Decomposing Interventional Causality into Synergistic, Redundant, and Unique Components
Authors: Abel Jansma
Venue: NeurIPS 2025 spotlight ()
Keywords: causality
synergy
redundancy
attribution
Möbius inversion
partial information decomposition
explainability
XAI
Primary area: probabilistic_methods
TL;DR: We introduce a framework to decompose causal effects into synergistic, redundant, and unique components, revealing how causal power is distributed among variables in complex systems.

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.

Decomposing Interventional Causality into Synergistic, Redundant, and Unique Components
Abel Jansma
Dutch Institute for Emergent Phenomena, the Netherlands
Institute for Logie, Language and Computation, University of Amsterdam Institute of Physics, University of Amsterdam a.a.a. jaa.aaaaammmuuv...l
Abstract
We introduce 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, argue that proper causal decomposition must be interventional in nature. We develop a mathematical approach that systematically quantifies how causal power is distributed among variables in system, using 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, chemical reaction networks, and a transformer language model. 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.
Introduction
Causal language is ubiquitous throughout the natural and social sciences. A ball falls towards the earth because of gravitational force, inflation rises because of excessive money supply, climate changes because of greenhouse gas emissions. In each case, we seek not just to describe what happens, but to understand why it happens- identify the causal mechanisms underlying the observed phenomenon. Implicit in causal language are claims about the effect of interventions or counterfactual scenarios (we can mitigate climate change by reducing greenhouse gas emissions, and inflation would not be as high if there had been no excessive money supply). ). When we study something that is embedded in web of complex causal interactions, attributing causality can become more difficult, so sophisticated mathematical and computational techniques have been developed that all rely on one of two things: either one is able directly intervene in the system and study its response, or one requires a priori knowledge of the causal dependencies in the system. In this study, we focus situations in which causal effects are identifiable, and address the question of causal attribution 220 7 Given that we know the effect of interventions on various variables, how do we attribute the causal power to the different variables? In particular, we are interested distinguishing between three types of attribution: unique, redundant, synergistic causal power. Decomposition into these three classes is commonly done in information theory, where it known as the Partial Information Decomposition (PID, (25)). Given two coin flips, for example, the answer to the question 'was there carried purely synergistically by the pair of coins, since each coin individually carries no information whatsoever about the answer.
number of heads' is
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Here we extend this approach to causal quantities and define a 'partial causality decomposition', Disentangling these three different components of causality is crucial good understanding and control of complex systems. DiFrisco and Jaeger [ for example, already emphasised the importance of identifying and distinguishing redundant and synergistic causality in gene regulatory systems. In machine learning it is common to obtain predictions from opaque models whose internal reasoning not clear. Understanding and attributing causal power, blame, or responsibility to input features is crucial to safe and reliable deployment of such models. In machine learning, this is commonly done using Shapley values [24]]] (2).. which are closely related to the decomposition presented here 13], but fail to disentangle unique, redundant, and synergistic contributions.
Redundant and synergistic causality are related to the established notions of causal sufficiency and necessity 120 9 [Z]. If set of variables carries redundant causal power, then any of the variables suffice to affect the outcome, whereas exerting the synergistic causal power of set of variables necessitates a joint intervention. However, sufficiency and necessity are usually defined with respect to individual outcomes (rung of Pearl's ladder of causation), whereas in this study we decompose an average causal effect (namely, the maximum average causal effeet in Equation (8), which inhabits rung 2). A brief example how one could apply the framework to study sufficient and necessary causes is included in Appendix C
Recently, Maatinez--áácchee et measure (based on mutual information) that they refer to as causal. We, however, are of the opinion that a measure of causality should necessarily be interventional, and not accessible from the joint distribution alone [20]_ Still, Martinez-Sánchezz et al. [19] raise an interesting question, namely, is it possible to disentangle the unique, redundant, and synergistic causal power of a set of variables? We will show that this is indeed possible, drawing inspiration from the Partial Information Decomposition [25
demonstrated a technique decompose an observational
and the use of Möbius inversions in complex systems [13
2 Background
2.1 Partial orders and Möbius inversion
To show the algebraic structure of the decomposition, first need introduce number of concepts. First are partially ordered sets, or posets:
Definition (Partially Ordered Set). A partially ordered set is tuple (S, 5). where S is set and < is a binary relation on S that is reflexive, antisymmetric, and transitive.
When the ordering is clear from context, we sometimes use the shorthand S to denote the poset. An example of partial order is (P(T), S). the power set of T ordered by inclusion. Given a poset S and a, b E S, the interval [a, b) is the set {æ a I S b). If all intervals on S are finite sets, then S called locally finite. We also define the following:
Definition 2 ((Anti)chains). Let (S, <) be a poset. A subset T C S is a chain for all a, b e T, then T is an antichain.
a < b or b < a. for all a, b T. a < b implies
Note that in particular any single element subset of a poset is both a chain and an antichain.
Functions on S can interact with the partial order in various ways. One such function is the Möbius function:
Definition 3 (Möbius function). Let (8,5) be a locally finite poset. Then the Möbius function
ps S x S Z is defined
=z=y us(2,2) 2) if < y
Σ 2
Ms(x,
=
(1)
otherwise
This allows us to state the Möhius inversion theorem (MIT):
Theorem 1 (Möbius inversion theorem, Rota (II). Let (S, <) be a locally finite poset. Let f.g. + R. be functions on S, and let μs be the Möbius function on S. Then
f(b)= Σ g(a)
+
g(b)= Σ ()( b
(2)
b
2

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The Möbius inversion theorem states that sums of function over a partial order can be inverted using the Möbius function. Because the l.h.s. of (2) can be interpreted as discrete integral over the poset, the convolution with the Möbius function on the r.h.s. is often considered as a generalised discrete derivative. Indeed, applying the MIT to the natural numbers with their usual ordering recovers a discrete version of the fundamental theorem of calculus. As many quantities in complex systems correspond to integrals or derivatives over partial orders, the Möbius inversion theorem is powerful tool in the analysis of such systems [13].
