A New Approach to Backtracking Counterfactual Explanations: A Unified Causal Framework for Efficient Model Interpretability
Authors: Pouria Fatemi
Ehsan Sharifian
Mohammad Hossein Yassaee
Venue: ICML 2025 poster (2025)
Keywords: Interpretability
Explainable Artificial Intelligence
Causal Inference
Counterfactuals
Structural Causal Models
Primary area: social_aspects->accountability_transparency_and_interpretability

Abstract: Counterfactual explanations enhance interpretability by identifying alternative inputs that produce different outputs, offering localized insights into model decisions. However, traditional methods often neglect causal relationships, leading to unrealistic examples. While newer approaches integrate causality, they are computationally expensive. To address these challenges, we propose an efficient method called BRACE based on backtracking counterfactuals that incorporates causal reasoning to generate actionable explanations. We first examine the limitations of existing methods and then introduce our novel approach and its features. We also explore the relationship between our method and previous techniques, demonstrating that it generalizes them in specific scenarios. Finally, experiments show that our method provides deeper insights into model outputs.

A New Approach to Backtracking Counterfactual Explanations: A Unified Causal Framework for Efficient Model Interpretability
Pouria Fatemi 12
Ehsan Sharifian
3 Mohammad Hossein Yassaee 4
Abstract
Counterfactual explanations are a widely used tool for inter- pretability. They address two main questions:
Counterfactual explanations enhance interpretabil- ity by identifying alternative inputs that produce different outputs, offering localized insights into model decisions. However, traditional methods often neglect causal relationships, leading to un- realistic examples. While newer approaches inte- grate causality, they are computationally expen- sive. To address these challenges, we propose an efficient method called BRACE based on back- tracking counterfactuals that incorporates causal reasoning to generate actionable explanations. We first examine the limitations of existing methods and then introduce our novel approach and its fea- tures. We also explore the relationship between our method and previous techniques, demonstrat- ing that generalizes them in specific scenarios. Finally, experiments show that our method pro- vides deeper insights into model outputs.
1. "Wy did the model produce this outcome?"
2. "What changes can lead to different outcome?" (Karimi et al., 2022).
These explanations offer localized insights by highlighting minimal modifications to input features that would alter the model's output (Wachter al., 2017; Karimi al., 2020a). For instance, in the context of loan applications, coun- terfactual explanation could recommend increasing one's income reducing debt to secure approval.
Despite their benefits, traditional counterfactual methods often overlook causal relationships between features, which can lead impractical or unrealistic suggestions (Slack et al., 2021). For example, advising someone lower their income while increasing savings ignores the causal depen- dency between these factors. This limitation reduces the practical value of such explanations. Causal algorithmic recourse (Karimi et al., 2021) incorporates Interventional Counterfactuals (ICF) to produce more realistic outputs, but this approach is typically computationally expensive and difficult to scale.
1. Introduction
Machine learning (ML) has become a core technology in areas such as healthcare, finance, and autonomous sys- tems (Bhoi al., 2024; Xie et al., 2024; Sancaktar al., 2022). Although ML models are generally very effective, their limited interpretability is still significant obstacle (Jethani et al., 2021). Understanding why model gener- ates a specific prediction is crucial for trust, fairness, and accountability (Miller, 2019; Zhang & Bareinboim, 2018; Von Kügelgen et al., 2022; Karimi et al., 2023). This need especially clear in high-stakes domains like medical diag- nosis or loan approval, where decisions can lead to serious consequences (Doshi-Velez & Kim, 2017).
Backtracking Counterfactuals (BCF) (Von Kügelgen et al., 2023) present new way to define counterfactuals in causal inference. We propose a new framework for generating counterfactual explanations using backtracking counterfac- tuals. Our method combines causal reasoning with compu- tational efficiency, enabling to produce actionable expla- nations scale. The key contributions of our work are as follows:
We analyze the limitations of existing counterfactual methods, including their inability to handle causal de- pendencies and their high computational costs.
'Department of Mathematics, Technical University of Mu- nich, Germany "Munich Center for Machine Learning, Ger- many 'Department of Electrical Engineering, École Polytechnique Fédérale de Lausanne, Switzerland 'Department of Electrical Engi- necring, Sharif University of Technology, Tehran, Iran. Correspon- dence to: Mohammad Hossein Yassace <yassacc @ sharif.
We introduce our novel method, BRACE: Backtracking Recourse and Actionable Counterfactual Explanations, that leverages backtracking counterfactuals to provide actionable and meaningful explanations.
We show that our new approach unifies existing meth- ods in certain scenarios.
Proceedings of the nnnd International Conference Machine Learning. Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
We demonstrate through experiments that our method provides better insights into model behavior.

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A Unified Causal Framework for Efficient Model Interpretability
This paper is organized as follows. Section introduces fundamental concepts in causal inference, counterfactual reasoning, and the problem definition. Section 3 reviews prior work on counterfactual explanations and interpretabil- ity. Section details our proposed framework. Section 5 explores the relationship between backtracking and interven- tional counterfactuals, while Section examines how our method connects to existing approaches. Section 7 discusses metric selection and our optimization method. We present experimental results in Section 8, followed by conclusions and future directions in Sections 9 and 10, respectively.
( ,.2" "....... 2 CF) must agree with CF on all indices in A. Intuitively, xFF addresses the question: "What would the variables X have been if X took the values XA in- stead of the observed values xAA""
We focus on two main ways to form such counterfactuals, both described by random variables X CFF; the interventional approach and the backtracking approach. Below is a concise explanation of these two methods.
Interventional Counterfactuals. In the interventional method, we force the antecedent x by modifying the system's structural functions S to create new set SCF (SCF, 1 ........ F.FF). Specifically, we fix each I.P to be 2 for E A. while keeping f = fi for all i & A. This process is similar to making a direct change in the causal mechanism of the variables in A, referred to as a hard intervention.
2. Preliminaries and Problem Statement
In this section, we review the notion of Structural Causal Models (SCMs) (Pearl, 2009), discuss interventional ver- sus backtracking counterfactuals, and formally define the problem setting.
Backtracking Counterfactuals. By contrast, backtrack- ing counterfactuals preserve the original structural assign- ments S and instead adjust the latent variables U. To enforce CF XA. introduce a modified set of latent variables UCF These are drawn from backtracking conditional distribution P&(UCT U) (Von Kügelgen et al., 2023), which controls how closely UCF resembles the original U. Once we obtain Uer we derive the resulting distribution of X CF (given x and x xFF by marginalizing over all possible values of UCF
2.1. Structural Causal Models (SCMs)
A SCM C == (S, Pu) describes set S of causal relation- ships among variables through structural equations:
X; = f(()))t) Ut) i i= 1...,,, n,
(1)
where X Xpa(i) are the parents variables (direct causes) of X,, and Uj are independent noise terms sampled from a distribution Pu. These relationships are represented by Directed Acyelic Graph (DAG) G, which governs the observational distribution PK (Peters et al., 2017).
Both interventional and backtracking perspectives provide valuable insights into counterfactual reasoning but rely on distinct causal reasoning paradigms. Here, we only gave a brief overview of these two approaches. Their precise definitions appear in Appendix A.
The acyelic structure of G ensures that each X, can be expressed as a deterministic function of U. This results in a unique mapping from U to X, denoted by:
X F(U).
2.3. Problem Definition
We examine a complex model (e.g., a deep neural network) designed for classification tasks. This model is represented as h: Rdd > {0, 1,. m,). where for a given input x, the model predicts h(x) y-
commonly referred to the reduced-form expression. The function F() translates the distribution of latent variables U into the distribution of observed variables X. We as- sume causal sufficiency, implying no hidden confounders are present.
The input x is assumed to follow a SCM C = (S, Pt), where the structural equations S are fully known. Formally, if U denotes the latent (noise) variables of the SCM, the input X is generated as X = F(U), with the function F(.) explicitly defined. The components of are mutually independent and F.)) is invertible. The goal is find a counterfactual input CF that satisfies:
Additionally, we adopt a Bijective Generation Mechanism (Nasr- Baasr-Esfahanyee et al., 2023), which assumes that fo fi (xxx(t):))) is invertible for fixed patt)- This ensures the existence of the inverse mapping F (.), allowing to recover:
U- = F (X).
