interventions on n 2 . (1) Prediction error comparison illustration on a single data point. In Fig. 1 , we evaluate the performance of counterfactual outcome estimation on a particular sample as an illustration. The green point represents the value of (n 1 , n 2 ) after doing change on n 2 in natural counterfactuals, the purple point indicates the value of (n 1 , n 2 ) after doing hard intervention on n 2 in non-backtracking counterfactuals, and the blue scatter plot shows the ground-truth support of (n 1 , n 2 ). We calculate the absolute error of n 3 's outcome given the value of the green point and the purple point, respectively, and find the error of the green point (with natural counterfactuals) is much smaller than that of the purple point (with non-backtracking counterfactuals), since the green point is a more feasible intervention compared with the purple one. In Fig. 1 (a), we also observe that the two points have the same value of n 2 , while our natural counterfactuals use necessary backtracking to backtrack to n 1 and allow interventions on n 1 . (2) Prediction error comparison on whole Test Set. As shown in Fig. 1 (b), a larger number of points with non-backtracking counterfactuals deviated from the line y = x, meaning that n 3 's prediction outcome is very different the ground-truth value. On the contrary, our outcomes are mainly located around the line y = x (a small number of points may be far away from the line y = x since the learned SCM is not perfect even within support). Quantitative Results. In Table 1 , when putting change or do on n 2 , Our MAE error is better than the non-backtracking by 61.6%, verifying the effectiveness of our approach.We also do changes or interventions on n 1 , where our natural counterfactuals will not do backtracking since n 1 is the root cause. Even in this setting, our result outperforms the non-backtracking, since our approach exclusive the points not satisfying -natural generation.Considering More Casual Graph Structures. We also consider other causal graph structures and hence generate three more simulation datasets, i.e., Toy 2 -4. Toy 2 -4 contains 2, 4, and 3 nodes respectively, The results are reported in Table 1 . More details are in Sec. A.1. In this section, we investigate two types of counterfactuals using the MorphoMNIST dataset, comprising three variables: (t, i, x). The causal graph, as presented in Fig. 2 (a) of the Appendix, indicates that t (the thickness of the digit stroke) influences both i (intensity of the digit stroke) and x (images). Additionally, i serves as the direct cause for x. A sample from this dataset is illustrated in Fig. 2 (b ). The dataset encompasses 60,000 images in the training set and 10,000 in the test set.Our approach aligns with the experimental settings of the simulation experiments detailed in Sec. 5.1, with two notable exceptions. Firstly, for learning counterfactuals, we deploy two state-of-the-art deep learning models, specifically V-SCM (Pawlowski et al., 2020b) compute the absolute error between the parent values post either hard or feasible intervention and their predicted equivalents. This calculation is based on the images that the learned SCM produces when provided with (t, i) inputs.Quantitative Results of change(i) or do(i) and Ablation Study on Naturalness Threshold ϵ.We employ two models, V-SCM and H-SCM, to execute counterfactuals with varying values of ϵ using change(i) or do(i). As depicted in Table 2 , our error diminishes as ϵ increases when using the same inference model, because a higher ϵ opts for more feasible interventions. For instance, when considering H-SCM alongside change(i) and do(i), our method reduces errors from 48% at ϵ = 10 -4 to 59% at ϵ = 10 -2 , compared to backtracking.In this task, we utilize the 3DIdentBOX dataset from the collection cited in (Bizeul et al., 2023) . Specifically, we focus on Weak-3DIdent and Strong-3DIdent. Both share the causal graph displayed in Fig. 4 We adopt the exact same setting as in the MorphoMNIST experiments. Here, with ϵ = 10 -3 , we employ H-SCM as the inference model. We intervene or alter (d, β, γ) . The results are presented in Table 3 . In both datasets, our approach outperforms the non-backtracking method. However, in Strong-3DIdent, our method has a larger margin over the non-backtracking approach since nonbacktracking faces more unfeasible interventions when executing hard interventions using Strong-3DIdent. We also visualize results on Strong-3DIdent, and these are displayed in Fig. 5 , confirming the effectiveness of our approach. 
conclusion and discussion: 
 This study presents novel concepts of naturalness criteria and minimal changes for generating practical natural counterfactuals. These prioritize feasible, data-backed interventions. By devising an optimization problem that minimizes world scenario distances while ensuring ϵ-natural generation, we offer a structured approach to natural counterfactual reasoning. Our method's efficacy is underscored by diverse deep learning-based case studies. In the case of arbitrary machine learning systems that do not incorporate our causal graph, users have the option to rely on established background causal knowledge acquired through experiments, expert insights, or other authoritative sources. Alternatively, users can choose to employ data-driven methods to discover causal relationships, as long as these methods are implemented with appropriate assumptions and considerations taken into account. It is important to note that our approach significantly diverges from prior-based backtracking counterfactuals (von Kügelgen et al., 2022) . In prior-based backtracking counterfactuals, interventions are limited to noise variables, potentially resulting in unnecessary alterations. Conversely, our approach selectively employs backtracking only when essential, aiming to minimize changes when direct intervention is unviable. For a more comprehensive understanding of these distinctions, please refer to Sec. C and Sec. B.Eqn. 7. We report the Mean Absolute Error (MAE) between our learned counterfactual outcomes and ground-truth outcomes on n 2 or/and n 3 on the test dataset repeated multiple times with multiple random seeds. Notice there may be no feasible interventions for some changes, as we have claimed, and thus we only report outcomes with feasible interventions, which is within the scope of our natural counterfactuals.Findings on Toy 1. We first do changes or interventions on n 2 . (1) Prediction error comparison illustration on a single data point. In Fig. 1 , we evaluate the performance of counterfactual outcome estimation on a particular sample as an illustration. The green point represents the value of (n 1 , n 2 ) after doing change on n 2 in natural counterfactuals, the purple point indicates the value of (n 1 , n 2 ) after doing hard intervention on n 2 in non-backtracking counterfactuals, and the blue scatter plot shows the ground-truth support of (n 1 , n 2 ). We calculate the absolute error of n 3 's outcome given the value of the green point and the purple point, respectively, and find the error of the green point (with natural counterfactuals) is much smaller than that of the purple point (with non-backtracking counterfactuals), since the green point is a more feasible intervention compared with the purple one. In Fig. 1 (a), we also observe that the two points have the same value of n 2 , while our natural counterfactuals use necessary backtracking to backtrack to n 1 and allow interventions on n 1 . ( 2) Prediction error comparison on whole Test Set. As shown in Fig. 1 (b), a larger number of points with non-backtracking counterfactuals deviated from the line y = x, meaning that n 3 's prediction outcome is very different the ground-truth value. On the contrary, our outcomes are mainly located around the line y = x (a small number of points may be far away from the line y = x since the learned SCM is not perfect even within support). Quantitative Results. In Table 5 , when putting change or do on n 2 , Our MAE error is better than the non-backtracking by 61.6%, verifying the effectiveness of our approach.We also do changes or interventions on n 1 , where our natural counterfactuals will not do backtracking since n 1 is the root cause. Even in this setting, our result outperforms the non-backtracking, since our approach exclusive the points not satisfying -natural generation.Considering More Casual Graph Structures. We also consider othrt causal graph structures and hence generate three more simulation datasets, i.e., Toy 2 -4. In Toy 2, there are two variables (n 1 , n 2 ) and n 1 causes n 2 . As shown in Table 5 , even in such a simple case, our approach still advances the non-backtracking when putting do or change on n 1 . Toy 3 contains four variables, where n 1 is the confounder of n 2 and n 4 and n 2 causes n3 which is also the parent node of n 4 . Three variables are in Toy 4 and they form a chain, i.e., n 1 causes n 2 and then n3. Similarly, in these experiments, ours also performs better. 
 