2.2 Decompositions into antichains
Principled decomposition into synergistic, redundant, and unique components was first explored by Williams and Beer (25 in the context of information theory under the name Partial Information Decomposition (PID). Their approach forms our main inspiration, so we briefly describe it here. Consider the mutual information X2; Y) that two variables (X1.Xy carry about variable Y. If X2; Y) = v, then one might ask: How are the v bits about Y carried by the pair (2), One could imagine that both variables carry the same v bits, so that the information is carried redundantly, or that both variables uniquely carry 5. Another option that neither variable has information by itself, but it is the joint state of the pair that carries the bits synergistically. For two variables, one therefore writes
(X,,, X2; Y) (o((x)); + (X(X2)))
X2]; + (+([(X)) (x2)))) (X2)))
(3)
where the 'partial' information Io((x;); is the information that variable X; uniquely carries about Y. Is( (s(((X)) (X2]);() is the information that is carried redundantly by the two variables, and la((X1, X2; Y) the information that carried synergistically by the joint state of the pair. Note, however, that the situation becomes more complex if more variables are added. Three variables 1X,Xx, X3). for example, can carry information redundantly between the joint state of {X1, X2) and the state of (X3}. or synergistically between all three variables, etc. The central insight of the PID is that information can be carried redundantly among all incomparable subsets of variables. Given a set of variables S, the incomparable subsets S are the antichains of (P(S), S). denoted A(S). Some elements of A((X1, X2, X3)) and their interpretation are:
(2,1, - synergy among X2,
X2X,, (X2, X+H1 redundancy between (X1, X2) and (X2, (X2), {Xa}} + redundancy between X1, X2
We sometimes use simplified notation of the type ((X1, X2), (X2, X3)) = X2X3x The set [X1,
{[X,, X2]. [X,, X2, }3}} is not an antichain, because [X1,X9) C [X1, X2, Xs). Terms like this should be excluded because the redundancy between {X1, X2] and (X1, X2) is simply the contribution of {(X,, }2)}. We will usually denote sets with uppercase Latin letters, and antichains with lowercase Greek letters.
A nice property of antichains is that they can be ordered with respect to redundancy by setting S 3 when it is least as 'easy" to access the information in a as that in B. For example, we set {{X1, X2 ). [Xa}} < X2), (Xa, (1), because you can access the information in ([X1,X2), {X3}] ( by observing the pair X2) or the single variable X3, whereas accessing the
information in { {2). (X3, Xa)} necessarily requires observing a pair. This ordering can be
formally defined as follows:
as
VB E F.E AÇB
(4)
The redundancy ordering is shown for n 2, 3, 4 in Figure T and allows us to write the decomposition into antichains as
I(X:Y)=
Σ
Iola:Y)
(5)
oEA(X)
which can then be inverted using the MIT to find the contribution of each of the types of information:
To(3: Y) Σ
(a,
(T)(()))
(6)
3

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[0.1.2
(0. 1.
[10)
(02)
[14]
{0, 1
(0))
(12)
(01
{0}
{1}
000
(1)
(02)
[1)
(2)
(21)
(0){1)
(0)2))
(122)
[0](1((2)
Figure 1: The transitive reduction (Hasse diagram) of the redundancy ordering from Equation for the n = 2, 3, variables. The posets contain respectively 4, 18, and 166 elements.
On the r.h.s. of Equation (6. added subscript to the mutual information (a Y), since is now evaluated both on sets of variables, but also on other antichains. On a set of variables, (n; (S, Y), but on general antichain a, In(a; Y) is the information that the variables in share about Y (n referring to 'sharedness') ). There are many ways to define this, and significant portion of the PID literature has focused on exploring different approaches to fix the definition of II 5)). For example, Williams and Beer [25) introduced the PID with Eyy() min I(Y = y; A), that quantifies the expected "specific" information that any souree A é a provides about outcomes of Y. Since this measure was found to behave unintuitively on some distributions, a common alternative is the minimum mutual information (MMI) mmmi(a; (a; Y) min
In (see e.g. 2 6
3
10
the measure min (a; Y)
second complication in performing an antichain decomposition is that the number of ways informa- tion be carried among n. variables grows superexponentially with (namely, the nith Dedekind number). This makes recursively solving the system equations (5) or recursively calculating the
Möbius function in (6), computationally very expensive. This has limited the PID approach mostly to systems with n < 3. Recently, however, a closed-form expression for the Möbius function on A(S) for any S was derived, which can offer double-exponential speedup and opens up the possibility to study larger systems [14]. However, to avoid the complexity of the computation obscuring the simplicity the method, we chose here focus on decomposing quantities of up to three variables. for which such speedups are not necessary.
3 Interventional causality
3.1 Average causal effects
A central object of study in causality is the average treatment effect (20). Given a set of variables Xs = [X₃ é E S}. where S is some indexing set, the average treatment effect (ATE) of Xs on an outcome Y is defined
ATE(Xs: Y) [[Y]do(Xs 21)) Edy(do(Xx= zo)
where the do-operator denotes an intervention, and and denote the 'treated' and 'untreated' state, respectively. The most direct way to estimate quantities like 7 is to perform randomised controlled trial, where the treatment X is randomly assigned to the subjects. In that case, P(Y [do(X z)) = P(Y I). and the ATE can be estimated from the joint distribution of X and Y. However, randomised controlled trials are expensive, and one commonly has to rely on observational data. In
that case, the ATE not directly observable, because PdY|do(X = =)) & P(Y = 2). In fact, probabilities under interventions are by definition not derivable from the joint distribution only. One always needs additional information about the causal structure, usually given in the form of a directed acyclic graph (DAG) called the causal graph. To estimate the effect of interventions, Pearl [20] developed rules that relate interventional quantities on the causal graph to observational quantities relative to transformed graph, collectively referred to as the do-calculus, By combining knowledge

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of the joint distribution and the causal graph, one can estimate the effect of interventions, or conclude that an effect is not identifiable. It is these true causal effects, defined terms of interventions, that our method decomposes.
Since aim capture the total causal power in a of variables, not just the effect of binary treatment, we slightly modify the ATE to a 'maximal average causal effect' by defining:
MACE(Xs; Y)
max z,z'EXs
=)] E[Y = z²)l)
(8)
where Xy is the set of possible values that Xs can take. Note, however, that this is just one way to characterise causal strength. For instance, one could instead maximise the difference []]]ssXX x)] - ]|] with the observational state. Many definitions are possible (see e.g. [15] or the alternative explored in Appendix ( each of which can be similarly decomposed using our method. Further note that the MACE is not a measure of the causal effect of Xs on Y, but rather a measure of the maximal causal power that Xş exert Y. This is an important distinction, it identifies which variables have causal power, but does not indicate which interventions will lead to which outcomes.
The MACE quantifies the maximal causal power that set of variables Xs can exert on Y. This definition has the advantage that it does not rely on a priori assumptions about the possible values of the variables, but it no longer captures the direction of the causal effect. It is this quantity that we wish to decompose into synergistic, redundant, and unique components. This makes sense, since while the MACE quantifies the total causal power of a set of variables, it does not tell us how this power is distributed among the variables. To understand and steer the behaviour of causal systems, can be crucial to know if the causal power is redundant, synergistic, unique.