(3)
1. CF is similar to x,
2.2. Interventional and Backtracking Counterfactuals
2. CF CF + y= l1(x), and
3. the causal structure of the input variables is maintained.
Let x be the observed value, and let CF = -x i E A) be an alternative set of values for a subset A C (1,2,...,nk A full counterfactual vector CF =
In essence, y CF represents the desired outcome of the model, and the task is to determine the nearest plausible input that
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A Unified Causal Framework for Efficient Model Interpretability
would produce this outcome. This problem definition sheds light on why the model predicted y instead of yfF and provides a localized understanding of the model's behavior around x.
variables while preserving causal dependencies, thereby en- suring consistency with the structural causal model. This approach has been extended through practical algorithms, such as Deep Backtracking Explanations (Kladny et al., 2024), enabling computation of backtracking counterfactu- als in high-dimensional settings.
3. Related Work
Our approach belongs to the category of example-based methods, focusing on counterfactual explanations. It is di- rectly comparable to methods such as Counterfactual Expla- nations (Wachter et al., 2017), Causal Algorithmic Recourse (Karimi et al., 2021), Backtracking Counterfactual Explana- tions (Von Kügelgen et al., 2023), and Deep Backtracking Explanations (Kladny et al., 2024). Below, we briefly review and critique these methods.
Methods for interpretability are generally classified into feature-based and example-based approaches (Molnar, 2020). Feature-based techniques attribute model predictions to input features, offering global or local interpretability. For instance, SHapley Additive exPlanations (SHAP) (Lundberg & Lee, 2017) decompose predictions into additive contri- butions of features. Extensions like Causal Shapley Values (Heskes al., 2020) and Asymmetric Shapley Values (Frye et al., 2020) incorporate causal dependencies or relax sym- metry assumptions, respectively, to enhance the interpre- tive granularity and address redundancies. Local surrogate models, such as Local Interpretable Model-Agnostic Ex- planations (LIME) (Ribeiro et al., 2016), provide localized, model-agnostic explanations by approximating the behavior of black-box models for individual predictions.
Counterfactual Explanations: The method in (Wachter et al., 2017) generates counterfactuals through the following optimization:
arg min xF
dx CF x)
(4)
s.t. )(x = CF
Example-based methods focus on understanding models through data points. Prototypes and criticisms (Kim et al., 2016) identify representative and atypical samples, while Contrastive Explanations (Dhurandhar et al., 2018) highlight minimal features that sustain or alter predictions. Counter- factual explanations (Wachter et al., 2017), a prominent example-based approach, aim to find minimal modifications to input features that result in different model outputs. These methods are inherently model-agnostic, localized, and intu- itive for decision support systems.
A key drawback of this approach is its failure to account for causal dependencies among input variables, often leading to counterfactuals that are unrealistic infeasible. For exam- ple, in loan approval scenario, it may suggest decreasing age while increasing education level, violating causal re- lationships. Although these counterfactuals minimize the distance to the original input, they offer little practical guid- ance for future improvements and fail to provide actionable insights.
Causal Algorithmic Recourse:
The method in (Karimi et al., 2021) addresses feasibility by optimizing the follow- ing:
In recent years, causality has played a growing role in in- terpretability. Causal Algorithmic Recourse (Karimi et al., 2021) generates actionable and realistic counterfactuals by respecting causal structures. This approach ensures the plausibility of counterfactuals by adhering causal de- pendencies in the data. Subsequent research has extended this framework to address various challenges. For instance, (Karimi et al., 2020b) weakens the assumption of fully known causal graphs and proposes methods for algorithmic recourse when causal knowledge is incomplete. Similarly, (Dominguez-Olmell Olmedo et al., 2022) focuses on generating robust and stable algorithmic recourse by introducing cost functions tailored to ensure resilience against adversarial perturbations. Additionally, advancements like (Janzing et al., 2020) refin feature attributions using causal insights, and (Jung et al., 2022; Wang et al., 2021) explore novel Shapley value formulations incorporating causality to create more meaningful interpretations.
arg min A
cost(A: x)
s.t.
h(x h(xxF) CP)
= CF
(5)
x CF = FA (F (x))
Here, xx) measures the intervention cost, and F (.) represents causal functions after intervening on A. While this method ensures actionable counterfactuals, it has two challenges. First, the optimization is combinatorial, requir- ing search over all subsets A, which grows exponentially with n input variables (2" subsets). Second, the method relies on interventional counterfactuals, which are often criticized for lacking causal intuition (Dorr, 2016). Back- tracking counterfactuals are considered a better alternative.
Backtracking Counterfactual Explanations: The method in (Von Kügelgen et al., 2023) formulates the problem as:
A novel direction involves backtracking counterfactual ex- planations (Von Kügelgen et al., 2023), which modify latent
arg max XCP
PB(x CF
"F,,,,y), v),
(6)
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focusing on the backtracking conditional distribution (UCFF U) which adjusts latent variables to produce counterfactuals. However, its main drawback is the depen- dence on Pg. Different choices of this distribution lead to varying counterfactuals, and selecting BP is left to the user. Additionally, solving (6) becomes computationally chal- lenging for complex 1/ distributions, as we must integrate over all values of this distribution compute backtracking counterfactuals (see Appendix A).
with the following optimization problem:
arg
min xUP .urr
dx (x,x CF) + + du (u,u
8.1.
1(x = CF
(8)
x =F CF)
x= F(u),
which can also be expressed as:
Deep Backtracking Explanations: The method in (Kladny et al., 2024) refines backtracking counterfactuals using this optimization:
arg min dx (x,x x CF) + A de Fx),, xCF s.t. M(x = y CF
(9)
or equivalently:
arg min o
dy (F-1(x(T) F (x))
arg min oF
dx (x, F(u + +Ady (x)'(x),
s.t.
h(x(P)= = CF
(10)
s.t. h (F(( = u
This method eliminates dependence on ( U) by focusing on the latent space distance dy. However, it ig- nores proximity between C and x in the observed space. As a result, the generated counterfactuals may lack intuitive interpretability and fail to meet the original goal of being close to
Intuitively, this optimization seeks the closest input x CF to x that achieves the desired output yF while preserving the causal relationships encoded in the input variables.
In (8), the objective function includes two terms: dx (FF. which ensures the counterfactual input re- mains close to the observed input, and dy (u. u( which ensures that the latent variables of the factual and coun- terfactual worlds are similar. The constraints enforce the desired counterfactual output (h(xCF) "F ) causal con- sistency (xCF FFu )), and the relationship between the observed input and the latent variables (x F(u)).
While these methods offer valuable insights, they have no- table limitations. In the next section, we propose a new approach that addresses these issues and provides a more effective solution.
4. Our method
As du (u, u CF) increases, the counterfactual latent vari- ables u CF deviate further from the factual latent variables u, making the counterfactual less connected to the factual ob- servation. The parameter À regulates the trade-off between maintaining proximity in the latent space and ensuring the counterfactual remains close to the original input.
In this section, we propose our method called BRACE: Back- tracking Recourse and Actionable Counterfactual Explana- tions. As discussed earlier, one of the main limitations of backtracking counterfactuals is their reliance on the condi- tional distribution I (B (UCF [U. This dependency arises because the choice of PB (UCFF U) significantly influ- ences the resulting counterfactuals, and its specification is left entirely to the algorithm. Such distribution is essential when a probabilistic representation of backtracking coun- terfactuals is required. However, in our scenario where X = FUU) and F(.) is invertible, a simpler perspective can be adopted. Here, U can be treated as a deterministic vector, which simplifies the formulation considerably.
When À = 0, the proximity of latent variables is ignored, resulting in solutions that lack causal consistency and focus solely on minimizing the distance between x and CF Con- versely, as X + oo, the optimization prioritizes minimizing dy (u, u (F)) which ensures minimal deviation in the latent space but disregards proximity in the input space. The ideal solution balances these objectives, ensuring that the counter- factual is both causally consistent and close to the original input.
