Abstract

Aggregate queries are essential for summarizing data and obtaining condensed information. Explaining such queries-by identifying how specific predicates influence the result-provides deeper insights into the factors shaping query outcomes. However, existing statistical, interventional, and game theoretic explanation methods lack causal grounding, while causal methods require complete causal graphs, which are rarely available in large databases. To address this, we propose Causal Banzhaf Value (CBV): introducing causal awareness into Banzhaf values, our CBV method delivers explanations even in the absence of full causal graphs. Experiments on real world data demonstrate that CBV is computationally eficient, aligns with human intuition, and is consistent with causal explanations.

1. Introduction

The increasing reliance on data-driven decision-making in ifelds such as business, healthcare, and science amplifies the importance of query explanations in understanding patterns, trends, and anomalies [ 1, 2, 3 ]. Aggregate queries, such as averages or sums, play a pivotal role in summarizing highdimensional data but pose challenges for understandability. Analysts often need explanations, such as the contributions of individual data segments or predicates1 to understand results, especially in high-dimensional datasets where interactions and dependencies are complex [ 4, 5 ].

Example

Consider the Stack Overflow Developer Survey [ 6 ] with features such as age, developer role, education level, and salary. An analyst might pose the aggregate query

SELECT AVG(Salary) FROM StackOverflow; to retrieve the average salary. Still, understanding why it takes a specific value requires additional explanation; e.g., predicate {Role = C-level Executive} might increase salary, while {Age = 25} might decrease it. Aggregate query explanation breaks down results into additive contributions of predicates, enabling analysts to identify key factors and make informed decisions or policy recommendations.

However, accurately attributing the importance of individual predicates is challenging due to feature interdependencies and causal relationships. Existing techniques, such as DIFF [ 2 ] or MacroBase [ 4 ], fail to capture these dependencies, while game-theoretic methods like Shapley [ 7 ] and Banzhaf values [ 8 ] lack causal awareness [ 9 ]. Causal approaches, including XInsight [ 1 ] and CauSumX [ 10 ], rely on fully specified causal models, which are often computationally expensive and impractical to construct [ 3 ].

To address these limitations, we introduce Causal Banzhaf Values (CBV), a novel approach for causally informed query explanations. CBV integrates causal knowledge of feature dependencies and employs conditional sampling [ 11 ] to estimate contributions accurately. E.g., it accounts for interdependence between Age and Education, guages and Analytical Processing of Big Data, co-located with EDBT/ICDT ∗Corresponding author. 0000-0002-4786-0572 (P. Khani); 0000-0002-1091-9948 (I. Assent) 1Please note that we use the term ”predicate” to refer to feature values for explanations rather than to selection predicates in SQL queries. © 2025 Copyright for this paper by its authors. Use permitted under Creative Commons License ensuring causal information is reflected in the explanation. Unlike methods focusing on entire features, CBV evaluates the importance of predicates, ofering more granular insights. For example, identifying that Role = C-level drives salary outcomes is more informative than attributing importance to feature Role. Moreover, CBV does not require a fully specified causal graph, making it suitable for scenarios where defining complete causal models is infeasible. Finally, CBV is computationally eficient, balancing accuracy with practicality for real-world applications.

2. Related

Work

Statistical query explanation, e.g. [ 2, 4 ], finds associations; relational methods, e.g. [ 12 ], trace data transformations through relational operations like joins. Intervention-based approaches, e.g. [ 13 ], detect outliers in aggregate queries. All lack the ability to provide causal explanations. Gametheoretic methods like Shapley [ 14 ] and Banzhaf values [ 15 ], capture interactions between features, but assume feature independence and disregard causal relationships [ 9 ]. Also, the computational demands of simulating interventional scenarios grow exponentially with dimensionality, limiting scalability. OLAP explanations mostly focus on predefined query structures (cubes) rather than analyzing feature interactions. [ 16 ] extend OLAP operations with abstract highlevel interpretability mechanisms, such as unexpectedness. [ 17 ] annotate OLAP cubes, focusing on statistical rather than causal relationships [ 18 ]. Causal approaches, such as XInsight [ 1 ] and CauSumX [ 10 ], assume fully specified causal models between all features, which are often dificult or infeasible to construct [ 3 ], or even completely unavailable, depending on the domain. Still, some knowledge about causal feature interactions is usually available, e.g., from domain knowledge or causal knowledge discovery approaches. Our approach can leverage partial causal knowledge, meaning CBV can operate even when no complete causal graph is available. This makes it more practical than fully interventional causal methods, while still incorporating causal awareness beyond statistical or game-theoretic methods.