3.2 Decomposing the causal effects into antichains
In order to decompose the MACE over the antichains, necessary extend its domain to all antichains. For antichains of cardinality one, the definition from Equation (8) can be used. For general antichain a, we define this 'redundant' MACE, denoted as MACEn Y), as the minimum of the MACEs of elements of the antichain, since this is the causal power that they share:
MACEn Y) min MACE(X Y)
(9)
ACo
min max AEa EXA
(olY)do(XA = x)] - EE[Y do = x²)])
(10)
Note, however, that one could come up with different definitions of redundant causality, as long as it reduces to the chosen definition of the causal effects on antichains of cardinality one. With this definition, we can decompose the MACE of of variables Xr C Xs over the lattice of antichains as
MACE(X;; Y) MACE, ((T))
Σ C(aY)) aS(X7}
(11)
where C(a; Y) denotes the 'partial causal effect' of antichain a to the full MACE. This corresponds 'partial causality decomposition'. A Möbius inversion then shows that the partial causal effects can be written as
C(3 Σ (axxs)) (a;C) as3
(12)
where (A(Xs) is the Möbius function of the antichain poset (A(Xs) S) which can be calculated using the fast Möbius transform [14). One nice property of the MACE is that it is monotone function on the antichain poset:
Lemma 1. The function MACE An RR is monotonic with respect the redundancy ordering on An.
Proof. See Appendix A
This lemma serves to lend credibility to our claim that the definition of MACE is sensible, but also implies the following:
Theorem 2. The partial causal effects C(a; Y) are nonnegative.
5

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OR
XOR
AND
COPY
1.0
X
X
X1
Cla:
0.5
x
X1
0.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
p
Figure 2: The causal contributions C(a; Y) of the antichains of logical gates.
Proof. Immediate by inserting Lemma | innto the proof of Theorem 5 in 25
Nonnegativity of causal effects makes intuitive and practical sense. Consider the following:
syn(S:V) = Σ
C(a:Y)
(13)
aEA(S)
AAEa:|A
This quantity SSn(S; Y) is the total aggregated synergistic causal power within the set of variables $ on Y. It captures all causal power that requires coordination among multiple variables. Nonnegativity of the C(o; Y) implies that even when not all these terms in the sum can be calculated, partial sum will still give lower bound on syn(S;
4 Decomposing causality in practice
Code to reproduce the figures this section is available at
4.1 Causal power in logic gates
The causal graph of a logic gate with two inputs is simply collider structure: X1 + Y + X2. In this case, the do-operator reduces to the see-operator (by the back-door criterion P(Y|do do)))o(())) ()) = P(Y[X)
if Y IL X in Gx, the graph with all arrows out of X removed), that is, E(Y do((1)) E(Y]X)). so we can just use the conditional expectation to calculate the MACE.
Let the two inputs be independently drawn from binomial distribution with P(X, = 1) P(X 1) p. and the output be their logical OR, AND, XOR, or COPY, where COPY(X,, X2) X1.
The MACE for each of these, as function of p. can be easily calculated by hand. For example,
(ACE X1 OR X2) B(X, OR X₂ dX 1)] E[X1 OR X2|do(X1 0)]
(14)
= 1 - p
(15)
((); XOR X2) 11 2pp
(16)
The causal contributions C(a; Y) are then simply the Möbius inversion of these values, and shown in Figure 2 This shows that the causal contributions indeed disentangle the unique, redundant, and synergistic interventional power the variables. the OR gate, at low the causal power is very redundant: changing either output to I will probably change the output state. In contrast, at high only way to affect the outcome is generally to set both variables to which requires synergistic causal control. The inverse is true for the AND gate. Controlling the XOR of the inputs at p requires full synergistic control, whereas for large or small the causal power is mostly redundant because flipping either to 1 at low p. to 0 at high p, tends to activate the output. The unique causal power of an input vanishes in all gates that are symmetric under permutations the inputs. In contrast, the causality in the COPY gate is completely built from the unique causal power of X1-
4.2 Causal power in cellular automata is context-dependent
more interesting causal structure is that cellular automaton. Consider ID cellular automaton with N cells, where each cell can be 0 1. The state of cell at time + 1 is determined by the state of itself and two neighbours time I. We consider the causal power cell B and its two neighbours at time t have over B at time t 1. The causal graph takes the following form:

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A
B
Ct-1
A
B
C
B
The MACE of A can be written M Be+1) = max (E| |o( Ar = s]] E(B (|o( = s'Il) e,8"
which shows that we need access to quantities like pl BB+1 |do( A, = 8)). Note that there are a lot of backdoor paths from A, to Bt+1, but all are blocked by the set (Be, C² so we can write:
pl Bet |d( Ag = at)) = Σ p(B.+1|A = az. B, = be, Co = c)p(Be = be, C. = cc)
berct
p( |o( A, az, a)) Σ p( B az, B. be. C. a) )p( = b,)
p(Br+1 =
a,, B. B, b,, C. = c.) - B, C= c.)
However, we do not have a clear prior probability distribution on B, and C., so there are multiple ways to evaluate these expressions. First, we consider the 'maximum entropy" solution, which simply assumes that the input states are drawn from a uniform distribution. For a single intervention, this implies:
B( AA = s)) = Σ P(B = Br = be, Ca =t)
(17)
Decomposing causal effects under this assumption essentially decomposes the causal power under maximal ignorance. To calculate the MACE we just need to calculate the average effect of A on B++1, conditional on the possible states of B, and C₁, which can be immediately read off from the rule specification. Another option to let the inputs always be zero, so that p( Bt. C.) ,',,00 which gives the causal power in the context of the empty state.
Alternatively, could estimate the prior distribution of B, and C. based on data from simulated automaton, which gives the causal power decomposition of the dynamics conditional on an initial state. We will consider two possible initialisations: a random initialisation, where each cell drawn from Bernoulli distribution with p 0.5, and a middle-1 initialisation, where all cells are zero except for the middle cell, which
to one.
The studied rules and their causal decompositions are shown in Figure 3] for each of the priors. To get empirical estimates, simulated automata with 100 cells and periodic boundary conditions that evolved over 10k steps, where the first 500 states were discarded.