When U deterministic, we may treat UCF as another deterministic vector close to U, preserving the essence of backtracking without resorting to P7 (UCF U). In inter- pretability tasks, one typically seeks an input CF near x that remains faithful to causal constraints. Thus, viewing CP as deterministic naturally aligns with this goal.
5. Relation Between Backtracking and Interventional Counterfactuals
Our causal model is represented as X = FUU,, where F(.) is an invertible function. Consequently, the distribu- tion of the noise variables conditioned on X x becomes
Based on this reasoning, we propose our method, BRACE,

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A Unified Causal Framework for Efficient Model Interpretability
deterministic. Specifically, all the probability mass of the posterior distribution (T X x) is concentrated at u u= Fx). (x).
solution, x!CF Therefore, interventional counterfactuals can be considered a specific case of backtracking counter- factuals.
As outlined in Section 2.2, backtracking counterfactuals aim to produce a desired counterfactual outcome by keeping the causal graph unchanged while minimally modifying the noise variables U after observing x. Although backtracking counterfactuals are typically defined using the backtracking conditional distribution P (UCF) U). when F(.) is invert- ible, the deterministic nature of U eliminates the necessity for statistical modeling. Instead, we directly analyze the connection between backtracking and interventional coun- terfactuals via their respective causal equations.
This reasoning can be generalized to any subset of variables A. For any counterfactual query involving A. we can con- struct uC such a way that backtracking counterfactuals align with interventional counterfactuals.
To the best of our knowledge, Theorem 5.1 is the first result that relates backtracking and interventional counterfactuals, and it holds independent significance. Theorem 5.1 demon- strates that when F()) is invertible, backtracking counter- factuals inherently include interventional counterfactuals as a specific case. Furthermore, interventional counterfac- tuals, typically expressed as x = F (u), where F (.(.) represents the structural equations post-intervention, can equivalently be reformulated as x1Cr F(u bridging the gap between the two paradigms. By the construction (13) in Theorem 5.1, we can see ICF = F (F (x)).
Theorem 5.1. In structural causal models that adhere to the Bijective Generation Mechanism (i.e., F()) is in- vertible), backtracking counterfactuals generalize interven- tional counterfactuals. Specifically, one of the solutions derived from the backtracking counterfactual formulation always coincides with the interventional counterfactual.
Proof. Consider a counterfactual query involving a subset of variables X = X²X. Under the Bijective Generation Mechanism, the posterior distribution (c X = x) assigns probability one u F Fx) making u deter- ministic. The interventional counterfactuals for this query are defined by the following system of equations:
6. Connection Between Our Method and Previous Approaches
Our method BRACE unifies other existing methods in certain scenarios. In our optimization problem (8), setting À = 0 simplifies the problem to Counterfactual Explanations (4), where causal relationships are disregarded, and the objective becomes finding cF that is closest to x while modifying the model output.
ICF = f((x ICF
pa(s)> uz). Vi E A,
(11)
!CF = x;,
E
When À + co, (8) reduces to Deep Backtracking Explana- tions (7), which exclusively focuses on finding CF closest to u without considering the proximity between CF and x, while ensuring the model's 's output changes.
In contrast, the backtracking counterfactuals are determined by:
1
BCF
Vi A, [xB u BCF)= Vi A.
[xXXC BCF), pat) us BOP
(12)
BCF
Our solution (8) can also be interpreted as a special case of Backtracking Counterfactual Explanations (6). Specifically, it can be shown that employing the backtracking conditional distribution:
The key difference between (11) and (12) lies in the adjust- ment mechanism. Interventional counterfactuals modify the causal graph to enforce X xn, whereas backtracking counterfactuals achieve the same result by adjusting the noise variables.
PB (u CF
u) x exp
(- [dx (F(u),F(uCF) -x. (u,
(14)
Due to the DAG assumption in the causal graph, it is clear that equation (11) has a unique solution. After performing interventions on the set A, we obtain multiple DAGs from which we can derive the unique solution for ICF by starting from the source nodes.
renders (8) equivalent to Backtracking Counterfactual Ex- planations (6). Detailed derivations are provided in the
Appendix B, leveraging the theoretical framework from (Von Kügelgen et al., 2023).
Given that F()) is invertible, we define
While the connections to Counterfactual Explanations, Deep Backtracking Explanations, and Backtracking Coun- terfactual Explanations are established, a significant ques- tion remains: how does our solution (8) relate to Causal Algorithmic Recourse (5)? The following theorem provides an answer.
ICF = F (X
(13)
We can also rewrite equation (12) as x BCCF = F(u BCF).
By definition, if we substitute u BCF u tracking equations (12), we arrive at the same counterfactual
ICF into the back-
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Theorem 6.1. Assume that the distance fanctions dx (,x) and ( u) are convex, F(.) and h())) are linear functions, and the cost function is given x) = dx (xCP x). Then, our method BRACE outperforms Causal Algorith- mic Recourse. Specifically, for a fixed distance ar in the latent space dy (u u) = a, there exists a X such that the solution of (8) yields a counterfuctual x CF closer to the observed input x than the solution of (5).
the optimal du (uCF, u) =a.
ufF from (8) satisfies the distance constraint
To approach this, consider the following vector optimization problem:
arg min x .F
(dx (x, xFF) du (u,
h(x = y OP,
s.t.
(19)
x OF = =F(u
Proof. We start by reformulating Causal Algorithmic Re- course (5) into form analogous our proposed solution (8). Using Theorem 5.1, Causal Algorithmic Recourse (5) can be rewritten as:
x= F(u).
The optimization (19) simultancously minimizes dx (x, xCF) and dy (u, u ) However, in certain cases, reducing one term may result in an increase in the other.
dx (x ,)
arg min xFP, A
h(x (Fx = =y CF,
s.l.
x CF = F (u (CF)
(15) To resolve this, we utilize the concept of Pareto optimality (Boyd & Vandenberghe, 2004). A well-known result for convex problems is that scalarizing the objective:
ICF = F-1 (FA (u)),
A
x = F(u).
dx (x.x + A do (u,
(20)
The main question is whether there exists a Ả such that the optimal xF in our method (8) coincides with the optimal xCP CF in (15). Suppose the optimal solution to (15) is attained for ICF Let represent the distance between u ICF and u:
yields all Pareto-optimal solutions by varying À > O. Specif- ically, every optimal solution of the scalarized optimization corresponds to a Pareto-optimal optimal point of the vector opti- mization. Moreover, since the vector optimization problem (19) is convex (from the assumptions of the theorem), all Pareto-optimal points can be achieved.
du (u
ICF A
u) = a.
(16)
We now define the following optimization problem:
Returning to our optimization, note that the solution to (17) is Pareto-optimal point of (19) because with constraint dy (uCF u) we minimizes dx (x x). Thus, the solution cannot be further improved along the dx (xCF, x) axis
arg
min x cue
dx (x x))
1(x OP) = y CF
s.l.
x CF = (u CF)
(17)
du (F u) a,
Thus, by varying X, it is possible to identify a A such that the solution of our method BRACE (8) matches the solution of (17), ensuring dy (u u) = a. Consequently, as demon- strated in (18), our proposed method yields a XCF that is closer to x compared to Causal Algorithmic Recourse, while preserving the fixed distance between the latent variables u and u
x= F(u).
Let the optimal solution to (15) be x ICP and the optimal solution to (17) be x BCF Then, it follows:
dx (xe BCF x) S dx (xx .x).
(18)
The convexity of the distance functions, along with the linearity of F(-) and ht(.), are assumed primarily to ensure the existence of by utilizing the convexity of the vector optimization problem (19). However, similar conclusions can still be derived without these assumptions if there exists suitable À such that dy (u u)
This inequality holds because ICF satisfies the con- A' straint du CF u) = a, while other feasible values of CF within the same constraint may reduce the objective dx (xer, x) further. Hence, the Causal Algorithmic Re- course formulation (15) may not always yield the closest CF to x among all CF satisfying the distance constraint from u.