3. Predicate Attribution

The Banzhaf Value quantifies the influence of each predicate (feature-value pair) on a query outcome by computing its marginal contribution across all possible subsets of predicates. Given a function ∶ 2 → ℝ that assigns a value to CEUR

ceur-ws.org each subset of predicates, representing the query result (e.g., average salary), the Banzhaf Value of predicate ( = ) (feature with value ) is: (=) = 2−11

∑ ⊆ ∖{(=)} [ ( ∪ {( = )}) − ()

Banzhaf Value is particularly suited here because it considers subsets, so is inherently order-independent. This aligns with query explanations, where order of predicates does not influence their contribution (e.g., {Education = Master's, Developer Role = CTO} is equivalent to {Developer Role = CTO, Education = Master's}). In contrast, Shapley Value [ 14 ] relies on coalition-based marginal contributions which inherently consider order. Banzhaf Value computation is thus more eficient, as it avoids the factorial complexity of Shapley Value [ 15 ].

Banzhaf Value (BV) ofers a systematic framework for attributing contributions by evaluating all possible subsets of predicates. However, its limitations become evident when applied to aggregate query explanations. BV assumes independence among predicates, overlooking dependencies such as between Age = 10 and Education level = Doctoral Degree. This lack of causal awareness leads to misrepresentation of contributions. Also, BV distributes contributions symmetrically, failing to account for the hierarchical and asymmetric nature of causal chains (e.g., Education afects Role, which influences Salary). As a result, foundational predicates are undervalued. BV also incurs high computational overhead by evaluating all subsets exhaustively, which becomes impractical for high-dimensional datasets. CBV addresses these issues by integrating causal knowledge via partial DAGs to respect predicate dependencies, enabling accurate attribution. Unlike BV, CBV attributes contributions to specific predicate-value pairs, capturing their individual impacts while reflecting causal asymmetry. By focusing on causally valid subsets and using sampling techniques, CBV achieves eficiency without sacrificing accuracy. For predicates without causal ancestors, CBV defaults to BV, ensuring consistency.

4. Causal Banzhaf Value

Traditional Banzhaf Value fairly attributes contributions by averaging marginal efects but assumes feature independence. For instance, in a causal chain where Education

Years Coding

Age Major

Salary

Student

Education Developer role

Level influences Job Role, which then afects Salary, it treats Education Level and Job Role as independent contributors, potentially misestimating their true impact.

We propose the Causal Banzhaf Value (CBV), which incorporates causal knowledge into attribution by working with partial causal graphs to leverage available knowledge without requiring complete DAGs. CBV focuses on causally valid subsets to maintain consistency and employs conditional sampling to estimate contributions eficiently. This combination of partial causal integration, flexibility, and eficiency makes CBV a practical alternative to causal methods. Unlike Banzhaf value in Explainable AI [ 19 ], which focuses on features, CBV evaluates the contributions of feature values (predicates). This aligns with query explanation needs, where specific feature values, such as Role = Clevel, drive outcomes. Here, causal relationships are at attribute level (e.g., Education), and contributions at predicate level (e.g., ”Master’s”). Predicate-level causal relationships are left for future work.

CBV incorporates causal dependencies by considering causally valid subsets of features, defined based on a partial directed acyclic graph (DAG) : for each feature , we identify its ancestors = { ∈ ∣ ≺ } , where ≺ represents the causal ordering in (Algorithm 1, line 2). Then, we identify the set of all causally valid subsets for feature , denoted as (Algorithm 1, line 3):

← { ∣ ⊆ , ⊆ ∖ {, }}, Thus, a subset is valid if it contains all causal ancestors of ( ), and contains neither feature itself nor the target variable . This ensures that attributions respect the known causal structure, while otherwise adopting the assumption of order independence as in the Banzhaf value for feature combinations where order is not known to impact outcomes. For example, ( Gender = Male ∣ Education = Bachelor's, Age = 25) is identical to ( Gender = Male ∣ Age = 25, Education = Bachelor's).