The results give nuanced and context dependent description of the causal power inside the automata. For example, if you want to control the 'Rule 90' automata with a single intervention, then this is possible under a middle-1 initialisation, but not under random initialisation. More details on the decomposition for each of the shown rules are available from Appendix B
4.3 Rate-dependency in the causal decomposition of a chemical network
We are interested in seeing how the causal decomposition of a system can change as one varies a parameter. To illustrate this, consider the following chemical network. Two chemicals X, and Xg can spontaneously form a molecule Y at rates k and k2, but can also come together to form Y at rate kg- The molecule Y then spontaneously degrades at rate k4, which set equal to 1. The concentration of Y is described by the following rate equation and steady-state concentration:
d[Y] Au [X]
Ko X2] + keg [X hu
(18)
dt
Y (X1, X2) = [
[
Kg X [x,]
(19)
7

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54
86
ABYAC
AJ8/C
AB|AC
AC.BC
AIBIC
A|C
110
190
240
250
0.3
8C
A|B
A|ac
sjc
AIPIC
ABACICC
ABC
AC|8C
ABBCC
AIC
Zeros
Middle-1
Uniform
Random
Figure 3: The causal decomposition of the cellular automata function of the probability of the inputs. Rule number indicated above each figure. Error bars indicate standard deviation across 20 random initialisations. Only antichains with causal power of least 0.01 in least context shown.
X2
Cla:
X|X]
10
10"
10a
Kg
Figure 4: The causal power of perturbations in chemical system. Parameters are set to k1 10,
lg 1. [X1]= [Xa] =
imagine that the experimenter able to modify the concentrations of X, and X₂ by adding an amount 6; to the concentration of X,, and that the target variable is the normalised concentration y = y X'] Note that the causal structure of this system is still just collider X; Y X₂,
so calculate the effect of an intervention on one of the concentrations, we just need calculate the conditional expectation of Y given the intervention, which is the steady state concentration (assuming that the system equilibrates quickly).
( =()) Mas + X2)
(20)
Y1.s (X1,
The MACE of an antichain of variables {X1, X2) Y is then again simply given by Equation (T). where the inner maximisation is trivial because we assume that the intervention is either of size or
The different causal contributions are shown in Figure as function of the rate k3. low k3. spontameous synthesis of Y dominates, and since k1 > k2 almost all causal power lies uniquely with X1. At high h3, the combinatorial synthesis dominates, which is reflected by the fact that the causal power lies mostly with synergistic control. At high ks, the asymmetry in spontaneous synthesis of Y no longer relevant, so -ynerriitis causal power is left redundant.
4.4 Synergistic and redundant semantics in transformer language model
Finally, we demonstrate our causal decomposition in a more realistic scenario: study the causal effect of string completions on sentiment analysis scores by language model [23), as made avail- able through the Transformers Python library [26]. Let the baseline sentence be "this movie is' Let intervention correspond to appending word to the baseline sentence, and define the

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Completion: "not bad"
"horribly bad"
"really bad"
Component Value
0
-10
Syn
Red
Un(1)
Un(2)
Syn
Red
Un(1)
Un(2)
Syn
Red
Un(1)
Un(2)
Figure 5: The causal decomposition of sentence completions in a language model for sentiment analysis reveals that semantic synergy, redundancy, and uniqueness accurately captured. The box plots show the distribution of the causal components across 25 base sentences (see Appendix(D)
causal effect of appending string A to be:
CB(A; (("hhis movie is" A) ("this movie is")
(21)
where is the model's (logit) positivity score, and addition represents string concatenation. Since the causal effect is now signed, slightly modify the definition of the redundant causal effects to preserve the sign:
mea CE(A;Y)
if E a :E((A; > 0 if a CE(A; <0 otherwise
CE, Y)
maxaEa CE(A:Y)
(22)
That is, the redundant causal effect the strongest signed effect that all elements from a can achieve. With this, we decompose the effect of sentence completions into synergistic, redundant, and unique effects of words. We investigated examples of each of these three components: the completion "horribly bad" represents redundant semantics (both words generally signal negative sentiment), "not bad" represents synergistic semantics (the combined meaning is different from that of the individual words), and "really bad" represents unique semantic content (the sentiment fully contained in the word "bad"). The results are summarised Figure The decomposition matches with the above intuitions about synergistic, redundant, and unique semantic content, as the decomposition for "horribly bad" shows strong negative redundancy, "not bad" shows strong positive synergy that reflects the semantic negation (as well negative redundancy, because the model associated both "not" and "bad" with negative sentiment), and "really bad" shows strong unique negative sentiment for the word "bad".
5 Discussion
We have presented principled decomposition of causal effects into synergistic, redundant, and unique components. Decomposing causal effects in logic gates, cellular automata, chemical networks, and language models revealed the synergistic, redundant, and unique causal power of different variables. Furthermore, we illustrated how the causal decomposition can depend on the dynamical the value of its parameters. These insights allow for more nuanced
context of the system, understanding of the causal structure system, and can used to guide interventions in a system to achieve more desirable outcomes.
The formalism similar in spirit and algebra that of the partial information decomposition, but by decomposing an interventional quantity we are able to disentangle synergistic and redundant causal power. The approach outlined here would in principle work for any quantity where notion of synergy and redundancy can be defined, in particular for other definitions of causal power than the one Equation ( We encourage and anticipate further exploration of this approach, as our definition of the redundant MACE is deliberately simple to keep the presentation of the general formalism transparent. Two examples of extending the definition of redundant causality were already presented: one that preserves the sign of the effect of an intervention (Section 4.4). and one that replaces the

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MACE by a counterfactual outcome (Appendix C. Both were inspired by the MMI measure from the PID literature, but one can imagine deriving different measures of redundant causality from other PID functions, or even completely novel ones. Causal decompositions in other contexts, like in climate systems with feedback loops, may require a more sophisticated notion of redundant causality. Janzing et al. [5], for example, define causal effects in terms of Kullback-Leeibe divergence, which can be similarly decomposed [[3]. More generally, our approach fits into the wider context of deriving Möbius inverses of observable quantities to obtain fine-grained description of complex systems 13. which might be used to derive different decompositions of causality in the future.
While the aim of this study is similar to that of Martinez-Sánchez et al. [19], we have taken a very different approach. We believe that decomposing causality fundamentally requires an interventional quantity, so decomposing a quantity derived purely in terms of the joint probability distribution, as done by Martínez-Sānchez et al. [9] for mutual information, cannot yield true causal insight. While Martinez-Sánchezz et al. [19 consider it an advantage of their method that it does not scale with the Dedekind numbers, we believe that the superexponential growth of the number of antichains is feature, not bug. It is reflection of the fact that the number of ways information can be carried among n variables fundamentally grows superexponentially with n. Bartinez-Sánchhe et al. [19] only consider redundancies among individual variables, not considering possible redundancies between pairs. For example, they do not include redundancies of the form { {X,,,x, X2). {X3) }} }. or {2X, {X3, ). both of which are significantly nonzero in the cellular automata studied here. In general, their approach decomposes the total mutual information into 2(2" - 1) + n terms. By not including the full set of redundancies, but still requiring that the sum of the terms adds up to the total, their method conflates multiple sources and does not disentangle the causal structure. While their results suggest that their method can be useful in understanding the structure of complex systems, we believe that it does not disentangle redundancy from synergy, and that they do not capture causal quantities.