 Appendix: In this section, we study two types of counterfactuals on the dataset called MorphoMNIST, which contains three variables (t, i, x). From the causal graph shown in Fig. 2 (a), t (the thickness of digit stroke) is the cause of both i (intensity of digit stroke) and x (images) and i is the direct cause of x. 2023), First, trained on the dataset, parent predictors given a value of x can predict parent values, i.e., (t, i)'s, and then measure the absolute error between parent values after hard intervention or feasible intervention and their predicted values, which is measured on image the Learned SCM generates given the input of (t, i). Quantitative Results of change(i) or do(i).We use V-SCM to do counterfactual task of change(i) (where ϵ = 10 -3 ) or do(i) with multiple random seeds on test set. In Table 6 , the first column shows the MAE of (t, i), indicating our results outperform that of nonbacktracking. Next, we focus on the rest fourcolumn results. In both types of counterfactuals, we use the same value i in do(i) and change(i). Hence, after inference, we know which image satisfying ϵ-natural generation in the two types of counterfactuals. In "NCO=1" of the table, NCO indicates the set of counterfactuals after natural counterfactual optimization. Notice that NCO set does not mean the results of natural counterfactuals, since some results do still not satisfy ϵ-natural generation after natural counterfactual optimization. "NCO=1" mean the set containing data points satisfying ϵ-natural generation and "NCO=0" contains data not satisfying ϵ-natural generation after natural counterfactual optimization. Similarly, "NB=1" means the set containing data points satisfying naturalness criteria. (NCO=1, NB=1) presents the intersection of "NCO=1" and "NB=1". Similar logic is adopted to the other three combinations. The number of counterfactual data points are 10000 in two types of counterfactuals.In (NCO=1, NB=1) containing 4135 data points, our performance is similar to the non-backtracking, showing natural counterfactual optimization tends to backtrack as less as possible when hard interventions have satisfied ϵ-natural generation. In (NCO=1, NB=0), there are 3135 data points, which are "unnatural" points in non-backtracking counterfactuals. After natural counterfactual optimation, this huge amount of data points become "natural". In this set, our approach contributes to the maximal improvement compared to the other three sets in Table 6 , improving 60.7% and 47.6% on thickness t and intensity i. The number of points in (NCO=0, NB=1) is zero, showing the stability of our algorithm since our approach will not move the hard, feasible intervention into unfeasible intervention. Two types of counterfactuals perform similarly in the set (NCO=0, NB=0), also showing the stability of our approach. Ablation Study on Naturalness Threshold ϵ. We use two models,V-SCM and H-SCM, to do counterfactuals with different values of ϵ. As shown in Table 7 , our error is reduced as the ϵ increases using the same inference model, since the higher ϵ will select more feasible interventions.In this task, we use more practical public datasets, 3DIdentBOX which contains multiple datasets (Bizeul et al., 2023) . We use one of them, Weak-3DIdent and Strong-3DIdent. Their share the same We follow the exactly same setting as in the MophoMNIST experiments. Here, with ϵ = 10 -3 , we use H-SCM as inference model. We do intervention or change on (d, β, γ). The results are shown in Table 8 , in both dataset, our approach can perform better than the non-backtracking one but , in Strong-3DIdent, ours exists bigger margins over the non-backtracking method, since nonbacktracking encounter more unfeasible interventions when doing hard interventions using Strong-3DIdent. We also do visualization on Strong-3DIdent. In Fig. 5 Non-backtracking counterfactuals only do a direct intervention on target variable A, while our natural counterfactuals do backtracking when the direct intervention is implausible. Notice that when the direct intervention on A is already plausible, our procedure of natural counterfactuals will be automatically distilled to the non-backtracking counterfactuals. In this sense, non-backtracking counterfactual reasoning is our special case.(1) Intervention Approach and Resulting Changes:Prior-based Backtracking Counterfactuals: These counterfactuals directly intervene on noise/exogenous variables, which can lead to unnecessary changes in the counterfactual world. Consequently, the similarity between the actual data point and its counterfactual counterpart tends to be lower. In short, prior-based backtracking counterfactuals may introduce changes that are not needed.Natural Counterfactuals: In contrast, our natural counterfactuals only engage in necessary backtracking when direct intervention is infeasible. This approach aims to ensure that the counterfactual world results from minimal alterations, maintaining a higher degree of fidelity to the actual world.(2) Counterfactual Worlds:Prior-based Backtracking Counterfactuals: This approach assigns varying weights to the numerous potential counterfactual worlds capable of effecting the desired change. The weight assigned to each world is directly proportional to its similarity to the actual world. it is worth noting that among this array of counterfactual worlds, some may exhibit minimal resemblance to the actual world, even when equipped with complete evidence, including the values of all endogenous variables. This divergence arises because by sampling from the posterior distribution of exogenous variables, even highly dissimilar worlds may still be drawn.Natural Counterfactuals: In contrast, our natural counterfactuals prioritize the construction of counterfactual worlds that closely emulate the characteristics of the actual world through an optimization process. As a result, in most instances, one actual world corresponds to a single counterfactual world when employing natural counterfactuals with full evidence.(3) Implementation Practicality:Prior-based Backtracking Counterfactuals: The practical implementation of prior-based backtracking counterfactuals can be a daunting challenge. To date, we have been prevented from conducting a comparative experiment with this approach due to uncertainty about its feasibility in practical applications. Among other tasks, the computation of the posterior distribution of exogenous variables can be a computationally intensive endeavor. Furthermore, it is worth noting that the paper (von Kügelgen et al., 2022) provides only rudimentary examples without presenting a comprehensive algorithm or accompanying experimental results.Natural Counterfactuals: In stark contrast, our natural counterfactuals have been meticulously designed with practicality at the forefront. We have developed a user-friendly algorithm that can be applied in real-world scenarios. Rigorous experimentation, involving four simulation datasets and two public datasets, has confirmed the efficacy and reliability of our approach. This extensive validation underscores the accessibility and utility of our algorithm for tackling specific problems, making it a valuable tool for practical applications.In Sec. 4, we propose various theories for natural counterfactuals, including ϵ-natural generation's definition, which A-realization set G(A) should follow when doing interventions on causally earlier variables, and tools for minimal changes containing various distance definitions to measure the difference between actual world and counterfactual world. We also provide how to do natural counterfactuals by the Natural Counterfactual Optimization (NCO) framwork. Here, we specify one method to apply to machine learning system, which can be as an example to help future researchers or practicers apply these high-level theories.Problem Setting. Given data collected from real world and known causal graph, one can learn a SCM to model the data distribution. Given the full evidence E = e, we want to know, if there had been A = a * , what would have happened to B using the learned SCM?A Specific Optimization Method. Below is the equation of optimization: w k L (1) (P (u k ), P (u * k ))EQUATIONWhere the optimization parameter is changed from exogenous values into the values u * G(A) of noise U * G(A) , which is corresponding variables of G(A). For simplicity, we use A as subscript as indicator of terms related to A, instead of number subscript. In practice, the Lagrangian method is used to optimize our objective function and the corresonding loss is as below:L(u * G(A) \ u * A ; u G(A) ) = u k ∈u G(A) ,u * k ∈u * G(A)w k L (1) (P (u k ), P (u * k ))+ w ϵ u * k ∈u * G(A)max((ϵ -P (u * k ), 0) + max(ϵ + P (u * k ) -1, 0))EQUATIONwhere the first term is the measure of distance between two world and the second term is the constraint of ϵ-natural generation. where w ϵ is a constant hyperparameter to punish noise values in the tails of noise distributions. Notice that u * A is not optimized explicitly, since the value pa * A is fully determined by other noise values u * G(A) \ u * A and a * is an constant, with reversible function f -1 A , u * A is fully determined by u * G(A) \ u * A and a * . In the next section, we use Eqn. 7 to do natural counterfactual optimization in multiple case studies of machine learning.The term "naturalness" can be defined in various ways. In our current context, our definition of naturalness is exclusively determined by the information contained within Structural Causal Models (SCMs), rather than being influenced by the particular methods employed to carry out interventions. However, we are interested in investigating alternative definitions of naturalness in our future research. 