This indicates that, even in cases where the convexity of dx x), der u), or the lincarity of F-) and h(.)) are not assumed, our proposed method often provides a CF that is closer to x than Causal Algorithmic Recourse, while preserving the fixed distance a between the latent variables u and n CF
Next, we examine whether there exists a À such that the optimal solution of our method (8) aligns with the opti- mal solution of (17). In essence, we seek a À such that
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Importantly, convexity and linearity assumptions are not necessary for the core insight remain valid. Even without assuming convexity of the distance functions or linearity of F()) and h(-), we can always find a solution that out- performs Causal Algorithmic Recourse among the Pareto optimal points of the vector optimization problem (19). The assumptions of linearity and convexity are only required to guarantee that this Pareto optimal point can be captured by some value of X.
X2
X
X3
Y
X4
It is worth noting that our method is substantially more ef- ficient computationally than Causal Algorithmic Recourse, since that approach relies on a combinatorial optimization procedure. Moreover, our method also surpasses Back- tracking Counterfactual Explanations computational effi- ciency, especially when dealing with complex distribution Pg.
Figure I. Causal graph of the bank's high-risk detection model. X1 is gender, X2 is age, Xx is loan amount, and X4 is repayment duration in months. The model's output Y indicates high or low risk for loan approval.
Heuristic approaches also provide practical alternatives; for instance, shortest path searches in empirical graphs (Poyi- adzi et al., 2020) expanding-sphert searches (Laugel et al., 2017) offer approximate solutions in specific scenarios.
7. Metric Selection and Optimization Approach
8. Experimental Evaluation
To solve the optimization problem in Eq. (8), it is essential to define the distance metrics dx xCF) and du (u, u FF For (.,),, which measures proximity in the observed space, the ε norm is a natural choice as it minimizes the number of modified features, making the counterfactuals more interpretable and actionable:
8.1. Simulation Setup
To evaluate the proposed method, we adopt the experiment in the Causal Algorithmic Recourse paper (Karimi et al., 2021), as it serves as critical baseline for comparison. Since the primary focus is on assessing the interpretability of the proposed method, we use simple model for l(-).. This ensures that the exact solutions to the optimization problem can be computed and aligned with our intuitive understanding of the task.
dx (x, x (T) = xxx
(21)
For the latent space, (.: .) evaluates how plausible a coun- terfactual is relative to the original latent representation. The l2 norm ensures smoothness and proximity:
We consider model h(-) designed to classify individuals as high- low-risk for loan approval. The input vector X is assumed to follow the given causal structure:
du (u, CF
u 1u--u CF
(22)
X1 = U1.
By combining these metrics, the optimization problem can be reformulated in meaningful way.
X2 = U2.
(24)
X3 = fx(X1.X2) Uz.
Solving the optimization problem in Eq. (8), particularly for complex models such as neural networks (Katz et al., 2017) or additive tree models (Ates et al., 2021), is generally NP- hard. Gradient- traseent-based methods are effective when both the objective and the constraints are differentiable. For example, the constraints can be integrated into the objective function as penalty terms:
X4 = f4(X)) + UU4
where the system's output is given by h(X,, Xg, X4). Figure 1 illustrates the causal graph associated with the problem.
For this simulation, the causal graph is assumed to be known, while the functions fa()). fa(-), and l((-) are estimated using real-world data from the German Credit Dataset (Hofmann, 1994). We assume f-(-) and fa(:) are linear functions and h(-) is logistic regression. Following (Peters et al., 2017), the coefficients of the causal model can be derived using linear regression when the causal functions are linear.
arg min XUE
dx x (F) + X do (F (x). C))
(23)
+ + (F(x ), CF)
where Loss(.) is a common classification loss, such as cross- entropy. To approximate solutions in practice, 33 is gradually increased until the counterfactual CF satisfies the desired output class OF (Szegedy et al., 2013).
The feature X1 (gender) is one-hot encoded during logistic regression for h(() and is kept fixed when generating coun-
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A Unified Causal Framework for Efficient Model Interpretability
Table 1. Counterfactual solutions from different methods for an individual originally classified high-risk (x
(female, 24, 84308, 48)).
Each method modifies the features flip the prediction low-risk.
Method
Gender
Age
Loan amount
Duration
Original (High-risk)
female
24
$4308
48
BRACE (Our Method, À 1)
female
24
$4087
33.0
BRACE (Our Method, À = 1.2)
female
24
$3736
33.3
Counterfactual Explanations (Wachter et al., 2017)
female
24
$4308
32.8
Causal Algorithmic Recourse (Karimi al., 2021)
female
24
$4308
32.8
Deep Backtracking Explanations (Kladny et al., 2024)
female
27.2
$2727
35.7
terfactuals. Since X2 is categorical, modifying it is avoided, as changing gender does not provide actionable insights. Additionally, all features are normalized by their standard deviations to improve performance.
duration, the repayment becomes significantly more chal- lenging for the user, thereby making the explanation less actionable.
To put it quantitatively, repaying $4308 over 48 months corresponds to monthly payment of $89.75. Any expla- nation that deviates considerably from this monthly rate is less actionable. Counterfactual Explanations and Causal Algorithmic Recourse yield a monthly repayment of about $131.3 (i.e., $4308 divided by 32.8 months). In contrast, our solution results in a monthly repayment of $ $123.8 when X 1 (i.e., $4087 divided by 33 months) and $112.2 when À == 1.2 (i.e., $3736 divided by 33.3 months). Thus, while Counterfactual Explanations and Causal Algorithmic Re- course increase the monthly repayment by 46.3%, our ap- proach leads to increases of 37.8% and 25.0% for À = and = 1.2, respectively, making our explanations more actionable for the user.
8.2. Optimization Problem and Results
We consider an individual with features x (female, 24, $4308, 48) classified
high-risk by the model h(-). Using the causal model and the learned func- tions f-(-) and f-). we derive the latent representation u.
The following optimization problem is then formulated:
CF
(F)
U4
arg min CP
CP
Oi
0
2=
s.t.
h(x CF low-risk,
(25)
CF CF
22 u2
23 CF = fa CF CF
.12
+ ug
CF
Table 2 provides the counterfactual outcomes for another example with initial features x (male, 27, $14027, 60), where similar trends across methods are observed.
CF [4( + u4
CF
CF
44
We solve the optimization problem in (25) with À = 1 and = 1.2. Table reports x CF from our method and other prominent approaches for comparison.
8.3. Sensitivity Analysis
Estimating causal functions in the input graph often involves approximations, introducing potential noise. To evaluate the robustness of our method, we add zero-mean Gaussian noise with a standard deviation of to the coefficients of fa(:) and solve the optimization problem (25) with À 1.2 for an individual with features x (female, 24, $4308, 48). The resulting counterfactuals
As shown in Table 1, both Counterfactual Explanations and Causal Algorithmic Recourse focus solely reducing the repayment duration, which is not actionable for the user and fails to provide meaningful guidance for future improve- ments. In contrast, the Deep Backtracking Explanations alters all features, leading to a significant departure from the original observation and reducing local interpretability. Our approach finds a balance by adjusting both the loan amount and repayment duration while maintaining sparsity and interpretability, offering more intuitive and actionable explanation for the user.
x CF
(female, 24, $3839, 33.2),
CF
(female, 24, $3572, 33.5),
(26)
x = (female, 24, $3618, 33.4).
Despite significant noise, the results remain stable, with age unchanged and explanations staying sparse. This highlights the robustness of our method, showing that even approx- imate causal functions produce reliable and interpretable counterfactuals.
When the user's initial features are given by = (female, 24, $4308, 48), this suggests that 48-month loan repayment is appropriate for the user. Therefore, if we ad- just this feature vector solely by reducing the repayment
8

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A Unified Causal Framework for Efficient Model Interpretability
Table 2. Counterfactual solutions from different methods for an individual originally classified high-risk.
Method
Gender
Age
Loan amount
Duration
Original (High-risk)
male
27
$14027
60
BRACE (Our Method, À 1)
male
27
$13686
36.9
BRACE (Our Method, À 1.2)
male
27
$13149
37.4
Counterfactual Explanations (Wachter et al., 2017)
male
27
$14027
36.6
Causal Algorithmic Recourse (Karimi al., 2021)
male
27
$14027
36.6
Deep Backtracking Explanations (Kladny et al., 2024)
male
31.9
$11599
41.1
9. Conclusion
ensure that the noise variables UCP in the counterfactual world remain mutually independent. In SCMs, this indepen- dence is important for maintaining the causal interpretation of the noise variables. Extending the definition enforce this independence would enhance both the theoretical con- sistency and practical usefulness of backtracking counter- factuals, making them more aligned with core principles of causal reasoning.
