Cyclic Counterfactuals under Shift–Scale Interventions
Authors: Saptarshi Saha
Dhruv Vansraj Rathore
Utpal Garain
Venue: NeurIPS 2025 poster ()
Keywords: Counterfactuals
Causality
cyclic SCM
Primary area: theory

Abstract: Most counterfactual inference frameworks traditionally assume acyclic structural causal models (SCMs), i.e. directed acyclic graphs (DAGs). However, many real-world systems (e.g. biological systems) contain feedback loops or cyclic dependencies that violate acyclicity. In this work, we study counterfactual inference in cyclic SCMs under shift–scale interventions, i.e., soft, policy-style changes that rescale and/or shift a variable’s mechanism.

Cyclic Counterfactuals under Shift-Scale Interventions
Saptarshi Saha
Dhruv Vansraj Rathore
Computer Vision and Pattern Recognition Unit
Indian Statistical Institute
Indian Statistical Institute
Kolkata, West Bengal 700108. India
Kolkata, West Bengal- 700108, India saptarshi.. sahaasaaaaaaaaees r@isical.aaeee in
cs2306Gisical. ac.aalacc.ii. in
Utpal Garain
Computer Vision and Pattern Recognition Unit
Indian Statistical Institute
Kolkata, West Bengal- 700108. India utpal@isical. ac.
Abstract
Most counterfactual inference frameworks traditionally assume acyclic structural causal models (SCMs), i.e. directed acyclic graphs (DAGs). However, many real-world systems (e.g. biological systems) contain feedback loops or cyclic dependencies that violate acyclicity. In this work, we study counterfactual inference in cyclic SCMs under shift-scale interventions, i.c, soft, policy-style changes that rescale and/or shift variable's 's mechanism.
Introduction
Most research counterfactual reasoning (Pawlowski et al., 2020; Sanchez and Tsaftaris, 2022; Saha and Garain, 2022; Komanduri et al., 2024; Melistas et al., 2024; Wu et al., 2025; Kügelgen al., 2023; Kladny al., 2024) assumes that the underlying causal structure among variables be represented by a Directed Acyelic Graph (DAG). However, this acyclicity assumption is often violated in real-world systems. For instance, gene regulatory networks frequently exhibit feedback loops, leading to cyclic dependencies that DAGs cannot capture. Such cycles are integral to the dynamic behavior of biological systems and are crucial for understanding processes like cell differentiation and immune responses. Given the prevalence of cycles in such systems and the availability of detailed perturbation data, there is compelling case for extending counterfactual inference frameworks accommodate cyclic structures.
However, progress in this area has stalled due to the need for new theoretical breakthroughs, as many properties that hold acyclic models no longer apply when feedback loops are present. A fundamental issue is that set of structural equations with cycles may not have unique solution for the endogenous variables. Solvability refers to the existence of at least one solution (equilibrium), and unique solvability means there exactly one solution (almost surely). If an SCM is uniquely solvable -i.e., not a simple SCM-it might generate multiple different outcome distributions or undefined behavior under interventions. Although the class of simple SCMs includes acyclic SCMs as special case, the theory remains much less developed for the cyclic setting.
In causality, hard (structural) intervention (Pearl's do-operator) (Pearl al., 2016) sets variable to a fixed value, severing its dependence on its usual causes. In contrast, shift and scale interventions
first author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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are types of soft (parametric) interventions that modify the value of a variable by some function (such as adding or multiplying by a constant) without removing its original imput links. In particular, hard intervention is just degenerate case of soft intervention (Massidda et al., 2023). This generality means one can ask nuanced "what-if" questions: What if everyone received 20% more of the drug? What if we lowered each student's s class size by 5? Such policies cannot be represented as simple do(X = x) since they depend on individuals' original X. Soft interventions are strictly more expressive in defining counterfactual worlds. More specifically, shift can implement dynamic-like policies ("increase dose for those who had high risk") that static do-interventions cannot capture. Shift interventions have been used to learn causal cyclic graphs (Rothenhäusler et al., 2015), to match a desired causal state (Zhang et al., 2021). Lorch et al. (2024) use shift-scale intervention in causal modeling with stationary diffusions. However, the theoretical foundations supporting their use remain underdeveloped.
Contributions. In this work, we develop theoretical framework for counterfactual inference in cyclic causal models under shift-scale interventions. Our main contributions are as follows:
We show that SCMs satisfying a global contraction condition are simple-ic... uniquely solvable with respect to every subset of variables- in the presence of cycles.
We prove that under shift-scale interventions with bounded scale coefficients (i.e., layl S 1), the intervened twin SCM remains uniquely solvable, ensuring the well-posedness of counterfactual queries.
We establish that this class of shift-scale interventions is closed under composition, making it algebraically stable for sequential interventional analysis.
Under an additional Lipschitz regularity condition in the exogenous noise, we derive sub- Gaussian tail bounds for counterfactual functionals, showing that the distribution counter- factual outcomes concentrates sharply around their mean.
2 Background and Problem Setup
This section provides a brief overview of SCMs and establishes the notational framework adopted throughout the paper. Our exposition aligns with the formalism presented in Bongers et al. (2021), Definition (Structural Causal Model). A Structural Causal Model (SCM) is defined as a tuple
(I, J, X, &,
where I denotes a finite set indexing the endogenous variables, denotes a disjoint finite set indexing the exogenous variables, X IIseI X, is the joint domain of the endogenous variables. with each X, being a standard measurable space, E = [Ies Ej is the joint domain of the exogenous variables, with each E also standard measurable space. f X + X is measurable function representing the causal mechanisms that determine the values of endogenous variables from both endogenous and exogenous inpuls, Pe Ilje Pe, is product probability measure over &, describing the joint distribution of the exogenous variables.
Definition 2 (Solution of an SCM). A solution of M is a pair of random variables (X, E) defined on a common probability space (Ω, FP)) such that:
(i) E 12-+ has distribution Pe;
(ii) X 2 satisfies the structural equations
X E) P-almost surely.
For convenience, we say that random variable X is solution of M if there exists exogenous random variable E such that the pair (X, E) constitutes a solution of M Definition 3 (Parent). For E I, an index & E IUU I is called parent of i iff there does not exist measurable map f. R. x Exp + X, such that for Pe-almost every E E and all X,
x, f(
=
Iz == (((
Exogenous variables have no parents. We write pa(i) for the set of parents of i and extend to sets by pa(O)= =-ioo pa(i).