Decision: The reviewers and authors held a vigorous discussion. Overall the consensus is that the paper presents a novel approach to an important problem. The approach is especially promising in the era of big generative model, as the problem of generating "natural counterfactuals" comes up in many applications of such models. During the discussion multiple comments were made that would improve the paper, including its motivation and presentation - I trust the authors will follow through on these in the finalized version.

Reviews: review: paper_summary: paper_summary: This paper addresses a key limitation of non-backtracking counterfactual reasoning in causal inference. The authors argue that Pearl's framework often generates unrealistic scenarios. To solve this, they propose "natural counterfactuals," which allow controlled backtracking to ensure scenarios remain realistic.They introduce Feasible Intervention Optimization (FIO), a novel framework that generates natural counterfactuals by incorporating a naturalness constraint. This ensures counterfactual instances are plausible given the observed data distribution. The authors also propose distance measures to minimize backtracking while achieving the desired outcome.
strength_weakness: strength_weakness: - FIO involves several parameters and choices (naturalness criteria, distance measures), which may require careful tuning and consideration for different applications.

- The paper mainly focuses on contrasting natural counterfactuals with non-backtracking ones. A more thorough comparison with existing backtracking approaches is missing in the experiment section.

- The paper mainly focuses on contrasting natural counterfactuals with non-backtracking ones. A more thorough comparison with existing backtracking approaches is missing in the experiment section.
questions: questions: - How sensitive is the proposed method to violations of the assumptions about the underlying SCM?

- What is the computational complexity of FIO and how does it scale with the size of the causal graph and the dataset?
limitations: limitations: See weaknesses and questions
paper_summary: paper_summary: The paper takes the recently developed idea of backtracking counterfactuals and applies it to improve the realistic generation of counterfactuals from data, which is known to be a hard task as the standard, non-backtracking, counterfactuals lie out of the distribution and generative models perform badly on those.
strength_weakness: strength_weakness: There are some technical issues which need to be addressed before the paper can be accepted.
questions: questions: Full disclosure: I reviewed a previous version of this paper for last year's conference, and the current version is much much better. Still, there are some issues which the authors should address. Some of them might be mistakes on my part due to my misunderstanding, others are questions for clarification, others are problems that need to be solved. 

First I present some larger issues, followed by some minor issues as they appear in the order of the paper.

1: I find the motivation rather odd, which is that the supposed benefit of natural counterfactuals is that they are easier to learn. But surely the main focus should be: which counterfactual (backtracking, non-backtracking, partial backtracking) is appropriate/meaningful in a given situation, rather than “which one can we learn”. Simply put, we shouldn't just focus on learning something because it's easy to learn, the thing learned should also be sensible. The paper actually remains silent on this issue, or rather, it seems to imply that non-backtracking counterfactuals are always the only sensible ones, and their natural counterfactuals are therefore just a heuristic. The reason I say this is implied is because in the experiments the non-backtracking one is taken as the ground truth. This is at odds with the related literature cited, which in fact argues that sometimes the backtracking counterfactual is the "ground truth", i.e., semantically it is what is meant with a counterfactual query. It would be good if the authors could be more explicit on where they stand on this issue.

2: There are several implicit assumptions that should be addressed. 

- It is assumed that there is a unique an(A)*, yet the definitions given do not guarantee unicity. 

- Initially it seems as if it is also assumed that an an(A)* always exists, which is not the case. Although this is made clear later, it would be good to flag this early on. 

3: 159: This I don’t understand, because it violates the idea that C has to include A. Take the trivial case in which a*=a.  Then you will get the empty LBF intervention, which of course does not include A.

4: Building on the previous point, what if we have A=a, and yet A=a was extremely unlikely (so in the far-end of the tail of the distribution). Now consider the counterfactual a*=a. Will we not get that there is no solution? And if so, isn't that strange? This could be mitigated by combining the distance criterion and the naturalness criterion in a more weighted fashion, rather than having the latter be a necessary criterion and only then apply the distance one. This would bring it more in line with [30], see their 3.17 in particular. 

5: Continuing, note that Choice (2) seems to result in a variation of 3.17, except that the distance is here between endogenous variables. Note though that a distance on endogenous variables can easily induce a distance between exogenous variables, by considering the two endogenous states that would result from two exogenous states, and thus this is not an essential difference.

6: Th4.1 is described as being about the identifiability of counterfactuals, but it is not, because it already assumes knowledge of do(C=c*), and that is part of what needs to be identified. Furthermore, it is then said that the theorem confirms that counterfactuals fall within the support of the observational distribution, but this ignores the earlier point that an LBF need not exist, and thus we are only guaranteed to identify counterfactuals in the case that it does. 

7: I did not have time to closely examine the experiments, except for the following: all the results are only about those counterfactuals which happen to be in the scope of the natural counterfactuals, and the others are excluded. Yet it seems crucial to know how many were excluded in this manner in order to evaluate how useful in practice this method is, so this should be reported as well.

Minor issues:

81: "a most recent paper" I’d say a more accurate description is that [30] was the first paper to formally introduce backtracking counterfactuals within the causal models framework, and thus the current paper builds on that one. (Also, the usefulness for counterfactual explanations is also something that is explicitly discussed in [30].)

100: So the paper is limited to assuming independent and unique exogenous variables. Note that this is not the case in [30].

121: "In this paper..." Why? The generalization seems trivial.