In this work, we presented a new framework BRACE for counterfactual explanations based on backtracking coun- terfactuals. Our approach overcomes the limitations of interventional counterfactuals by introducing an optimiza- tion problem that generates actionable and causally con- sistent explanations. By solving single unified objec- tive parameterized by A, BRACE recovers four established classical Counterfactual Explanations (A = 0), Deep Backtracking Explanations (A + x)), Backtracking Counterfactual Explanations via a specific backtracking conditional distribution, and Causal Algorithmic Recourse under a convexity assumption. Additionally, we demon- strated that method is both easier understand and more computationally efficient compared to causal algorith- mic recourse. Through simulation experiments, we verified that the proposed method produces explanations that are more intuitive for users and more practical for real-world applications.
paradigms classia
Acknowledgements
This research was conducted while Pouria Fatemi and Ehsan Sharifian were affiliated with Sharif University of Technol- ogy.
Impact Statement
Our work advances the field of machine learning by propos- ing framework for counterfactual explanations that hances interpretability while ensuring causal consistency and actionability. This method unifies multiple existing approaches, including counterfactual explanations, deep backtracking explanations, causal algorithmic recourse, and backtracking counterfactual explanations, while improving computational efficiency.
10. Future Work
This work opens several directions for further research:
Relaxing Assumptions for Connection with Causal Algo- rithmic Recourse: Our approach depends on convexity as- sumptions to establish a connection with causal algorithmic recourse. Future work could investigate alternative condi- tions that do not require these assumptions, allowing the theory to be applied to non non-lincar and non-convex models frequently found in real-world applications.
The potential impact of our work is most relevant to high- stakes applications such as finance and healthcare, where reliable and transparent décision-making is essential. By incorporating causal reasoning into counterfactual explana- tions, our approach contributes to making Al-driven deci- sions more interpretable and aligned with real-world con- straints. While our method does not introduce direct ethi- cal concerns, its application in automated decision-making should be carefully evaluated to ensure fair and responsible use.
Testing on Complex Models: This study focused on sim- pler models for fi..) to ensure intuitive understanding of the task. A valuable next step to test our method on more complex models, such as deep neural networks, and com- pare its performance with existing state-of-the-ant methods. This would help demonstrate the method's effectivvness in handling challenging real-world scenarios.
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A Unified Causal Framework for Efficient Model Interpretability
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A Unified Causal Framework for Efficient Model Interpretability
A. Formal Definition of Interventional and Backtracking Counterfactuals
To formally understand the differences and computation processes underlying interventional and backtracking counterfactuals, we outline their respective definitions and procedural steps below.
A.1. Interventional Counterfactuals
1. Abduction:
Update the distribution of the noise variables U in the causal model from Po to the posterior distribution Pujx=x using the observed factual data
2. Action:
Perform hard intervention do(X, 2i CFF) for i E A, modifying the structural equations of the causal model.
Denote the modified structural equations as SCF, while retaining the original equations SCF Si for & A.
3. Prediction: Using the updated causal model CCF (SCr, Pujx=x), compute the distribution over the desired counterfactual outcomes YCF.
A.2. Backtracking Counterfactuals
1. Cross-World Abduction: Update the joint distribution PB (U
U) PB (UCF
U) using the variables
(xCF , x) to obtain the posterior distribution PB(UCP, U U x):
PB(u OF u) FFF A (u (P) = =xA CP (()((u) = x]
PB
CF u
CF x)
(27)
P Px x)
Where
P CF x) = /PrssPuuu CF 1u) x
- 1(F((() = du du du
(28)
Calculating ( x) becomes computationally challenging for complex PB (UC U) distributions, as we must
integrate over all values of this distribution.
2. Marginalization: Marginalize over U compute the posterior distribution Py (U
CF x):
A
Pn CF
xx)x
CF u XA
x)
du.
(29)
3. Prediction: Using the updated causal graph with noise distribution (B(UCP CF x), compute the probability of the desired counterfactual event:
Pn(y
CF
CF x) = P CF
CF |x
( y
CP du
(30)
B. Relation of Our Solution to Backtracking Counterfactual Explanations
We aim to demonstrate that our solution (8) can be connected to the backtracking counterfactual explanations framework presented in (Von Kügelgen et al., 2023), which is formulated as the optimization problem (6), by considering a specific
choice of CF U). This connection is established by following the three steps of backtracking counterfactual computation:
1. Cross-World Abduction: Compute the posterior distribution of the latent variables in the causal graph. Given that the function F()) is invertible, we have:
Pg (U
CF , u CF ,x) = Pg (u
CF
u, CF x) PB (u y CF x) u. OF) PB(u|x
(31)
Pg (u CF
(32)
Pa CF
u. OF) (F- (x) (x)= n
(33)
12

---

A Unified Causal Framework for Efficient Model Interpretability
2. Marginalization: Compute the marginal posterior distribution over u CF concentrated the point F-1 (x), we have:
Since the entire probability mass is
PB cr yr x) PB (u u F (x) 1 c .x)
(34)
PB CF F (x),
(35)
3. Prediction: Compute the distribution XCF
CF Since F(.) is deterministic, have:
CF UCP
F (x) .9
x CF
=(uu
(36)
Now, consider specific choice for the backtracking conditional distribution:
CF
u))
exp (-dx (F(u), F(u X de (u, u [F)
(37)
Substituting this into the posterior distribution of UCF [x+ CF we obtain:
{ Texp f (x, F(u du (x),, (x)),(())) }
if hxx y
UCF F (
OF x
(38)
0.
otherwise.
Taking the logarithm both sides, have:
-x -x(xx,F(uFF) - X du (F-1(x), CF),
if FF
log Pg (u CF F- Fx1(x),
(39)
-00;
otherwise.
Thus, we have:
arg max log Pn (u CF DCP
FF (x), = arg min oPF
dx (x, F(u + A du (x)) (F).
s.6.
1 CF
(40)
As shown in (40), the optimization problem (10) aligns with backtracking counterfactual explanations (6). Therefore, our solution provides a valid interpretation based on backtracking counterfactuals.