The Causal Banzhaf Value (CBV) C for a predicate ( op ) of feature , a comparison operator such as equality =, and value is defined as:

C 1 ( op ) = 2−1 ∑ ∑ [ ( r ∪ {( op )}) − ( ∈ r∈ℛ() r)] , where ℛ() is the set of all possible realizations (value assignments) of features in subset , and ( r) is the expected value of the target variable conditioned on subset with realization r. For simplicity, we adopt the equality operator, i.e. predicate ( = ) , in the following presentation.

To estimate ( r) and ( r ∪ {( = )}) , we employ conditional sampling [ 11 ], which maintains feature dependencies and provides high estimation accuracy. For each subset , realizations r() are sampled from the empirical distribution ( )̂ (Algorithm 1, line 7), while features not in ∪ {} are sampled conditionally as x() ∼ ( ∖̂( ∪ {}) ∣ = r() ) (Algorithm 1, line 8). These samples allow the estimation of the expected value ( r) (Algorithm 1, line 13): ( r) ← 1

∑ ( r() , x() ), =1 and similarly for ( r ∪ {( = )}) , where ( = ) during sampling (Algorithm 1, line 14): is fixed ( r ∪ {( = )}) ← 1

∑ ( r() , = , x(=) ).

=1 Here, ( r() , x() ) is the estimated value of target variable given a sampled realization r() of the subset (Algorithm 1, line 9), and x() , which includes the sampled values for all features not in ∪ {} . This term reflects the target outcome based on the sampled configuration of the subset and the conditionally sampled remaining features. Similarly, ( r() , = , x(=) ) captures target variable under the same realization of , but with feature explicitly set to value (Algorithm 1, line 10). The term x(=) corresponds to the sampled values of the remaining features conditioned on = r() and = . By setting = , this estimate reflects the impact of the specific value for feature on the target variable , considering the dependencies defined in the data distribution (Algorithm 1, line 11).

Example Respecting causal dependencies (Fig. 1), for e.g. Education = Master's , CBV finds causally valid subsets Education, and for each ∈ Education, marginal contribution ( r∪{( Education = Master's )})− ( r), where ( r) is the average salary conditioned on r, the realization of predicates in . CBV aggregates these contributions across all subsets and realizations to quantify causally consistent importance.

Enumerating all subsets and performing conditional sampling can be computationally intensive, especially for highdimensional datasets. To address this, We propose to approximate CBV by employing Monte Carlo sampling [ 20 ], which involves randomly selecting a suitable number of subsets from and estimating the corresponding expected values. This approach balances eficiency and accuracy (Alg. 1).

5. Experimental Evaluation

We implement Banzhaf Value (BV) [ 14 ] and Causal Banzhaf Value (CBV) using PyTorch with GPU acceleration on an NVIDIA T4 GPU, with 500 Monte Carlo samples for marginal contribution estimates via conditional sampling. Experiments are conducted on Stack Overflow Developer Survey [ 6 ], ofering insights into developer demographics, education, roles, and salaries. A partial causal DAG is created from the complete version in [ 10 ] by excluding one feature and several edges, see Figure 1.

Figure 2 illustrates BV and CBV contributions for different features (plots (a)-(g)), and for diferent equality and range predicates. For features without ancestors in the DAG (race/ethnicity, age in Fig. 2g, Fig. 2f), BV and CBV produce identical results, as causal dependencies are absent. We now discuss cases where CBV and BV difer.

For FormalEducation (Fig. 2a), BV underestimates contributions of advanced degrees (Doctoral, Master’s, Professional) and overestimates those of elementary and secondary Algorithm 1 CBV Require: Dataset , features = {1, 2, … , } , target variable ; partial causal DAG ; number of samples

CBV for all features ∈ and unique Ensure: Contributions (=)

values for that feature ∈ 1: for ∈ do 2: ← { ∈ ∣ ≺ } 3: ← { ∣ ⊆ , ⊆ ∖ {, }} 4: for ∈ do 5: for ∈ do 6: for = 1 to do 7: r() ∼ ()̂ 8: x() ∼ (̂ ∖(∪{}) ∣ = r() ) 9: () = ( r() , x() ) 10: x(=) ∼ (̂ ∖(∪{}) ∣ = r() , ( = )) 11: (( =)) = ( r() , ( = ), x(=) ) 12: end for ( r) ← 1