175: This assumes that all variables are real-valued. Yet later it is mentioned that not all variables need to be at the same scale. Isn't that inconsistent?

221: Why restrict to a single variable A all of a sudden?

Th4.1: Doesn't the independence of U_i and Pa_i already follow from the independence of the exogenous variables?

Th4.1: Why is there no mention of the distance criterion for do(C=c*)?


Typos:

110: "distribution" -> "distributions"

123: "date"

234: "encourage"

303: "datasets, which"
limitations: limitations: Yes
paper_summary: paper_summary: The paper presents a framework for generating "natural counterfactuals" that are more feasible within the support of  the training data distribution. This approach includes controlled backtracking through an optimization method  that uses a "naturalness" criterion as a constraint.
strength_weakness: strength_weakness: My first concern is with the term "natural counterfactual." Both Pearl's non-backtracking counterfactuals and backtracking counterfactuals use an algorithm that begins with the prior probabilities of the exogenous variables. In the abduction step, the counterfactual posterior is computed based on the observed facts in the non-backtracking case, or on both the observed facts and observed counterfacts in the backtracking case. However, this paper samples from the observational distribution of the training data. 

Theorem 1 builds on prior work $[17]$ where the posterior distribution has been discussed, mentioning that due to the monotonicity assumption of $f_i$ with respect to 
$U$, the posterior distribution is a point mass on $𝑈=u$. However, the experiments in this paper sample from the prior distributions of the structural causal model (SCM), which means it is not a true counterfactual distribution.

This leads me to the purpose of the paper. It uses optimization to find a point within a high-density area of the data manifold generated by the observational distribution, given a request to change a value. Which use case would require this result?
For example, Pearl’s non-backtracking counterfactuals address questions on actual cause, causal necessity, and personalized policy. If performing $do(A)$ is not feasible, it does not imply that a feasible data point within the training data distribution can answer these questions. Further, the concept of "naturalness" is not based on the immutability of the variable but on the probability density of the variable given its parents. For example, the variable "Age" might meet the criterion based on its probability density, but it remains immutable.

References:

[17] Chaochao Lu, Biwei Huang, Ke Wang, José Miguel Hernández-Lobato, Kun Zhang, and
      Bernhard Schölkopf. Sample-efficient reinforcement learning via counterfactual-based data
     augmentation. arXiv preprint arXiv:2012.09092, 2020.
questions: questions: (1) Line 46: “When interventions lead to unrealistic scenarios relative to the training data,  predicting counterfactual outcomes in such scenarios can be highly uncertain and inaccurate [12].” 
Reference [12] addresses selection bias in potential outcome counterfactuals, in particular,  when the treatment group does not match the control group. It is not related to SCM counterfactuals.

(2) Line 48 “This issue becomes particularly pronounced when non-parametric models are employed, as they  often struggle to generalize to unseen, out-of-distribution data [27]. “ 
Reference [27] discusses the issue of i.i.d. in machine learning, particularly when the training data distribution differs from the interventional distribution, leading to out-of-distribution scenarios. How is the current paper related  to the discussion in [27]?

(3) Appendix F: “Therefore, [30]’s backtracking counterfactual does not reduce to Pearl’s counterfactual even when $A^∗ =\emptyset$.”
This is incorrect. Neither non-backtracking nor backtracking is a special case of the other. Appendix A of [30] presents a unified framework for counterfactual reasoning that integrates both backtracking and non-backtracking counterfactuals, as suggested by an area chair at that conference. Also, if $A^∗ =\emptyset$, then there is no counterfactual query.

References:

[12] Negar Hassanpour and Russell Greiner. Learning disentangled representations for counterfactual
      regression. In International Conference on Learning Representations, 2019.

[27] Bernhard Schölkopf, Francesco Locatello, Stefan Bauer, Nan Rosemary Ke, Nal Kalchbrenner,
      Anirudh Goyal, and Yoshua Bengio. Toward causal representation learning. Proceedings of the
     IEEE, 109(5):612–634, 2021

[30] Julius von Kügelgen, Abdirisak Mohamed, and Sander Beckers. Backtracking counterfactuals.
      arXiv preprint arXiv:2211.00472, 2022.
limitations: limitations: NA
paper_summary: paper_summary: The authors propose a novel approach for generating causally valid and *natural* counterfactuals. These counterfactuals are *natural* in that they are close to the observed data manifold. This is achieved by allowing a certain amount of backtracking, which involves tracing back upstream causal effects on variables.

The authors essentially propose a more flexible version of Judea Pearl’s $do$-calculus, by replacing the explicitly non-backtracking $do$-operator with a $change$-operators that represents the notion of a *least-backtracking feasible intervention* (LBF). Intuitively, LBF interventions take into account the effects of causal ancestors instead of considering the intervention itself as entirely exogenous.

In order to compute LBF interventions, the authors propose a *feasible intervention optimization* that targets minimal changes to the causal ancestors of some variable $A$ while satisfying certain naturalness constraints. These constraints ensure that the generated counterfactual remains anchored in the observed data manifold. Through experiments involving both synthetic and real-world datasets, the authors demonstrate that their proposed method yields more “natural” counterfactuals than the non-backtracking alternative approach.

The proposed method has some clear links to the literature on counterfactual explanations, which are mentioned but not thoroughly explored here. This is an interesting avenue for future research and this work constitutes a solid first step in that direction. Overall, the paper seems to make a strong contribution to the field of Causal Inference, although I am not a subject expert.
strength_weakness: strength_weakness: - No standard errors reported in Tables, so difficult to assess how substantial the performance differences are after accounting for noise.
- Link to CE is mentioned, but the authors could have gone into more detail. In particular, you propose that the literature on CE could benefit from incorporating your definition of *naturalness* (which I don’t disagree with) but you fall short of comparing it to existing definitions in CE. See also my related questions.
- The figures and annotations are too small (and in general, both figures and tables currently make the paper look a little crammed).
questions: questions: ### Link to CE

- How does this approach compare to existing approaches to causal algorithmic recourse (e.g. [MINT](https://arxiv.org/abs/2002.06278))?
- How do the proposed *naturalness* constraints compare to *plausibility* (e.g.  [Artelt and Hammer](https://arxiv.org/abs/2002.04862), [REVISE](https://arxiv.org/abs/1907.09615)) and *feasibility* (e.g. [FACE](https://arxiv.org/abs/1909.09369)) constraints in CE/algorithmic recourse? Equation 2 in your paper looks quite similar to Equation 5 in [Artelt and Hammer](https://arxiv.org/abs/2002.04862).

### Other questions

- The sentence spanning from line 81 to 86 is very long and it’s easy to get lost in it. Consider splitting this into 2/3 sentences.
- Figures and annotations on page 8 are very small (annotations are barely legible).
- Could you highlight the single point from Figure 1 (a) also in panel (b)? To illustrate that in (b) you essentially repeat the experiment over all test samples.
limitations: limitations: The authors mention that the assumption of invertibility is a small limitation, but not theoretically problematic. The work may have other limitations that I have failed to recognized, as I’m not a subject expert.