Limitations and future work Our approach faces similar limitations to the partial information decomposition. Most pressingly. the number of ways in which causal relationships can be redundant and synergistic scales superexponentially with the number of variables. While this Dedekind scaling in theory limits our analysis to around five variables, we believe that in practice it does not strongly diminish the applicability of redundancy decompositions like the one introduced here. Our approach can yield very fast decompositions for up five variables, the computational bottleneck being the calculation of the MACE, for which more efficient proxies might be found. While the fast Möbius even larger systems, decompositions among more than five variables quickly become hard to interpret and are therefore not likely to be of practical relevance (partial causal effects quickly become hard to interpret, since the causal power among n variables can be distributed among up to (1x/2)) sets by Sperner's 's Theorem).
transform from 14 can be employed calculate parts of the decomposition
further limitation shared with the PID is the ambiguity of the definition of redundancy. We chose to base our definition of redundant causality on the (MMI) measure (I, which has been criticised for its simplicity 22 Still, these limitations have not hindered widespread application of the PID to real-world systems (Luppi et al. [17 and Luppi et al. [88 being two recent MMI-based applications of the PID to human brain data), so similarly should not hinder real- real-world applicability of our approach.
Another limitation is causal inference itself: our method can decompose causal effects, but does not provide easy way to estimate them. It is well-known that accurately estimating causal effects is hard problem, which can limit the applicability of our method. Still, the decomposition can be used to gain insights in variety of real-world systems where we can either do interventions, have models that can simulate them. In systems with complex control, like gene regulation or the climate, there might be a combination of both redundant causality (which could improve robustness) and synergistic causality (which could allow for more efficient or fine-grained control). Understanding how to intervene in living systems to achieve a desired outcome is also a major challenge in synthetic biology and medicine. Disentangling causal power in Al systems is another important application, as it can help understand, steer, and align their behaviour. Lindsey et al. [16) recently developed a method to intervene on the computational graph of a large language model, which could be a very natural setting to apply our method. In the social sciences, one could imagine that individuals can have both redundant and synergistic control over the behaviour of the group. How to attribute responsibility, blame, or rewards in such a social network both quantitative and a philosophicall/ehhiall question. We hope that our approach can provide some insight into the former.
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Acknowledgments and Disclosure of Funding
The author thanks Joris Mooij, Philip Bocken, Patrick Forré, Rick Quax, and Daan Mulder for helpful comments and suggestions concerning causality, synergy, and early draft of this manuscript, and acknowledges support from the Dutch Institute for Emergent Phenomena (DIEP) cluster via the Foundations and Applications of Emergence (FAEME) programme.
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54
86
180
Figure 6: The various cellular automata rules studied, initialised with single the middle of 100-cell array and evolved for 50 steps.
A proofs
A.1 Proof of Lemma 1
We first prove the following (obvious) fact:
Lemma 2. The function MACE PP() -+ R.. is monotonic on (P(X),
Proof. Assume that S 8. Let a" be the element of that achieves the maximum, such that MACEn Y) MACE(a" :Y). Then, since C B, we know that a" E B, so MACE, Y) 2 MACE(a"; Y) MACE, (a; Y). Therefore, B = MACEn(a; (); S MBCE,(8;Y)
We can now prove Lemma]
Lemma I The function MACE : An, + R is monotonic with respect to the redundancy ordering on An.
Proof. Recall that a 3 4b B, 3a Ea: a b. Now assume that given a S B, we have
(3,1) < MACE (a;Y) min MACE(a; aEa
(23)
min MACE(b; Y) BEA
(24)
By Lemma Equation 24) can only be true as long as $b E 8,0 a a C. since 3. know that Yb B, 3, E a < 3
However, which contradiction. Therefore.
MACE- Y) S MB
B Causal decomposition of cellular automata
The first few steps of a middle-1 initialisation of the automata rules are shown in Figure 6 We here provide further details on how the causal decompositions in Figure relate to these automata rules.
Rule 30 can be written B,+1 A, XOR (B, OR Ct), which shows that A indeed has some unique causal power, but that there also synergistic causality between A and C. However, in a context of only zeros, the XOR reduces simple OR gate, which makes the causal power fully redundant.
Rule 54 corresponds to B (A, OR C,) XOR B, so can be fully redundant in the context of zeros, but contains synergistic causality with B in all other situations. Note, however, that there is redundancy among the two possible pairwise synergies with B.
Rule 86 is the mirrored version of rule 30, which is reflected by the equivariance of their decomposi- tions under A C.
Rule 90 can be written as B.41 = A₁ XOR C,, which the causal decomposition correctly reveals either purely synergistic redundant causal power among A and C, depending the context. Only with middle-1 initialisation does the causal power become mixed, which the initialisation that makes rule 90 evolve fractal Sierpiński triangle.
Rule 110, famously complex and Turing complete, contains the most complex causal structure the rules studied here, though this is not visible from the causal decomposition in the context of zeros.
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Rule 190 can be written as Bi+1 = (A, XOR B₁) OR C. which is why the causal power is al- ways fully redundant in the context of zeros, but shows both pure synergy as well as redundancy between AB and C' in the uniform background. Under different initialisation, however, the causal decomposition become more complex.
Rule 240 is the exceptionally simple B,+1 = Ar, which is why every prior results in all causal power lying with A.
Rule 250 B++1 = A, OR Cr, and so, similar to the OR gate from Fig. at p = 0.5, contains equal parts synergistic and redundant causal power, except in the context of zeros, under random initialisation (which under this rule is equivalent to a context of only ones).
C Necessary and sufficient causes
To illustrate the flexibility of the framework for other causal quantities, consider decomposing the actual value an outcome Y, which is the value that Y takes when the input variables take their actually observed (binary) values Xg 2g. We extend the definition of the outcome to antichains as:
(a) min (do 1), Xai ISSA)
(25)
This now describes a counterfactual quantity, in contrast to the MACE, which puts it on the same 'rung' the usual notions of necessary and sufficient causes. We follow Halpern and Pearl (9) and define cause to be conjunction of primitive events. A Möbius inversion of Yn can identify necessary and sufficient causes as follows. We vi E S x4 0 for simplicity, and calculate the inversion as
D(B = == (a,
(26)
asS
Similar as before, this yields a nonnegative decomposition when Yn is monotone on (P(Xs), C). When ( Y) then the conjunctions {AWE B b B E B3} are the sufficient causes of Y A.A necessary cause is cause that appears in every sufficient cause.
