Title: Decomposing Interventional Causality into Synergistic, Redundant, and Unique Components
Venue: NeurIPS 2025
Accepted: yes
Keywords: causality

synergy

redundancy

attribution

Möbius inversion

partial information decomposition

explainability

XAI
Abstract: We introduce a novel framework for decomposing interventional causal effects into synergistic, redundant, and unique components, building on the intuition of Partial Information Decomposition (PID) and the principle of Möbius inversion. While recent work has explored a similar decomposition of an observational measure, we argue that a proper causal decomposition must be interventional in nature. We develop a mathematical approach that systematically quantifies how causal power is distributed among variables in a system, using a recently derived closed-form expression for the Möbius function of the redundancy lattice. The formalism is then illustrated by decomposing the causal power in logic gates, cellular automata, and chemical reaction networks. Our results reveal how the distribution of causal power can be context- and parameter-dependent. The decomposition provides new insights into complex systems by revealing how causal influences are shared and combined among multiple variables, with potential applications ranging from attribution of responsibility in legal or AI systems, to the analysis of biological networks or climate models.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose a way to attribute the effect of an intervention in a causal Bayesian network into components that are associated with different sets. There decomposition is shows what part of the causal influence is synergetic among different variables, which part is redundant and which one can uniquely attributed to a single variable.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\nThe decomposition described here is more principled that Shapley based approaches in the sense of being explicit about which part of the effect requires interactions, while Shapley averages over different background conditions. \n\nThe paper is very explicit about the distinction of association versus causality (rung 1 versus rung 2 in Pearl and Mackenzie’s ladder of causation). \n\nWeakness:\nThe advantage of being more principled than Shapley does not come for free, it renders the decomposition cumbersome with a heavy combinatorial explosion of terms. \n\nThe explanations are sometimes lacking clarity since crucial parts are implicit, see below.\n\nQUESTIONS:\n-\tThe decomposition into synergetic, unique, and redundant information is only explicit for the example with two variables, the general case is only implicit, which I found hard to follow. \n-\tI hope I got the decomposition into antichains but missed how this defines a decomposition into synergetic, redundant, and unique information. Doesn’t the antichain decomposition amount to a decomposition into redundant components, each of which is synergetic? I don’t get it. \n-\tOn a related note, I can roughly imagine what you mean by “A nice property of antichains is that they can be ordered with respect to redundancy by setting”, but I fail to understand it entirely.   \n-\tMention that poset stands for partially ordered set\n-\tLine 97: Y missing in I_delta\n-\tFigure 3 is suboptimal, the grouping could be more obvious. For instance, by dotted vertical lines that separate the antichains. \n-\t“The approach outlined here would in principle work for any macroscopic quantity, in particular for other definitions of causal power than the one in Equation (6)”. Why only macroscopic quantities? Maybe I just don’t understand the notion of macroscopic used here? \n-\tI don’t understand “Our definition of the redundant MACE is deliberatively naive to keep the presentation of the general formalism as transparent as possible”\n-\t“we believe that the superexponential growth of the number of antichains is a feature, not a bug”. This remark sounds like the mindset of pure theoreticians. Of course it is conceptually true, but it doesn’t help if one wants to apply it. \n-\tReference [2] does not occur in the text\n-\tReference [15] is incomplete\n-\tWhich properties does an information measure need to satisfy to enable your decomposition?\n-\twouldn’t the OR example in 4.1 better if X_1 and X_2 had different bias, that is, one has p_1 \\neq p_2?\n\nLIMITATIONS:\nPractical applications are not obvious", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a way to decompose the causal effect of a set of variables $X$ on $Y$ into different components: contributions from the individual variables in $X$, contributions arising only from controlling multiple variables jointly (\"synergistic\"), and contributions that could be achieved by intervening on several subsets of variables (\"redundant\"). Unlike earlier work with the same goal, this paper is based on the interventional notion of causality. Section 4 illustrates the behaviour of these measures on different simulation examples.\n\nSTRENGTHS AND WEAKNESSES:\nI find the work interesting, and see that it is potentially useful to be able to decompose causal effects as described.\n\nThe technical quality of the paper looks solid, and the writing and structure are clear.\n\nThe originality of the work is limited, as the PID it is based on has already been applied in similar settings.\n\nThe three examples in section 4 all consider cases where all X's are causes of Y. Latent confounding only arises in example 2, the cellular automata. However, this example is still quite artificial. While it is good to have an artificial example (like example 1, the logic gates) to demonstrate how the decomposition behaves, for purposes of establishing the practical utility of the decomposition it would be good to include a more realistic example.\n\nQUESTIONS:\nI have just some minor suggestions:\n- line 97: \"that carried\" should be \"that is carried\"\n- line 175: the reference to Eq. (8) seems wrong; should this refer to the equation at the top of the page?\n- the plots in Figure 2 should be labelled with the names of the gates. Right now, their order is OR - XOR - AND - COPY, which is not the order the reader expects.\n- line 243: something went wrong here\n- line 282: \"deliberatively\" should be \"deliberately\"\n\nLIMITATIONS:\nno", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper extends work in Partial Information Decomposition to the problem of decomposing causal effects under intervention. It takes advantage of Möbius inversion to provide a method for decomposing causal effects in terms of synergistic, redundant our unique power of interventions on causal variables.  \n\nThe paper shows that this decomposition can elucidate the causal power of interventions in a set of simple but increasingly realistic causal systems.\n\nSTRENGTHS AND WEAKNESSES:\nThe paper provides an excellent exposition of the relevant parts of PID and causal inference, and introduces a technique that can take advantage of the newly discovered fast Möbius transform. Although the available speedup is not sufficient at this point to improve the causal analysis of complex systems (> 5 variables) it provides a foothold for approximation techniques that might lead to analysis of large systems (including, perhaps, the causal structure of LLMs).\n\nThere is a small typo in the definition of the Möbius function:\n$\\mu_S : S \\times S \\to \\mathbb{R}$, should be $\\mu_S : S \\times S \\to \\mathbb{Z}$\n\nQUESTIONS:\nIs a combination of the purely intervention based notion of causal analysis here and the approach in the Martínez-Sánchez et al paper possible? Would such a combination allow for speedups. \n\nCan you think of clear paths towards using this technique to extend the internal circuit tracing analysis in Jack Lindsey, Wes Gurnee, Emmanuel Ameisen, Brian Chen, Adam Pearce, Nicholas L. Turner, Craig Citro, David Abrahams, Shan Carter, Basil Hosmer, Jonathan Marcus, Michael Sklar, Adly Templeton, Trenton Bricken, Callum McDougall, Hoagy Cunningham, Thomas Henighan, Adam Jermyn, Andy Jones, Andrew Persic, Zhenyi Qi, T. Ben Thompson, Sam Zimmerman, Kelley Rivoire, Thomas Conerly, Chris Olah & Joshua Batson (2025).\n“On the Biology of a Large Language Model.” https://transformer-circuits.pub/2025/attribution-graphs/biology.html\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-22"}