Comments: title: Thank You for Your Kind Feedback
Your Feedback Is Much Appreciated
Your Feedback Would Be Appreciated
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Response 1
Reply to rebuttal
Thank you again and please consider updating the score
Response 1-4
Response 1-3
Response 1-2
Response 1-1
Response to rebuttal
comment: Thank you very much for increasing your rating, and for sharing your remaining concerns. 

Regarding Question/Suggestion (1), we will follow your advice to focus more on the concept of "natural counterfactuals" and be clearer about the sense of backtracking used in this paper. With due respect, we do not think the term "backtracking" has become so fixed to a specific meaning in a specific approach (among many possible alternatives) that its general meaning of "not keeping the temporal or causal upstream fixed in making counterfactual supposition" should not be used anymore. But thank you for your reminder of the potential of causing confusion; we will make extra efforts to make the meanings of terms clear.

For Question (2), we agree that our approach does not fully address the original question in that example, if the original question is a non-backtracking interpretation of the counterfactual query and the non-backtracking interpretation turns out to violate our standard for picking out an intervention to realize the counterfactual supposition. That is NOT our aim. Our motivation is that in such cases in which we have good reasons to doubt that we can answer the non-backtracking interpretation of a counterfactual query (say, what if it were raining hard *and, implicitly, other variables in its causal upstream were kept the same*?) reliably based on available data, instead of returning a bad answer or simply saying "do not know", we can also (or even should) tell the user that if they adopt an answerable interpretation of the question that involves some backtracking (say, what if it were raining hard *and the atmospheric pressure were low*? We use atmospheric pressure instead of temperature in our example because atmospheric pressure is more likely to be a causal ancestor of weather in the given causal diagram), there is an answer we can reliably offer and here is the answer. Moreover, the user may have started with such an interpretation to begin with, as counterfactual queries are often ambiguous or under-specified. In any case, we reiterate that we are dealing with a distinctive interpretation of a counterfactual query and do not aim to address the "original" non-backtracking interpretation of the query, when that "original question" is demonstrably hard to answer given the available data.

Thank you once again for your time and excellent questions!
Thank you so much for all of your feedback and for appreciating the value of this work. We greatly enjoyed our discussions and learned a lot.

We will avoid the misleading characterization of fully backtracking counterfactuals as appealing to interventions on exogenous variables, and will make it clear that our aim is not to propose a "correct" semantics for counterfactuals, but instead one that is useful in some situations and for some purposes.

Once again, we are very grateful for your time and insights.
Based on your technical efforts, I’ve decided to increase my rating. However, in my humble opinion, the following two main concerns remain:

(1) The terms "non-backtracking" and "backtracking" have specific meanings in fields like ML, philosophy, and cognitive science. To avoid confusion, it might be better to talk only about a new concept "Natural Counterfactuals." For instance, backtracking refers to observational counterfactuals and deals with use cases like, "Had I observed $n_2=0.19$, what would $n_3$ 
be?". Here, you track the difference between factual and counterfactual values back to the parents. In contrast, your paper begins by requesting a change to $n_2=0.19$, and then your optimization framework finds new values for both $n_1$ and $n_2$, resulting in a simultaneous change for two variables. This approach doesn't align with the established definitions of backtracking or non-backtracking.

(2) What specific problem or use case does your approach address? For example, in lines 50-51, you mention autonomous driving. If the observational distribution is sunny weather with 40°C, then asking what would happen if the weather changed to heavy rain is a valid non-backtracking counterfactual, which would likely involve out-of-distribution samples. Replacing it with cloudy weather and 35°C, which is closer to the observational distribution, doesn't fully address the original question.

The XAI recourse literature uses optimization methods to find counterfactual values based on causal structures. While these techniques are typically applied in supervised learning, similar use cases may exist in other areas, such as generative AI (GenAI).
Dear Reviewer JBox,

Thank you once again for your valuable comments. Your suggestions on clarifying our motivations and inference procedure were very helpful. We are eager to know if our responses have adequately addressed your concerns.

Due to the limited time for discussion, we look forward to receiving your feedback and hope for the opportunity to respond to any further questions you may have.

Yours Sincerely,

Authors of Submission 4096
Thanks for these further clarifications. 

I still have some reservations about your description of fully backtracking counterfactuals, because their philosophical motivation was precisely to avoid anything even resembling an intervention, (strictly speaking, there is backtracking all the way to the initial conditions of the universe, it just so happens that in causal models the buck stops at the exogenous variables), but I think this is an issue that is mostly orthogonal to the paper and thus not that relevant. 

I agree that the epistemic perspective does in and of itself justify the value of your contribution, and I would even suggest that in the final version you try to remain as neutral as possible regarding what the correct semantics for counterfactuals is (or even what the correct semantics is in some specific context), because the contribution of the paper is not really about that and therefore the less you assume the better. For what it's worth, personally I do expect that in some contexts, your partially backtracking counterfactuals are also the correct counterfactuals, in the sense that people would indeed be inclined to interpret counterfactuals in that manner if the circumstances warrant. But perhaps that's something to evaluate in future work, together with some psychologists.
**1.2: "because they appeal to interventions..." Fully backtracking counterfactuals do not appeal to interventions at all, on the contrary, they're directly at odds with the interventional semantics of Pearl. Instead, they appeal to a change in the initial conditions, which is not an intervention because the initial conditions are not determined by causal equations and thus no "laws of nature" are broken. So aside from the issue as to which styles of counterfactuals are easier to estimate without access to the ground truth SCM, there is the more fundamental issue of which type of counterfactual is the formal representation of the informal scientific or even natural language query that we are formalizing.**

We agree. In our view, informal scientific and natural language queries in the form of counterfactual conditionals are very often ambiguous or under-specified (in their antecedents), the disambiguation or specification of which depends on contexts. We do not pretend to be able to resolve this fundamental issue. We also agree that fully backtracking counterfactuals do not appeal to interventions in the usual Pearlian sense, for those are restricted to endogenous variables and are associated with breaking endogenous mechanisms. However, formally speaking, changing the values of exogenous variables is analogous to intervening on exogenous variables, in the sense that they change those variables without affecting the mechanisms for other variables in the model (the invariance of the mechanisms for other variables is for many the crucial hallmark of an intervention). In fact, some authors explicitly advocate applying the notion of intervention to exogenous variables as well. It is in this spirit that we say fully backtracking counterfactuals appeal to "interventions" to exogenous variables. We now see that it is potentially misleading, and we will be more careful with the wording.


**1.3: To be clear, the above is not a criticism of the method itself, I'm simply trying to get conceptual clarity on what the assumed true underlying semantics for counterfactuals is. I guess this is the question that would resolve it: say you do have access to the ground truth SCM, and thus perfect inferences can be made. And now we want to reason about some counterfactual "If A were a\*". How would you compute it? Using do(A=a\*), or using change(A=a\*)?**

In such a case, yes, it will depend on how to interpret the counterfactual supposition, and we agree that in many contexts, the non-backtracking interpretation will probably stand out as the most salient (or as the philosopher David Lewis once wrote, "standard") resolution. In other words, in such a case, the epistemic motivation for adopting our interpretation is of course annulled. Still, the more ontic motivation described above, having to do with the feasibility of carrying out a non-backtracking intervention, may still be relevant. 