For example, when ((x)) Vies X; (cf. the disjunctive forest fire model from Halpem {8) then D(3; Y) = 1 if and only if B [[X,) E S}. indicating that activation of either of the variables a sufficient cause of Y, but there are no necessary causes.
In contrast, when Y(x)) AKES X; (cf. the conjunctive forest fire model from Halpern (8), then D(3; Y) = if and only if 3 = {Xs). indicating that the conjunction Ares X, = 1 is both a necessary and sufficient cause of Y = 1. However, in a context where x; = 1, D Y) = if and
only if B = {Xs((t)) J. indicating that smaller conjunction is sufficient cause.
When Y(x)) = 1 iff Σ; X; 2 2, then D)3;Y) if and only if B (Xr T C S, T = 2}. so conjunctions of variables are sufficient causes of Y 1.
As already noted by Halpern and Pearl 99. one might also allow causes to be formed from disjunctions. The set of all causes then be ordered by logical implication, yielding the free distributive lattice generated by Xs. This happens to be isomorphic to the redundancy ordering on the antichains [14, so under this definition of causes, the decomposition simply assigns value to every possible cause.
D Base sentences for sentiment analysis
The 25 base sentences prepended to the string completions in the sentiment analysis decomposition are:
"this movie is", "this book is", "I found this movie to be". "film rating:", ", "my opinion of the film is that it's", "the acting was", "the plot felt", "overall, thought the picture was". "the story is", "this film is considered to be". "I would describe this movie as", "the director's work is", "the script is", "the new series is", "the novel is", "what saw was", "the performance was", "this show is", "the reviever said it was", "my impression was that the film is", "the hotel room was", "my experience at the museum was", "I found the service", "Overall experience:", "In conclusion, the event was"
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Reviews:
[{"review_id": "8y3BUVcSBV", "reviewer": "Reviewer_dQV8", "summary": "The 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.", "questions": "Is 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", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 4, "strengths_and_weaknesses": "The 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 $\\mu_S : S \\times S \\to \\mathbb{Z}$", "was_revised": true, "final_justification": "The authors have given thoughtful responses to the reviewers, including revisions.  which only reinforces my view that the paper should be accepted. I have reviewed all of the author responses to my and other reviews, and believe that the paper should be accepted.  Significant small issues with clarity of exposition have been rectified, and additional experiments have been done to ground the theoretical work in a straightforward sentiment analysus setting.", "rebuttal": "Dear reviewer dQV8, thank you for your insightful review and positive but honest evaluation. \n\nRegarding your comments and questions:\n\n> Is 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\nYes, this is a very interesting possibility. Martínez-Sánchez et al never actually construct the partial order on which their decomposition is based, so they cannot exploit the fact that for a fixed partial order structure their decompositions can be reduced to a computationally efficient matrix-vector multiplication (the matrix being a representation of the Möbius function), as was done in [1]. If one manages to calculate the Möbius function on this partial order, this could modify and potentially drastically speed up their method to decompose the mutual information. \n\n> Can you think of clear paths towards using this technique to extend the internal circuit tracing analysis in Jack Lindsey [et al]\n\nYes. In their report, Jack Lindsey et al note that interventions on features in the attribution graph have causal effects on the output probabilities of certain tokens. We can treat the attribution graph as a causal graph (it’s a DAG), the probability of the desired output token as the target variable, and intervene on other features in the graph. This will reveal exactly how the features interact redundantly vs synergistically to predict the output token. This is an important application we envision, though constructing and intervening on attribution graphs is itself not trivial. \n\n> Typo\n\nThank you for catching the typo regarding the Möbius function's codomain; you are correct that it should be $\\mathbb{Z}$. This is now fixed. \n\n[1] Jansma, Abel, Pedro AM Mediano, and Fernando E. Rosas. \"Fast Möbius transform: An algebraic approach to information decomposition.\" Physical Review Research 7.3 (2025): 033049."}, {"review_id": "bEXDEPHqej", "reviewer": "Reviewer_EPS7", "summary": "This 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.", "questions": "I 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\"", "limitations": "no", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "I 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.", "was_revised": true, "final_justification": "I'd like to thank the authors for their clear and extensive answer to my follow-up question. Together with the other discussions on this page, it has convinced me that this work is a substantial addition to what is currently possible with PID, and that the new sentiment analysis example is a good illustration of this.\n\nThere are still choices that a user of this methodology will need to make, including in particular how to measure the causal effect of a set of variables, and how to generalize this to antichains. There a many possibilities for these, but I believe the authors have made a good choice of sensible options to present here. I think it is justified to leave further exploration to future work. I have updated my rating accordingly.", "rebuttal": "Dear reviewer EPS7, thank you for your insightful review and honest evaluation. We particularly appreciate your characterisation of our approach as interesting, potentially useful, technically solid, and clear. We are also grateful for your suggestion to study a more realistic scenario, which we have done by studying semantic synergy in sentiment analysis models, as outlined below. We believe this indeed demonstrates the flexibility and utility of the framework more convincingly. \n\nRegarding your comments and questions:\n\n>W1. The originality of the work is limited, as the PID it is based on has already been applied in similar settings.\n\nWe fully acknowledge that our work directly builds on the established PID framework. However, we believe our main contribution presents a novel method and a crucial conceptual shift: moving from decomposing an observational quantity (mutual information) to an interventional one (causal effects). Even in simple settings, we believe observational and interventional quantities should be clearly distinguished, and represent very different things. The ability to decompose causal quantities fundamentally changes what questions can be asked, and what insights can be gained. \n\n> W2. The 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\nThe simplicity of our examples is intentional; they were chosen to serve as clear, intuitively verifiable proofs-of-concept for the method. That being said, we recognise the additional benefit of showcasing the method on more realistic data, and thank you for suggesting this.  \n\nTo act on this, but keep the results intuitively verifiable, we have used the method to detect semantic synergy and redundancy in a sentiment analysis pipeline. As we are not allowed to share links to code or figures here, we briefly describe what we did, and summarise results in a markdown table. \n\nWe analysed the causal effect of string completions on sentiment analysis scores by the `distilbert-base-uncased-finetuned-sst-2-english` model. Let the baseline sentence be “this movie is”. Let an intervention correspond to appending a word to the baseline sentence, and define the causal effect of appending string A to be:\n\n$$\nCE(A;Y) = Y(\\text{\"this movie is\"} + A) - Y(\\text{\"this movie is\"})\n$$\n\nwhere addition represents string concatenation, and $Y$ represents the (logit) positivity score from the model. Since the causal effect is now signed, we define the redundant causal effect as\n\n$$\nCE_\\\\cap(\\\\alpha;Y) = \\\\begin{cases}\n\\\\min_{A \\\\in \\\\alpha} CE(A;Y) \\\\quad \\\\text{ if }  \\\\forall A\\\\in \\\\alpha: CE(A;Y)>0 \\\\\\\\\n\\\\max_{A \\\\in \\\\alpha} CE(A;Y) \\\\quad \\\\text{ if }  \\\\forall A\\\\in \\\\alpha: CE(A;Y)<0 \\\\\\\\\n0 \\\\text{ else}\n\\\\end{cases}\n$$\n\nThat is, the redundant causal effect is the strongest signed effect that all elements from $\\\\alpha$ can achieve. With this, we can decompose the effect of sentence completions into synergistic, redundant, and unique effects of words. We investigated examples of each of these three kinds. The results are summarised in a table below, and make intuitive sense:\n\n**Redundant semantics:** An example of a mostly redundant completion would be “this movie is horribly bad”—both words are negative. This is correctly reflected by the decomposition: Only the redundant effect between “horribly” and “bad\" is significantly nonzero (the redundancy is -7.13, reflecting a strong negative redundancy, whereas the synergy and unique components are all <0.03 in absolute value). \n\n**Synergistic semantics:** An example of a mostly synergistic completion would be “this movie is not bad”. Both “not” and “bad” by themselves decrease positivity in the model, but their combination is strongly positive. This is reflected by the decomposition: A synergy of 7.8, reflecting a strong positive synergistic effect. The significant redundancy reflects the negativity inherent in both words. The unique effects are much smaller. \n\n**Unique semantics:** An example of a completion where the full semantics are uniquely captured by a single word is “this movie is really bad”. The decomposition reveals that the sentiment is almost fully the result of the word “bad”: all components are smaller than 1 in absolute value, except the unique contribution of “bad”, which is strongly negative. \n\nSummary:\n\n| Text completion              | synergy   | redundancy | unique 1 | unique 2 |\n|-----------------------------|-----------|------------|----------|----------|\n| “horribly bad”              | 0.02      | **-7.13**  | 0.00     | -0.02    |\n| “not bad”                   | **7.83**  | -6.32      | 0.00     | -0.83    |\n| “really bad”                | -0.62     | 0.00       | 0.62 | **-7.15**    |\n\n\n\n\n\nWe believe this example, while still just a proof-of-concept, demonstrates the practical utility of the method in a more realistic setting, and we will add it to the paper. For the camera-ready version, we will put error bars on all these values by decomposing the completion relative to a range of baseline sentences (“this movie is”, “this book is”, “I found this movie to be”, “film rating:”, etc.), to attenuate context sensitivity (our preliminary checks showed all results to reproduce across these contexts). \n\nWe will also add to the discussion how the framework could be applied to more complex systems. For example, in line with suggestions by reviewers dQV8, we will discuss how one could use the framework to disentangle causal contributions in the attribution graphs of LLMs [1].\n\n\n>Q1. line 97: \"that carried\" should be \"that is carried”\n\nFixed.\n\n>Q2. line 175: the reference to Eq. (8) seems wrong; should this refer to the equation at the top of the page?\n\nCorrect, this was a referencing issue that is now fixed. \n\n>Q3. 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\nFixed, the plots are now labelled. \n\n>Q4. line 243: something went wrong here\n\nCorrect, this is now fixed and reads “The studied rules and their causal decompositions are shown in Figure 3, for each of the priors.”\n\n>Q5. line 282: \"deliberatively\" should be “deliberately\"\n\nFixed.\n\n> Limitations: no\n\nJudging by the reviewer guidelines, this response seems to signal that the reviewer could not find a discussion on the limitations of our method. We appreciate you raising this concern, and sincerely apologise that the discussion of limitations was not sufficiently clear. We tried to emphasise limitations of the method in the Discussion, specifically regarding our definition of redundant MACE (sentence starting on line 282) and the computational challenges posed by the superexponential growth of antichains (sentence starting on line 304). We apologise if this was not sufficiently highlighted and will add a dedicated and properly signposted \"limitations\" subsection to the discussion.\n\n[1] Ameisen, et al., \"Circuit Tracing: Revealing Computational Graphs in Language Models\", Transformer Circuits, 2025."}, {"review_id": "r1TIfVUgVu", "reviewer": "Reviewer_Kx4Q", "summary": "The 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.", "questions": "-\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?", "limitations": "Practical applications are not obvious", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strength:\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.", "was_revised": true, "final_justification": "after crucial points clarified in the rebuttal, I can recommend the paper as solid work.", "rebuttal": "Dear reviewer Kx4Q, thank you for your detailed review. We appreciate your honesty in flagging points of confusion and reflecting this in your confidence score.\n\nWe in particular recognise that the relationship between synergy, redundancy, antichains, and their ordering can be confusing to those unfamiliar with the partial information decomposition (PID). Thank you for pointing this out—we hope that our response below can clear most of this up.\n\nRegarding your comments and questions:\n\n> W1: The 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\nWe appreciate you contrasting our method with Shapley values. It is true that Shapley values are averages over ‘coalition’ synergies. However, even if not averaged, these values still conflate redundancies and synergies because the coarser decomposition into sub coalitions does not distinguish the two. \n\nWhile this makes a combinatorial explosion inevitable, this has not hindered widespread adoption of the PID, which suffers from the same combinatorics (see also our response to Q9). Furthermore, the recently developed ‘Fast Möbius Transform’ has also enable a significant speedup of antichain decompositions, and reduced the complexity of some special components, like the pure synergy, to exponential (matching the complexity of Shapley value calculation) [1]. \n\n\n> Q1. The 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\nWe have indeed only fully written out the 2-variable decomposition in the text, only highlighting 3 examples from the total of 18 terms in the 3-variable decomposition (in lines 105-106). We hope our answer to the next question clears up what the general case would look like.