13


Reviews:
[{"review_id": "Iu62fY00xZ", "reviewer": "Reviewer_7Lfk", "summary": "The authors propose a new framework for counterfactual explanations based on backtracking counterfactuals by introducing an optimization problem that seeks the nearest possible input modification needed to achieve the desired counterfactual outcome while preserving the causal relationships encoded in the input variables. They evaluate their framework on a simulated setup (structured causal model within a bank’s high-risk detection module) to show how their counterfactual explanations can be more intuitive for users and practical for real-world applications.\n\n## update after rebuttal\n\nSince my questions have been addressed, I have raised my score accordingly.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your thorough analysis and constructive feedback on our paper. We appreciate the opportunity to clarify the points raised and to provide additional insights into our research.\n\n**Answer for Q1:** In our quest for inputs that are actionable for the user, it is crucial to take the user profile into account. When the user’s initial features are given by x = (female, 24, \\\\$4308, 48), this suggests that a 48-month loan repayment is appropriate for the user. However, if we adjust this feature vector solely by reducing the repayment duration, the repayment becomes significantly more challenging for the user, thereby making the explanation less actionable.\n\nTo put it quantitatively, repaying \\\\$4308 over 48 months corresponds to a monthly payment of \\\\$89.75. Any explanation that deviates considerably from this monthly rate is less actionable. Counterfactual Explanations (Wachter et al., 2017) and Causal Algorithmic Recourse (Karimi et al., 2021) yield a monthly repayment of about \\\\$131.3 (i.e., \\\\$4308 divided by 32.8 months). In contrast, our solution results in a monthly repayment of \\\\$123.8 when λ = 1 (i.e., \\\\$4087 divided by 33 months) and \\\\$112.2 when λ = 1.2 (i.e., \\\\$3736 divided by 33.3 months). Thus, while Counterfactual Explanations and Causal Algorithmic Recourse increase the monthly repayment by 46.3%, our approach leads to increases of 37.8% and 25.0% for λ = 1 and λ = 1.2, respectively, making our recommendation more actionable for the user.\n\n**Answer for Q2:** According to our optimization formulation (see Equation 26), by “significant departure” from the input we refer to the L1 norm difference, namely, $\\left\\| \\mathbf{x} - \\mathbf{x}^{\\mathrm{CF}} \\right\\|_1$. This metric captures both the number of features modified and the aggregate relative change across those features. As you correctly mentioned, the Counterfactual Explanations method just minimizes this L1 norm, resulting in the smallest overall deviation from the original input. However, as highlighted in the Problem Definition point 3 (line 102), it is also essential to incorporate causal relationships among the input features. Consequently, our solution seeks an alternative that is not only close to the original observation in terms of the L1 norm but is also consistent with the underlying causal graph, thereby providing an actionable insight for the user.\n\n**Answer for Q3:** The following results demonstrate the performance of the Deep Backtracking Explanation method when identical noise is introduced:\n\n(female, 27.17, \\\\$2696, 35.8)\n\n(female, 27.14, \\\\$2752, 35.7)\n\n(female, 27.18, \\\\$2831, 35.6)\n\nAs shown, the deviations are minimal in this method. \n\nGiven that the causal algorithmic recourse method employs combinatorial optimization—which is typically sensitive to noise—we initially expected the introduction of noise to yield more dramatic changes in this method.\n\nIt is important to note that when the counterfactual explanation only modifies the sink nodes of our causal DAG (like the example in our paper), the outcomes of the counterfactual explanation and the causal algorithmic recourse methods become identical. This is because interventions on these sink nodes are sufficient for interventional counterfactuals in causal algorithmic recourse optimization. In such cases, adding noise is less likely to alter the set of required interventions, so causal algorithmic recourse would be more robust to noises.\n\nHowever, as the added noise alters the causal graph, the solution of the causal algorithmic recourse shifts slightly. Under the same noise conditions, the new solutions of causal algorithmic recourse are:\n\n(female, 24, \\\\$4353, 32.7)\n\n(female, 24, \\\\$4192, 32.9)\n\n(female, 24, \\\\$4432, 32.6)\n\nAlthough these shifts are not dramatic, we anticipate that in more complex cases—where combinatorial optimization plays a larger role—the impact of noise could be more pronounced.\n\n**Answer for Q4:** For a high-risk individual with attributes (male, 23, \\\\$15,672, 48), our solution for explaining the model's behavior and providing actionable insights—using $\\\\lambda = 1.2$—is (male, 23, \\\\$15,116, 33.7). Under the same noise conditions described in our paper, our method yields the following results:\n\n(male, 23, \\\\$15,253, 33.6)\n\n(male, 23, \\\\$15,055, 33.8)\n\n(male, 23, \\\\$14,953, 33.9)\n\nFor another high-risk individual with attributes (female, 24, \\\\$7,408, 60), our solution with $\\\\lambda = 1.2$ is (female, 24, \\\\$6,273, 30.9). When the same noise is applied, the method produces:\n\n(female, 24, \\\\$6,214, 31.0)\n\n(female, 24, \\\\$6,504, 30.7)\n\n(female, 24, \\\\$6,371, 30.8)\n\nSimilarly, for another high-risk individual (male, 27, \\\\$14,027, 60), our solution for $\\\\lambda = 1.2$ is (male, 27, \\\\$13,149, 37.4). Under identical noise conditions, our method yields:\n\n(male, 27, \\\\$13,077, 37.5)\n\n(male, 27, \\\\$12,984, 37.6)\n\n(male, 27, \\\\$13,274, 37.3)\n\nThese results underscore the stability of our approach across different input instances.", "claims_and_evidence": "The claim of the proposed framework being more intuitive and practical needs more analysis and evidence to be convincing (explained more in the later sections – see \"Experimental Designs or Analyses\").", "methods_and_evaluation_criteria": "The proposed method and evaluation criteria makes sense for the problem at hand.", "theoretical_claims": "There are a couple of theoretical claims (in Section 5 and 6), but they are backed by sound proofs.", "experimental_designs_or_analyses": "The experiment design is well-structured as a simulated setup; however, the accompanying analyses lack sufficient depth. Section 8 primarily describes the simulated setup and presents only a single case scenario, reporting counterfactual explanation results from various approaches, including their own. However, it does not provide enough qualitative or quantitative metrics to substantiate claims such as those in lines 409-410, where the authors assert that their approach “finds a balance … offering a more intuitive and actionable explanation for the user.” Similarly, in section 8.3, while the experimental setup is sound, lines 427-428 just state that “the results remain stable” without providing sufficient comparative context with other methods. To properly assess the robustness of the explanation, the experiments should explore multiple cases within this setup – ideally extending to other causal graphs to demonstrate generalizability. At a minimum, the study should thoroughly test variations of the stated causal graph across multiple individual instances, incorporating more comprehensive qualitative and/or quantitative evaluations.", "relation_to_broader_scientific_literature": "The proposed counterfactual explanation framework presents a promising approach to generating closer counterfactual explanations while maintaining causal relationships with input variables. This addresses the gap of ensuring stability in such explanations and can be explored further in more complex settings to strengthen its practical use.", "essential_references_not_discussed": "The related work provides a sufficient foundation for understanding the approaches evaluated in the paper.", "other_strengths_and_weaknesses": "Strengths:\n* The paper provides a thorough background and problem setting.\n\nWeakness:\n* The experiment analyses lack sufficient depth (see “Experiment Designs or Analyses” and “Questions For Authors” sections), making it less convincing in demonstrating the improvement of this approach compared to existing methods.", "other_comments_or_suggestions": "My primary concern is the depth of the experimental analysis. If more clarity and detail are provided, I would be open to reconsidering my score.", "questions_for_authors": "* Q1: Lines 403-404: I am confused as to why “reducing the repayment duration” is not an actionable item? Does this not mean that the individual can maybe repay faster to be considered a low-risk?\n\n* Q2: Lines 407: Can we quantify how much of a “significant departure” it is? What is considered significant here? Is it how many features are changed or does it also look into the sum of relative change of each feature? In both cases, the counterfactual explanations (Wachter et al.) approach seems to have the least change.\n\n* Q3: Line 427: What are the results for other methods? Did they change, and if so, to what degree?\n\n* Q4: Lines  430-431: To better assess the robustness and reliability of the method, additional cases with more instances are necessary to understand how consistently the method remains stable. Ideally, to evaluate its generalizability, the method should be tested on various causal graphs. This would help determine the extent to which the approximate causal functions hold, such as how many features can be causally connected while still maintaining reliable performance.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Xfw5nMe23m", "reviewer": "Reviewer_6rSy", "summary": "This paper proposes a new method for counterfactual explanation of model behavior for models that fall under the additive noise constraints. The new framework is based on backtracking counterfactuals, that find settings for exogenous variables that produce endogenous variables with a desired counterfactual value. They demonstrate in an experiment that their method for creating counterfactual inputs outperforms previous methods for causal algorithmic recourse on a small dataset.", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your insightful remarks regarding our paper. We truly appreciate your thoughtful feedback and look forward to exploring your suggestions further to enhance our work.