Thank you again for making these important and subtle questions so clear. We will improve our statements of the motivations accordingly.


**Re: 3: You should change the wording, because it does not match your explanation. The difference between a and a is the emptyset.**
 
That's right. It is our sloppiness and thank you for catching it. We will rephrase carefully.


``We look forward to receiving your feedback and would appreciate the opportunity to answer any additional questions you may have.``
We are grateful for your time and insights, and really appreciate the opportunity to further clarify our responses to your questions.

**1.1: You speak of the non-backtracking counterfactual being "unrealistic", but here unrealistic seems to mean: if we try to estimate the standard, Pearl-style, interventional counterfactual without access to the ground truth SCM, we get bad answers. So the underlying semantics is still the standard interventional one, it's just that we don't have a good estimator for it. (If I understand correctly, the ground truths for both the non-backtracking and the natural counterfactuals are both expressed in terms of an interventional counterfactual, it's just that they are compared to different interventions.)** 

This is an excellent point. Our semantics is indeed still based on interventions, and you are exactly right that the difference between ours and the standard Pearlian semantics is that they (often) pick out different interventions to realize the counterfactual supposition in the antecedent of a counterfactual conditional. In other words, given a counterfactual query: what if $A = a^*$?, we have different interpretations on how this change is supposed to be realized (or how the supposition $A=a^*$ is to be further specified). In Pearl's semantics, the interpretation is simple and elegant: The change is to be realized by an intervention on A alone (or the supposition is to be specified as "$A=a^*$ *and every variable in the causal upstream of A remains the same*"); in our semantics, by contrast, we impose a standard for picking out an intervention to realize the supposition, which may require intervening on some of $A$'s causal antecedents as well (and in this sense involve backtracking). So, although they are both based on interventions, they still constitute different interpretations of the original counterfactual supposition. 

Regarding our claim that the non-backtracking counterfactuals are sometimes "unrealistic", you gave a very nice statement of one important reason why we think this. For the purpose of this paper, perhaps this epistemic motivation is the most relevant and easiest to see, and we will consider focusing on this motivation and elaborating on its significance. In your original review, you seemed to indicate that this motivation is not sufficiently compelling, on which we respectfully disagree. If reliable answers to a counterfactual query interpreted in a certain way are very difficult or even impossible to obtain, it seems to us an excellent reason to develop a different but related interpretation that can still serve the desired functions to a good extent and admit more reliable answers. Imagine someone comes with a question: If his weight were decreased to 70kg, what would be his risk of diabetes? Suppose, given the causal assumptions and data we have, we cannot reliably answer the non-backtracking interpretation of this question. It seems that instead of just saying that we do not know, we can or even should adopt an answerable interpretation of the question and tell him something we can reliably offer, say, if he decreased weight to 70kg *and exercised for more than 150 minutes a week*, his risk would be such and such.   

For what it is worth, we also think a perhaps more important reason for sometimes regarding non-backtracking counterfactuals as "unrealistic" is that their way of interpreting the counterfactual supposition, say, $A=a^*$, as intervening on A alone while keeping every causal ancestor in the model intact, may be extremely difficult or practically impossible to realize (e.g., having a person stand still on a bus while keeping the sudden braking in the toy example we mentioned in the paper), as indicated by the rarity of the scenario in the data, and for that reason is not very relevant or helpful for guiding actual actions or decision-making. Imposing a naturalness criterion is one way to ensure that the picked out interventions are at least feasible to carry out in light of the evidence we have.
I thank the authors for their rebuttal. It has prompted some more questions. 

Re1: 
Thanks for the clarification. It still does not really address the question what the correct semantics for counterfactuals is though. 

1.1: You speak of the non-backtracking counterfactual being "unrealistic", but here unrealistic seems to mean: if we try to estimate the standard, Pearl-style, interventional counterfactual without access to the ground truth SCM, we get bad answers. So the underlying semantics is still the standard interventional one, it's just that we don't have a good estimator for it. (If I understand correctly, the ground truths for both the non-backtracking and the natural counterfactuals are both expressed in terms of an interventional counterfactual, it's just that they are compared to different interventions.)

1.2: "because they appeal to interventions..." Fully backtracking counterfactuals do not appeal to interventions at all, on the contrary, they're directly at odds with the interventional semantics of Pearl. Instead, they appeal to a change in the initial conditions, which is not an intervention because the initial conditions are not determined by causal equations and thus no "laws of nature" are broken. So aside from the issue as to which styles of counterfactuals are easier to estimate without access to the ground truth SCM, there is the more fundamental issue of which type of counterfactual is the formal representation of the informal scientific or even natural language query that we are formalizing.

1.3: To be clear, the above is not a criticism of the method itself, I'm simply trying to get conceptual clarity on what the assumed true underlying semantics for counterfactuals is. I guess this is the question that would resolve it: say you _do_ have access to the ground truth SCM, and thus perfect inferences can be made. And now we want to reason about some counterfactual "If A were a*". How would you compute it? Using do(A=a*), or using change(A=a*)?

Re: 3: You should change the wording, because it does not match your explanation. The difference between a and a is the emptyset.
Dear Reviewer JBox,

Thank you once again for your prompt feedback. We hope your concerns have been addressed. Your comments have substantially helped improve our paper. 

If you think your questions have been properly addressed, we would be immensely grateful if you could *reconsider and update your recommendation*. 

If there are other questions, we hope to have opportunities to respond to them.


Sincerely,

Authors
Thank you for your elaborate response! I do not have any further questions at this point.
**W(2)-2: It's also worth noting that this approach isn't performing or interventions but rather finding the closest points within the observational distribution.**

As explained in Questions W(1)-1 and W(2)-1, our method also performs interventions, following the same three-step inference procedure as non-backtracking counterfactuals: updating the posterior distribution, modifying the SCM by intervention, and conducting inference on the modified SCM. The key difference is that we perform a least-backtracking feasible intervention on set $C$, which usually includes some of $A$'s causal ancestors in addition to $A$. If an intervention on $A$ alone is feasible, then $C = A$.

In the example from W(1)-1, we perform interventions on both $n_1$ and $n_2$ simultaneously, whereas non-backtracking counterfactuals perform an intervention only on $n_2$. Specifically, in natural counterfactuals, both $n_1$ and $n_2$ have their links to their respective parents severed. Since $n_1$'s only parent node is $u_1$, the link between $n_1$ and $u_1$ is disconnected, and $n_1$ is directly assigned the value $-0.02$. Similarly, $n_2$'s links to $u_2$ and $n_1$ are severed, and $n_2$ is set to $0.19$.

As noted in Question W(2)-1, when intervention $do(A)$ alone is feasible, i.e., $C = A$, our counterfactual outcome matches the non-backtracking outcome. For example, given the evidence $(n_1, n_2, n_3) = (-0.59, 0.71, -0.37)$, $do(n_2 = 0.50)$ is feasible, meaning that $(n_1 = -0.59, n_2 = 0.50)$ falls within the support of the observable distribution. In this instance, the inference is the same for both natural and non-backtracking counterfactuals. During the second step of inference, $n_2$'s links to $u_2$ and $n_1$ are severed, and $n_2$ is set to $0.50$.