\n\n> Q2. I 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\nYour intuition is mostly correct: most antichains (namely, those that contain more than one set) indeed correspond to a redundant component of the decomposition. The remaining antichains—those that contain a single set—correspond to either unique of synergistic components. To summarise:\n\n- **Unique components** correspond to antichains with a single variable, e.g. $\\\\alpha = \\\\{\\\\{X_1\\\\}\\\\}$, and quantify how much of the causal power can be uniquely attributed to $X_1$. \n\n- **Synergistic components** correspond to antichains with a single set (of size >1) of variables, e.g. $\\\\alpha = \\\\{\\\\{X_1, X_2, …, X_n\\\\}\\\\}$, and quantify how much of the causal power is *only* accessible from joint interventions. \n\n- **Redundant components** correspond to antichains with multiple elements, like $\\\\alpha = \\\\{\\\\{X_1, X_2\\\\}, \\\\{X_3\\\\}\\\\}$,  and quantify how much of the causal power is accessible from *either* intervention. \n\nRedundancies of the type $\\\\{\\\\{X_1, X_2\\\\}, \\\\{X_3, X_4\\\\}\\\\}$ can, as you suggest, be considered as a ‘redundancy between synergies’, though this is mostly a linguistic choice. The important thing is its operational interpretation: it quantify how much of the causal power is accessible by a joint intervention on $X_1$ and $X_2$, **or** by a joint intervention on $X_3$ and $X_4$. Our method provides a value for *every* such mode of interaction, providing a complete decomposition into all possible redundant, synergistic, and unique components. \n\n> Q3. On 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\nThe ordering defined in equation (4) is indeed not trivial, and a key contribution of the original PID paper [2]. To see this, note that a component $\\alpha$ is “more redundant” than $\\beta$ if the requirements to access the information/causal power in $\\beta$ are stricter than those you need for $\\alpha$. For example, to access $\\beta = \\\\{\\\\{X_1, X_2 \\\\}\\\\{ X_3, X_4\\\\}\\\\}$, you need to be able to do joint interventions on $X_1$ and$ X_2$ **or** on $X_3$ and $X_4$. In contrast, to access $\\alpha = \\\\{\\\\{X_1, X_2 \\\\}\\\\{ X_3\\\\}\\\\}$, you need joint interventions on $X_1$ and $X_2$ **or** only a *single* intervention on $X_3$. Since $\\alpha$ is thus 'easier' to access, we say it's more redundant and write $\\alpha < \\beta$. Note that *more* redundant corresponds to being ‘lower’ in the ordering (the standard in the PID literature). We will make sure to explain this more clearly in the camera-ready version. Note that while the ordering is crucial to carry out the calculation of the decomposition, it is not very important for conceptual understanding of the decomposition. \n\n> Q4. Mention that poset stands for partially ordered set\n\nThank you---this is now fixed. \n\n> Q5. Line 97: Y missing in I_delta\n\nNot only was there a missing $Y$, there was also some notational inconsistency for $I_\\\\partial$ and $Y_\\\\cap$ in this paragraph. This is now fixed. \n\n> Q6. Figure 3 is suboptimal, the grouping could be more obvious. For instance, by dotted vertical lines that separate the antichains.\n\nWe have shaded the background behind every other group to clearly visually separate them. This has made the figure much clearer. Note that in response to reviewer BJo1, we have also increased the font size in this figure to make everything more legible. \n\n> Q7. “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\nWe recognise and agree that the use of ‘macroscopic’ here does not aid clarity. It is borrowed from the literature on Möbius inversions more generally, but offers no insight in this case. We have changed the sentence to read “[...] would in principle work for any quantity where a notion of synergy and redundancy can be defined, in particular for of […]”.\n\n\n> Q8. I 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\nWe refer to our definition as naive because many applications will require more sophisticated measures of causality---ones that keep track of signed effects, possible outcomes, etc. An example of this is shown in the discussion of sufficient and necessary causes in Appendix C. Another example is the definition introduced in the context of semantic synergy in response to a comment by reviewer EPS7. However, to keep the main text as simple as possible, we decided to centre the main text around a straightforward and somewhat crude or ‘naive’ measure. \n\n\n> Q9. “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\nWe respectfully disagree: we consider the rich algebraic structure a 'feature', because it can reveal very complex and nuanced causal dynamics in even small systems. It is simply a fact that there is a superexponential number of ways in which causal power can be distributed among variables. Scientific applications of the PID on real data sets show that the combinatorial explosion does not make practical applications intractable (see e.g. [3, 4]). \n\n> Q10. Reference [2] does not occur in the text\n\nWhile reference [2] is not discussed in detail, it is referenced as one of the studies that introduced a redundancy measure in the parentheses on line 122. \n\n> Q11. Reference [15] is incomplete\n\nThank you for pointing this out. We have fixed this to read “Dominik Janzing, David Balduzzi, Moritz Grosse-Wentrup, and Bernhard Schölkopf. Quantifying causal influences. The Annals of Statistics, 41(5):2324–2358, 2013”\n\n> Q12. Which properties does an information measure need to satisfy to enable your decomposition?\n\nWhenever extending the definition of a measure of information or causality from sets of variables to antichains of variables is possible, then the decomposition applies. This means that one can apply this framework in any setting where a notion of ‘redundancy’ or ‘synergy’ can be defined. This is what allowed insights from the PID to be used to define a decomposition of causality. \n\n> Q13. wouldn’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\nOne could certainly perform this analysis. If we assume w.l.o.g. that $p_1>p_2$, then the redundancy  $C(\\\\{1\\\\}\\\\{2\\\\}; X_1 \\\\text{ OR } X_2)= 1 - p_1$ and the synergy $C(\\\\{1, 2\\\\}; X_1 \\\\text{ OR } X_2)=p_2$. We believe this adds little new insight while requiring more plots. The main advantage would be showing that $X_1$ now gains a unique component, but this is already demonstrated with the COPY gate. Our aim with these first examples is to familiarise the reader with the decomposition and to illustrate that it matches with intuition, which is why we chose to keep the example as simple as possible.\n\n\n[1] Jansma, Abel, Pedro AM Mediano, and Fernando E. Rosas. \"Fast Möbius transform: An algebraic approach to information decomposition.\" Physical Review Research 7.3 (2025): 033049.\n\n[2] Williams, Paul L., and Randall D. Beer. \"Nonnegative decomposition of multivariate information.\" arXiv preprint arXiv:1004.2515 (2010).\n\n[3] Luppi, Andrea I., et al. \"A synergistic core for human brain evolution and cognition.\" Nature neuroscience 25.6 (2022): 771-782.\n\n[4] Luppi, Andrea I., et al. \"A synergistic workspace for human consciousness revealed by integrated information decomposition.\" Elife 12 (2024): RP88173."}]

Author rebuttal:

Meta-review: The 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. The paper shows that this decomposition can elucidate the causal power of interventions in a set of simple but increasingly realistic causal systems. The proposed decomposition is more principled that Shapley based approaches in the sense of being explicit about which part of the effect requires interactions. The formalism is also very explicit about the distinction of association versus causality.
Because of these strengths, the paper is deemed of great interest for the NeurIPS community.
Decision: Accept (spotlight)