∑ () =1 ( r ∪ {( = )}) ←

() ← 2−11 [ ( r ∪ {( ==∑1)})(=−) ( 1 r)] 13: 14: 15: (C=B)V 16: end for 17: end for 18: end for school by treating all efects as direct, ignoring downstream roles like developer role and upstream factors like Age, which significantly drive salaries. A notable example is ”No Formal Education”, which BV incorrectly attributes as highly important for higher salaries. CBV reveals that its apparent importance is actually due to its causal ancestor Age—many individuals in the dataset with no formal education are older and hold senior roles. For UndergradMajor (Fig. 2b), BV overestimates contributions for fields like Mathematics and Social Science, which often afect salaries indirectly through roles or skills. CBV reduces these contributions while increasing contributions for fields like Computer Science, which have direct links to high-paying roles, aligning better with domain knowledge in the causal model. For DevType (Fig. 2c), BV misses the importance of roles like Marketing and Clevel. CBV redistributes contributions, assigning higher values to roles like C-level, influenced by education and age (experience), and lower values to roles like Marketing. While negative attribution to ”Marketing”, and minor attribution to ”Student” may be surprising, it actually captures the underlying causality: Age is ancestor of DevType. Students tend to be young, and with limited work experience, which is captured by CBV. People in marketing roles exhibit a diferent distribution concerning age and experience, where others with similar age and experience tend to have higher salaries. Unlike BV, CBV recognizes this pattern. For YearsCoding (Fig. 2d), BV attributes all contributions directly to coding experience, inflating the importance of mid-level ranges (12–14 years, 15–17 years). CBV incorporates the causal dependency of YearsCoding on Age, recognizing that older individuals naturally accumulate more experience, which indirectly influences salary. This adjustment results in more accurate contributions. For Student status (Fig. 2e), BV suggests similar contributions for full-time and part-time students, which contradicts domain knowledge, as full-time students typically have less time for work and lower salaries. This issue arises because BV ignores Age as an ancestor of student status in the DAG, which appears to be the main cause for salary levels, whereas CBV contributions are in (a) Predicates in Formal Education (b) Predicates in Undergraduate Major (c) Predicates in Developer Type (d) Predicates in Years Coding (e) Predicates in Student (f) Predicates in Race / Ethnicity (g) Predicates in Age line with domain knowledge. For race/ethnicity (RaceEthnicity, Fig. 2f) and Age (Age, Fig. 2g), BV and CBV results coincide exactly, as the DAG has no causal incoming edges for these features. Both methods highlight disparities in salary outcomes for e.g. Native Americans and higher age.

CBV is much faster than BV ( Fig. 3). By focusing only on causally valid subsets, CBV avoids unnecessary computations on infeasible configurations, substantially reducing runtime. This improvement is especially notable for features with many causal dependencies, such as developer role. Even with Monte Carlo sampling applied to both methods, CBV is faster as it limits sampling and evaluation to valid combinations, reducing overhead and efort. For features age and race/ethnicity, without causal parents, BV and CBV produce identical results represented as a single bar for both.

6. Conclusion and Future Work

CBV presents a novel eficient causally consistent method for predicate attribution for aggregate query explanations, addressing the limitations of traditional game-theoretic or causal methods. CBV integrates partial causal knowledge without requiring complete causal graphs, making it a valuable contribution in practice. In real-world applications where causal knowledge is incomplete, the quality of explanations naturally depends on how well the available causal information reflects actual data-generating mechanisms. While conditional sampling preserves feature dependencies and enhances accuracy, it introduces computational overhead, particularly in high-dimensional settings. CBV could be extended to handle continuous domains directly, without the need to define predicate ranges (e.g., for age).

7. Acknowledgments

This work is part of RELAX (Grant Agreement No. 101072456), supported by Marie Skłodowska-Curie Actions.

References

[1] P. Ma , R. Ding , S. Wang , S. Han, D . Zhang, Xinsight: explainable data analysis through the lens of causality , Proceedings of the ACM on Management of Data 1 ( 2023 ) 1 - 27 .