To reiterate, our approach is a *causal* approach, relying on a given causal structure represented by a DAG. It is not simply minimizing a non-causal distance within the observational distribution. 

**Questions:**

**Q(2): You wrote: "One of our concerns about non-backtracking counterfactuals, despite their novelty and elegance, is precisely their frequent requirement to do out-of-sample generalization." Out-of-sample generalization is not a concern of non-backtracking (interventional) counterfactuals; instead, it is a necessary feature of it.**

As explained in Question W(2)-1 above, being out-of-sample (in the sense used in our paper) is NOT a necessary feature of non-backtracking counterfactuals. For example, unlike $do(n_2 = 0.19)$ given the evidence $(n_1, n_2, n_3) = (-0.59, 0.71, -0.37)$, the intervention $do(n_2 = 0.50)$ is not out-of-sample, meaning that $(n_1 = -0.59, n_2 = 0.50)$ falls within the support of the observable distribution.

Additionally, out-of-sample scenarios can pose challenges under certain conditions. For example, non-parametric models routinely struggle and often fail to generalize to out-of-sample data points, as demonstrated in our experiments, among others.

**Q(3): I read [30] again. They have Property 1 (preference for closeness). In other words, the closest (most similar) world to an actual world is itself. Please note that this is not important for the main issues of the paper.**

Yes, property 1 in [30] states that the highest probability density is assigned to the same value. However, during inference, they will sample not only the data point with the highest probability but also other data points with lower probability density. Technically, since [30] assumes continuous variables, the probability of sampling the exact closest world is zero (e.g., the value $a$ when $a^* = a$), whereas the probability is positive for sampling values around $a$. But we agree this point is not important for their main purposes.


``Thank you again for your useful feedback and precious time! We are eager to have further discussions with you and address your remaining concerns, if any.``
**W(2)-1: I still don't see a clear use case or problem statement that can be effectively addressed using the results of your optimization framework. Even in the Toy 1 example, if we make the request $do(n_2=0.19)$, our interest lies in the interventional distribution. This request is necessarily out-of-distribution. How does finding values of $n_1$ and $n_2$ that satisfy the optimization actually solve the problem at hand?**

Thank you for elaborating on your concern. A general counterfactual query takes the following form: Given evidence $E$, what would the value of $B$ have been if $A$ had taken the value $a^*$? Non-backtracking counterfactuals, fully backtracking counterfactuals, and natural counterfactuals offer three different interpretations of this general query.

Specifically, in non-backtracking counterfactuals, the counterfactual supposition $A = a^*$ is interpreted to be realized by directly intervening on $A$ alone, i.e., using $do(A)$, as you mentioned. In fully backtracking counterfactuals, [30] interprets the counterfactual supposition to be realized by changing (which amounts to intervening on) exogenous variables. In our natural counterfactuals, we in general realize the counterfactual supposition by intervening on $C$, which includes $A$ and possibly some of $A$'s causal antecedents, to ensure that the intervention is feasible in our sense. As we can see, the three methods use different interventions about the supposition in a counterfactual query, and neither fully backtracking counterfactuals nor our natural counterfactuals interpret a counterfactual query as necessarily a query about the distribution resulting from intervening on $A$ alone.

In our framework of natural counterfactuals, we aim to generate counterfactuals that are "natural" with respect to the actual distribution but otherwise stay as close to non-backtracking as possible. Therefore, we use Feasible Intervention Optimization to find the least-backtracking feasible intervention on $C$ within the support of the actual distribution by minimizing the extent of backtracking, as explained in Question W(1)-1. For example, in Question W(1)-1, the set $C = (n_1, n_2)$, obtained through our optimization, is used to answer the query.

Additionally, similar to our natural counterfactuals, backtracking counterfactuals in [30] essentially use data points from the observed distribution to answer $A = a^*$ as well. The difference is that we intervene on endogenous variables, which is potentially still useful for guiding actions, while [30] intervenes on exogenous variables, which is not useful for guiding actions, as far as we can see.

You wrote: *"Even in the Toy 1 example, if we make the request $do(n_2 = 0.19)$, our interest lies in the interventional distribution. This request is necessarily out-of-distribution."*  In the sense of "out-of-distribution" used in our paper, we respectfully think this is not true. An intervention, though cutting out some mechanisms originally in place, does NOT necessarily result in an instance outside the support of the observed distribution. For example, although the request $do(n_2 = 0.19)$ in the Toy 1 example is indeed out-of-distribution in our sense, not every intervention is out-of-distribution. For example, given the evidence $(n_1, n_2, n_3) = (-0.59, 0.71, -0.37)$, $do(n_2 = 0.50)$ is feasible, resulting in the counterfactual $(n_1 = -0.59, n_2 = 0.50)$ being within the support of the observable distribution. In our natural counterfactuals, we actually distinguish between in-distribution $do(A = a^*)$ and out-of-distribution $do(A = a^*)$ and adjust out-of-distribution cases by performing a least-backtracking feasible intervention on $A$'s ancestors $C$.
**W(1)-3: However, the term "counterfactual" has been used in other contexts, similar to this paper, where an optimization method finds the closest point to a given data point. This approach is also seen in works like those by Wachter et al. and subsequent related papers.**

Thank you for the opportunity to clarify the differences between our paper and Wachter et al's work and much of the literature on counterfactual explanation. We have cited the work of Wachter et al. and related papers, noting that our paper and their work on counterfactual explanations fundamentally differ in purpose, techniques, and cognitive frameworks.

The term "counterfactual" in such counterfactual explanations is unrelated to "counterfactual" in causal inference, as they do not rely on causality or the SCM framework. For more details, please refer to Section III-C in [Wachter et al.](https://arxiv.org/pdf/1711.00399), which explains that counterfactual explanations do not employ causal assumptions. Typically, in counterfactual explanations, given an input (image) $x$ and an output (class) $y$ from a model, the goal is to find another input $x'$ that leads to a different output $y'$ while ensuring $x'$ is as similar as possible to the original input $x$ based on their value difference. Although counterfactual explanations are unrelated to causality, they borrow causal terminology, such as "counterfactual," to refer to $x'$. This helps explain the classification behavior of the given model. 

We can summarize the key differences between our paper and counterfactual explanations as follows:

- **Different Purposes and Cognitive Frameworks**: Based on a general machine learning framework, counterfactual explanations aim to explain model behavior by finding another input with minimal changes that results in a different class. In contrast, similar to Pearl's counterfactuals, natural counterfactuals perform counterfactual inference to obtain counterfactual outcomes based on a causal framework, where we also consider the feasibility of interventions and prefer the least backtracking.

- **Different Technologies**: Abstractly, our method infers an outcome given an input, while counterfactual explanations find an input given a target output.