\n\n> The current paper presents a method that is limited to additive noise models.\n\nFirst, we note that our method applies to any case where $\\\\mathbf{F}(\\\\cdot)$ is invertible- a condition that is more general than the Additive Noise Models (ANM). While we require the ANM assumption to prove our theorems and to establish the connection between our method and causal algorithmic recourse, the practical application of our method only necessitates that $\\\\mathbf{F}(\\\\cdot)$ is invertible.\n\n> Why is it bad that a method recommends reducing loan duration alone?\n\nFor this question, please first refer to **Answer for Q1** of our rebuttal to Reviewer 7Lfk.\n\nAdditionally, we observe similar behavior in other instances. For example, for a high-risk individual with attributes (male, 27, \\\\$14,027, 60), our solution for $\\\\lambda = 1.2$ is (male, 27, \\\\$13,149, 37.4). In comparison, the solution provided by Counterfactual Explanations and Causal Algorithmic Recourse is (male, 27, \\\\$14,027, 36.6), while the Deep Backtracking Explanations method yields (male, 31.9, \\\\$11,626, 41.1).\n\nFor another high-risk individual with attributes (female, 24, \\\\$7,408, 60), our solution with $\\\\lambda = 1.2$ is (female, 24, \\\\$6,273, 30.9). In this case, the solution from Counterfactual Explanations and Causal Algorithmic Recourse is (female, 24, \\\\$7,408, 29.8), and the Deep Backtracking Explanations method produces (male, 30.1, \\\\$4380, 35.5).\n\nWe believe these results underscore the practical relevance of our approach in delivering actionable insights.\n\nFor your question about robustness, please refer to **Answer for Q3** and **Answer for Q4** of our rebuttal to Reviewer 7Lfk.\n\n> Assumptions of linearity and convexity in models is increasingly limiting in the modern age of machine learning.\n\nIt should be mentioned that, to the best of our knowledge, Theorem 5.1 in our paper is the first result to relate backtracking and interventional counterfactuals, and it holds independent importance. Additionally, Theorem 6.1 is, as far as we know, the first result that relates causal algorithmic recourse with backtracking explanation methods.\n\nAssumptions of linearity and convexity in our models are necessary because we require the vector optimization (Equation 20) to be convex. This convexity ensures that by varying $\\\\lambda$, we can capture *all* Pareto optimal solutions, thereby guaranteeing that we also obtain the desired Pareto optimal solution in Equation 18. However, as mentioned in the paper (lines 295–305), even without any additional assumptions—relying solely on the ANM assumption—we can *always* find a better solution than causal algorithmic recourse among the Pareto optimal points of the vector optimization (Equation 20). We need the assumptions of linearity and convexity to ensure that we can *capture* this Pareto optimal point for some $\\\\lambda$. That said, many Pareto optimal points of a non-convex vector optimization can still be reached, and our required Pareto optimal solution (the solution of Equation 18) might be one of them. For example, consider [1, Figure 4.9]. In the figure, all Pareto optimal points are shown as a bold line. Although the problem is non-convex, we can capture $f_0(x_1)$ and $f_0(x_2)$ with $\\\\lambda_1$ and $\\\\lambda_2$, respectively; however, there is no value of $\\\\lambda$ that can capture $f_0(x_3)$. We know that one of the points on the bold line is our desired point (Equation 18) under the ANM assumption. The linearity and convexity assumptions ensure that we can capture all the Pareto optimal points and, as a result, our desired one.\n\n\n[1] Boyd, Stephen P., and Lieven Vandenberghe. Convex Optimization Book. Cambridge university press, 2004.", "claims_and_evidence": "Backtracking counterfactuals are argued to be more intuitive, and this is supported by deep tracking counterfactuals which seems clearly true because they don't relate x and x^CF. \n\nIt's really elegant how the set up simplifies to deep backtracking and counterfactual explanation as you vary the parameter! The whole thing seems really theoretically elegant and nice.", "methods_and_evaluation_criteria": "The comparison between methods seems a bit fraught to me. Given that this is a single dataset with only five variables, I expected there to be more clear success criteria. Why is it bad that a method recommends reducing loan duration alone? I don't see why recommending lower loan duration and a lower loan is any less actionable or worse by some other metric.  \n\nAdditionally, the robustness evaluations seem a bit ad hoc, why isn't there a comparison between different methods that systematically compares their robustness? \n\nCompared to the rest of the paper, the experimental results and analysis seems like an after thought.", "theoretical_claims": "I think I follow the 5.1 and 6.1 proofs in the paper, but I wouldn't say I am familiar enough with the material to check its correctness closely.", "experimental_designs_or_analyses": "The experiment is a very simple causal algorithmic recourse task. It fits the proposed method perfectly and adheres to the needed assumptions.", "relation_to_broader_scientific_literature": "Counterfactual explanation connects deeply to fairness and explainable AI, as we need counterfactual inputs to understand how models make decisions and what they would have done under different circumstances. However, the current paper presents a method that is limited to additive noise models, which limits its connections greatly.", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "The weakness of the paper is the assumptions needed for any of the results to go through. Assumptions of linearity and convexity in models is increasingly limiting in the modern age of machine learning. Especially when the motivation is algorithmic recourse, because the vast majority of AI models used in our society will be text/image/voice models that are highly non-linear and non-convex. \n\nAlso, even within the limited model class, the comparison to other methods seems limited in ways pointed out above.\n\nThe strength of this paper is that its well-written, easy to ready, and theoretically elegant.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "See methods and evaluation criteria.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "V2OwlFnBkn", "reviewer": "Reviewer_cSC4", "summary": "This paper proposes a new and efficient method for backtracking counterfactuals using causal reasoning to develop the explanations. The paper provides an analysis of the method’s limitations, discusses the relationship to the literature, and provides experiments that show promising results of their techniques.", "questions": "", "limitations": "", "rating": 4, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your kind feedback and positive review of our work. We appreciate your recognition of the theoretical contributions, experimental design, and overall rigor of our paper.\n\nRegarding your question on how our algorithm would perform on large state-of-the-art models, we acknowledge that this is an important aspect to consider. As mentioned in lines 422–428 of section 10 of our paper, we recognize the significance of evaluating our method on more complex models. While our current experiments were conducted on a benchmark dataset for algorithmic recourse, we plan to extend our investigations to include large-scale, state-of-the-art models in future work. A comprehensive evaluation in this context remains an exciting direction for our future research, and we are eager to explore how our approach performs in real-world, complex model scenarios.\n\nThank you again for your encouraging feedback.", "claims_and_evidence": "Yes, the paper does a good job explaining the methodology, relating it to other literature in the field, proving several of the attributes of the methodology – thus providing a theoretical basis for the research – and it gives a strong experimental analysis.", "methods_and_evaluation_criteria": "Yes. Though this is more of a theoretical paper, the experimental section also directly compares the algorithm proposed to three distinct previous algorithms in the literature on a benchmark dataset for algorithmic recourse.", "theoretical_claims": "The paper makes several theoretical claims, e.g., theorem 5.1 discusses that backtracking counterfactuals generalize interventional counterfactuals as one of the solutions derived from the backtracking counterfactual formulation can be made the same as the interventional counterfactual. The proofs provided appear to be correct.", "experimental_designs_or_analyses": "The experimental design for this paper is sound and is based on previous research by Karimi et al. This allows the paper to have a direct baseline to compare their results to which is an important feature. The paper suggests promising results that improve upon three different papers from the last 8 years for algorithmic recourse. They also run a sensitivity analysis on their results suggesting robustness of their methodologies.", "relation_to_broader_scientific_literature": "This paper contributes to the broader scientific literature by providing a new method for backtracking counterfactuals explanations using a causal reasoning approach.", "essential_references_not_discussed": "N/A. This paper does a thorough literature review.", "other_strengths_and_weaknesses": "This paper is strong and provides many of the attributes of a good paper. It contributes to the literature directly and analyzes its claims rigorously.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "How would this algorithm perform on large state-of-the-art models?