[2] F. Abuzaid , P. Kraft , S. Suri , E. Gan , E. Xu , A. Shenoy , A. Ananthanarayan , J. Sheu , E. Meijer , X. Wu , et al., Dif: a relational interface for large-scale data explanation , The VLDB Journal 30 ( 2021 ) 45 - 70 .

[3] A. Meliou , W. Gatterbauer , K. F. Moore , D. Suciu , The complexity of causality and responsibility for query answers and non-answers , arXiv preprint arXiv:1009 . 2021 ( 2010 ).

[4] P. Bailis , E. Gan, S. Madden , D. Narayanan , K. Rong , S. Suri , Macrobase: Prioritizing attention in fast data , in: Proceedings of the 2017 ACM International Conference on Management of Data , 2017 , pp. 541 - 556 .

[5] Z. Miao , Q. Zeng , B. Glavic , S. Roy , Going beyond provenance: Explaining query answers with patternbased counterbalances , in: Proceedings of the 2019 International Conference on Management of Data , 2019 , pp. 485 - 502 .

[6] Stack Overflow , Developer Survey, 2021 . URL: https: //survey.stackoverflow.co/ 2021 .

[7] L. S. Shapley , A value for n-person games, Contributions to the Theory of Games 2 ( 1953 ) 307 - 317 .

[8] J. F. Banzhaf III , Weighted voting doesn't work: A mathematical analysis , Rutgers Law Review 19 ( 1965 ) 317 - 343 .

[9] T. Heskes , E. Sijben , I. G. Bucur , T. Claassen, Causal shapley values: Exploiting causal knowledge to explain individual predictions of complex models , Advances in neural information processing systems 33 ( 2020 ) 4778 - 4789 .

[10] B. Youngmann , M. Cafarella , A. Gilad , S. Roy , Summarized causal explanations for aggregate views , Proceedings of the ACM on Management of Data 2 ( 2024 ) 1 - 27 .

[11] S. Chakraborty , E. Fischer , Y. Goldhirsh , A. Matsliah , On the power of conditional samples in distribution testing , in: Proceedings of the 4th conference on Innovations in Theoretical Computer Science , 2013 , pp. 561 - 580 .

[12] C. Li , Z. Miao , Q. Zeng , B. Glavic , S. Roy , Putting things into context: Rich explanations for query answers using join graphs , in: Proceedings of the 2021 International Conference on Management of Data , 2021 , pp. 1051 - 1063 .

[13] E. Wu , S. Madden , Scorpion: explaining away outliers in aggregate queries , Proc. VLDB Endow . 6 ( 2013 ) 553 - 564 . URL: https://doi.org/10.14778/2536354. 2536356. doi: 10 .14778/2536354.2536356.

[14] D. Deutch , N. Frost , B. Kimelfeld , M. Monet , Computing the shapley value of facts in query answering , in: Proceedings of the 2022 International Conference on Management of Data , 2022 , pp. 1570 - 1583 .

[15] O. Abramovich , D. Deutch , N. Frost , A. Kara , D. Olteanu , Banzhaf values for facts in query answering , Proceedings of the ACM SIGMOD International Conference on Management of Data 2 ( 2024 ) 1 - 26 .

[16] P. Vassiliadis , P. Marcel , S. Rizzi , Beyond roll-up's and drill-down's: An intentional analytics model to reinvent olap , Information Systems 84 ( 2019 ) 147 - 168 .

[17] M. Francia , M. Golfarelli , S. Rizzi , Describing and assessing cubes through intentional analytics , in: Proceedings of the 24th International Conference on Extending Database Technology (EDBT) , 2021 , pp. 3 - 14 .

[18] M. Francia , S. Rizzi , P. Marcel , Explaining cube measures through intentional analytics , Information Systems 121 ( 2024 ) 102338 .

[19] A. Karczmarz , T. Michalak , A. Mukherjee , P. Sankowski , P. Wygocki , Improved feature importance computation for tree models based on the banzhaf value , in: Uncertainty in Artificial Intelligence, PMLR , 2022 , pp. 969 - 979 .

[20] A. Shapiro , Monte carlo sampling methods , Handbooks in operations research and management science 10 ( 2003 ) 353 - 425 .