- **Differences from a Causal Perspective**: Although Wachter et al.'s counterfactual explanations are unrelated to causality, from a causal perspective, they often address anti-causal problems, such as classifying an image $x$ into a category $y$, where $y$ is treated as the cause of $x$, as commonly seen in causal inference literature. Since counterfactual explanations do not build on the SCM framework, even though they try to minimize the value difference between $x'$ and $x$, it is difficult to interpret the distance from a causal perspective, unlike in our approach, which is explicitly based on causal assumptions as given in a causal DAG.
We are truly grateful for your prompt feedback and for providing more details regarding your original concerns. We also appreciate your acknowledgment that "naturalness" can have various interpretations and that you liked the explanation for Reference [12]. We are pleased to address your remaining questions and look forward to your further feedback.

**W(1)-1: Let's consider the experiment involving the toy 1 example: ... $(n_1, n_2, n_3)=(-0.59, 0.71, -0.37)$. Due to the monotonicity of the functions with respect to U, I calculated the values $(u_1, u_2, u_3)=(-0.59, 0.36, -0.93)$. As a result, the posterior counterfactual distribution becomes a point mass (Dirac delta) centered at $(u_1, u_2, u_3)=(-0.59, 0.36, -0.93)$. This means that sampling from this posterior distribution will always gives us the same values, $(u_1, u_2, u_3)=(-0.59, 0.36, -0.93)$. However, if I understand correctly, you are sampling 10,000 times from the prior distribution of the exogenous variables, which is the standard Gaussian.**

Thank you for this question, and please see our response below.

First, *our paper does not mention sampling 10,000 times from the prior distribution of the exogenous variables.* Instead, our test set comprises 10,000 data points, and we form 10,000 counterfactual queries to assess the effectiveness of counterfactual inference. Specifically, for each query, we use one data point as evidence, such as $(n_1, n_2, n_3) = (-0.59, 0.71, -0.37)$, and randomly sample a value for $n_2$ from its distribution as the counterfactual supposition, such as $n_2 = 0.19$.

Second, for any query, our inference procedure can be seen as following Pearl's three-step process: (1) Update the noise distribution given the evidence to a posterior; (2) Modify the SCM using the identified "natural intervention"; (3) Conduct inference using the posterior noise distribution and the modified SCM. As stated in Line 146, the only difference from the non-backtracking inference lies in Step (2), where we perform a least-backtracking feasible intervention on the variable set $C$, which consists of some causal ancestors of the variable set $A$ in addition to $A$.  Notably, when the intervention on $A$ alone already satisfies our criteria, we have $C = A$, yielding the non-backtracking counterfactual. 

Specifically, in our method as well as in Pearl's, *the first step* involves updating the prior exogenous distribution $p(U)$ to the posterior distribution given the evidence, and in the example you mentioned, would yield a point mass distribution on $U=(-0.59, 0.36, -0.93)$, as you noted. In our implementation, normalized flows ensure that by inputting $(n_1, n_2, n_3) = (-0.59, 0.71, -0.37)$ into our model, we obtain the unique posterior value of $U$ (as functions are invertible).

*The second step* involves modifying the SCM. In non-backtracking counterfactuals, an intervention $do(n_2 = 0.19)$ is applied, without changing anything in $n_2$'s causal upstream. In contrast, natural counterfactuals may invoke some backtracking if needed, and in this case would apply a least-backtracking feasible intervention $do(n_1 = -0.02, n_2 = 0.19)$ obtained through our Feasible Intervention Optimization, which would yield an instance that satisfies our "naturalness" criterion. 

*In the final step*, both methods utilize the updated noise distribution (now a single point) and the modified SCM to perform inference and obtain the outcome value of $n_3$.

We have refined the presentation of our inference procedure in the updated manuscript. Thanks again for giving us the opportunity to clear misunderstanding.

**W(1)-2: In the absence of an SCM, certain estimates can be made under specific conditions if both observational and interventional data are available, as discussed in a paper by Jin Tian and Judea Pearl. I did not observe any application of non-backtracking or backtracking counterfactual reasoning in this paper.**

Sorry, we are not sure what you meant by "application of non-backtracking or backtracking counterfactual reasoning"--your further feedback would be helpful. In this paper, we made some simplifying assumptions, compared to the classical work on (non-backtracking) counterfactual inference by Pearl and his collaborators (Tian, Shpister, Bareinboim, etc.), especially the assumption of no latent confounding (our causal diagram is a DAG with no bi-directed arcs), along with the assumptions of access only to observational distribution and monotonicity. As a result, we need not invoke the full machinery of those seminal works. Still, as explained above, we are effectively following Pearl's three-step procedure.  

Perhaps we misunderstood your concern here and, if so, would very much appreciate your elaboration.
Thanks for the detailed rebuttal, I appreciate the effort and have no further questions at this point
Thank you for the responses; I appreciate your efforts.

Here is my response:

(1) Let's consider the experiment involving the toy 1 example:The prior distribution of the exogenous variables is assumed to be a standard Gaussian. The factual observation is given as $(n_1, n_2, n_3) = (−0.59, 0.71, −0.37)$. Due to the monotonicity of the functions with respect to U, I calculated the values ($u_1, u_2, u_3) = (-0.59, 0.36, -0.93)$. As a result, the posterior counterfactual distribution becomes a point mass (Dirac delta) centered at ($u_1, u_2, u_3) = (-0.59, 0.36, -0.93)$. This means that sampling from this posterior distribution will always gives us the same values, $(u_1, u_2, u_3) = (-0.59, 0.36, -0.93)$. However, if I understand correctly, you are sampling 10,000 times from the prior distribution of the exogenous variables, which is the standard Gaussian.

Counterfactual reasoning, whether non-backtracking or backtracking, generally requires knowledge of the Structural Causal Model (SCM). In the absence of an SCM, certain estimates can be made under specific conditions if both observational and interventional data are available, as discussed in a paper by Jin Tian and Judea Pearl. I did not observe any application of non-backtracking or backtracking counterfactual reasoning in this paper. However, the term "counterfactual" has been used in other contexts, similar to this paper, where an optimization method finds the closest point to a given data point. This approach is also seen in works like those by Wachter et al. and subsequent related papers.

(2) I still don't see a clear use case or problem statement that can be effectively addressed using the results of your optimization framework. Even in the Toy 1 example, if we make the request $do(n_2=0.19)$, our interest lies in the interventional distribution. This request is necessarily out-of-distribution. How does finding values of $n_1$ and $n_2$ that satisfy the optimization actually solve the problem at hand? It's also worth noting that this approach isn't performing $do(n_1)$ or $do(n_2) $ interventions but rather finding the closest points within the observational distribution. 

(3) I agree that there could be different different interpretation of "Naturalness". Thanks!

Questions:

(1) Thanks for the explanation.

(2) You wrote: "One of our concerns about non-backtracking counterfactuals, despite their novelty and elegance, is precisely their frequent requirement to do out-of-sample generalization."

Out-of-sample generalization is not a concern of non-backtracking (interventional) counterfactuals; instead, it is a necessary feature of it.

(3) I read [30] again. They have Property 1 (preference for closeness). In other words, the closest (most similar) world to an actual world is itself. Please note that this is not important for the main issues of the paper.