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "e8p4B8BjNF", "reviewer": "Reviewer_Fw7M", "summary": "The paper presents an unifying view on generating counterfactual explanations via backtracking. Namely, the authors propose an optimization objective integrating insights from causal algorithmic recourse and backtracking counterfactual explanations. The paper shows how this new objective subsumes the previous definitions in particular cases. Then, the approach is tested on a synthetic causal setting from the literature.", "questions": "", "limitations": "", "rating": 1, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your feedback on our paper. We look forward to exploring your suggestions further to enhance our work.\n\nWe do not dismiss the backtracking distribution. As stated in Equation (15) of our paper, our method can be viewed as a special case of Backtracking Counterfactual Explanations by using the backtracking conditional distribution presented there.\n\nRegarding your questions about our experiments, please refer to our **Answer for Q1, Answer for Q3, and Answer for Q4** in our rebuttal to Reviewer 7Lfk.\n\nConcerning your question about generalizing our problem to cases where the input SCM is not fully known or is non-invertible: suppose that $\\mathbf{F}(\\cdot)$ is non-invertible and not fully known. In this scenario, after observing $x$, the posterior over $U$ is no longer a point mass. Instead, we can compute the distribution $U \\mid x$ using the available causal relationships and the prior $P(U)$. An interesting scenario arises when the causal graph among variables is known, yet the specific functional forms are not. Following the approach in [1], we can assume a probability distribution (e.g., a Gaussian process) over each causal function. By combining this with the prior on $U$, we can compute the posterior distribution $U \\mid x$. In this case, both $\\mathbf{F}(\\cdot)$ and $U \\mid x$ become random, leading to the following optimization:\n\n\\begin{split}\n\\arg\\min_{u^{\\mathrm{CF}}} \\quad & E_{F} \\left[d_X\\left(x,F(u^{\\mathrm{CF}})\\right)\\right] + \\lambda \\ E_{U \\mid x}\\left[d_U\\left(U, u^{\\mathrm{CF}} \\right)\\right] \\\\\n\\quad \\text{s.t.} \\quad & E_{F} \\left[h\\left(F(u^{\\mathrm{CF}})\\right)\\right] \\ge \\alpha\n\\end{split}\n\nIn practice, heuristic methods are used to solve these optimization problems, as discussed in [1].\n\n> Can you better clarify the implications of the various assumptions made in Theorem 6.1 when they do not hold?\n\nIt should be mentioned that, to the best of our knowledge, Theorem 5.1 in our paper is the first result that relates backtracking and interventional counterfactuals, and it holds independent significance. Additionally, Theorem 6.1 is, as far as we know, the first result that connects causal algorithmic recourse with backtracking explanation methods.\n\nAssumptions of linearity and convexity in our models are necessary because we require the vector optimization (Equation 20) to be convex. This convexity ensures that by varying $\\lambda$, we can capture all Pareto optimal solutions, thereby guaranteeing that we also obtain the desired Pareto optimal solution in Equation (18). However, as mentioned in the paper (lines 295–305), even without any additional assumptions—relying solely on the ANM assumption—we can *always* find a better solution than causal algorithmic recourse among the Pareto optimal points of the vector optimization (Equation 20). The assumptions of linearity and convexity ensure that we can *capture* this Pareto optimal point for some value of $\\lambda$. That said, many Pareto optimal points of a non-convex vector optimization can still be reached, and our required Pareto optimal solution (the solution of Equation (18)) might be one of them. In essence, linearity and convexity assumptions ensure we can capture *all* the Pareto optimal points and, consequently, our desired one.\n\n> Can you better clarify the potential challenges arising when optimizing Equation 24 with respect to classical differentiable recourse?\n\nWhen optimizing Equation (24) generally, there is no need to predefine a feasible set of features. However, if we want to restrict the optimization to a specific subset of variables that we intend to change, we can select them and solve the optimization accordingly. This approach contrasts sharply with causal algorithmic recourse, where one must *actively search* for the optimal subset of variables to intervene on from all feasible options. Their method relies on combinatorial optimization; however, when we fix a set of feasible variables in our method, it essentially reduces to a hyperparameter choice. In other words, if one claims to solve causal algorithmic recourse while simply fixing the feature set instead of optimizing to identify the optimal variables for intervention, the solution lacks the clear intuition behind an interventional counterfactual.\n\n[1] Karimi et al., Algorithmic recourse under imperfect causal knowledge: a probabilistic approach. Advances in Neural Information Processing Systems, 33:265–277, 2020.", "claims_and_evidence": "The authors do not provide a sound justification why the proposed objective (Equation 8) improves from the limitations of (deep) backtracking counterfactuals and related methods. Namely, the authors dismiss too easily the backtracking distribution $P_B(\\mathbf{U}^{CF} \\mid \\mathbf{U})$. Indeed, in Equation 8, optimizing over the distance term over the exogenous variables $d(\\mathbf{u}, \\mathbf{u}^{CF})$ does not ensure we are finding reasonable changes to the exogenous variables. Thus, falling in the same issue as Equation (4).  \n\nTheorem 6.1 holds under very specific conditions (e.g., linearity of the classifier and convexity of the distance functions). It is not clear to me how the theorem is still valid when those conditions are not satisfied (line 294-305 after the proof). \n\nIt is not clear how optimizing Equation (24) should improve over the difficulties highlighted by causal algorithmic recourse (Section 5). Indeed, even if we are employing a gradient based approach, we still need to specify a feasible set of features to work on $\\mathcal{A}$ (as they also do in [1]).\n\n[1] Dominguez-Olmedo, Ricardo, Amir H. Karimi, and Bernhard Schölkopf. \"On the adversarial robustness of causal algorithmic recourse.\" International Conference on Machine Learning. PMLR, 2022.", "methods_and_evaluation_criteria": "Since the paper provides a new objective function, I believe a robust empirical evaluation is needed to better understand the practical implications of the proposed approach. \n\nThe experiments are very limited, and they do not provide any statistical guarantees or proper evaluation of optimizing Equation 24 with respect to the alternatives. Namely, the authors focus on a single simulated scenario, by showing us only the results of a unique instance. Moreover, it would be nice to see empirical evaluations of conclusions derived from the theorems (Theorem 6), which are also lacking.\n\nI would suggest the authors to increase the sample size, provide analytical evaluation metrics (e.g., validity, sparsity, distance, etc.), and test the approach over multiple (even synthetics) causal datasets. Further ablations could consider linear/non-linear ANMs (as done by Karimi [1]), more complex classifiers (as done by Dominguez [3]), and approximate SCMs fitted from the data. The assumption of knowing the full SCM does not hold in practice, and optimizing Equation 24 with an approximate SCM would provide interesting insights. \n\n[1] (Adult and COMPAS) Razieh Nabi and Ilya Shpitser. Fair inference on outcomes. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 32, 2018.\n\n[2] (Loan) Karimi et al., Algorithmic recourse under imperfect causal knowledge: a probabilistic approach. Advances in Neural Information Processing Systems, 33:265–277, 2020.\n\n[3] (Further datasets) Dominguez-Olmedo et al., \"On the adversarial robustness of causal algorithmic recourse.\" International Conference on Machine Learning. PMLR, 2022.", "theoretical_claims": "I checked the claims in Theorem 5.1 and Theorem 6.1 and they seem to be correct (although Theorem 6.1 does require strong assumptions).", "experimental_designs_or_analyses": "See “Methods And Evaluation Criteria”.", "relation_to_broader_scientific_literature": "The paper is surely interesting for the broad field of Explainable AI.", "essential_references_not_discussed": "None.", "other_strengths_and_weaknesses": "None.", "other_comments_or_suggestions": "None.", "questions_for_authors": "- Can you better clarify the implications of the various assumptions made in Theorem 6.1 when they do not hold?\n- Can you better clarify the potential challenges arising when optimizing Equation 24 with respect to classical differentiable recourse?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper introduces a new framework for counterfactual explanations based on backtracking counterfactuals, aiming for more intuitive and practical explanations by preserving causal relationships. cSC4 and 7Lfk (after rebuttal) highlight the promising nature of the approach and its theoretical basis (cSC4, 6rSy), with 7Lfk noting the thorough background. However, a significant weakness identified by multiple reviewers (7Lfk, 6rSy, Fw7M) is the limited experimental evaluation, with concerns about the depth of analysis, the use of a single simulated dataset (7Lfk, 6rSy, Fw7M), and a lack of robust comparisons and statistical guarantees (6rSy, Fw7M). Fw7M also raises concerns about the justification of the proposed objective function and the strong assumptions underlying the theoretical claims. Overall, while the core idea is seen as interesting and theoretically elegant, the reviewers generally express reservations about the strength of the empirical evidence and the scope of the evaluation, requiring more substantial experimental validation to fully support its claims. The rebuttal improved Reviewer 7Lfk's score, but the fundamental concerns of other reviewers regarding the experimental rigor and justification of the method were not fully addressed.
Decision: Accept (poster)