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Definition (Unique Solvability). An SCM M is uniquely solvable with respect to a subser O _ I there exists a measurable function
90 Pa x Epa(O) - Xo
such that for all E X and Plmmoos every E E,
90 (pa(O))
Jo =o= So(2, e)-
Definition 5 (Simple SCM). An SCM M is called simple if it is uniquely solvable with respect every subsel O ç I.
Acyclic SCMs are simple.
Definition (Twin SCM), Let M be a structural causal model. The twin SCM associated with is defined as
= = (IUT, J, XxX, PE).
where I' == (i': : is a disjoint copy of the endogenous index set, and f X xxxxxx - xxxx x is the measurable function defined by
=(= (f(r,))), f(E',e)
with x, x' E X and E
How the twin map f is constructed For any noise realisation E & and stacked endogenous state (x,, x') = (xi,...,20 TZz "1....,,zzz) E X' x X, the twin-SCM mechanism f X x X x E -
z X x is defined by f(z,x',e) = (f(x, f(x',e)). Written coordinate-wise
(f) jEI, f(()',)),
for some i I(i.e, j E I'),
iec the first copy (un-primed) follows the original mechanism X₃ (x, while the primed copy applies the same function its own state x'. compact notation
i,(,,'',e) = ()(,,),, jEI, (1(('',),, JEE
Definition (Counterfactual distribution ). Let M be its twin SCM. Consider a perfect intervention
P,Pe) be an SCM and let Mtwin
do(I, Ez), iCIUI, E
applied to Mtwin, and denote the intervened model by (Mtwin) )do(I.zz)" If this intervened twin SCM admits (measurable) solution (X, X') X' x X, then the joint distribution P(x,x') 3 is called the counterfactual distribution of M under the perfect intervention do(I, {7) associated with the pair of random variables X').
In the appendix, we delineate a clear correspondence between the twin-network formulation of structural causal models and Pearl's canonical action- abbountion abdduciion-prediction schema for counterfactual inference.
2.1 Shift-Scale intervention
For example, instead of forcing a treatment X to set value, shift intervention might increase each individual's natural treatment dose by fixed amount and a scale intervention might multiply each dose by a factor (e.g. 10% increase) all while allowing X to remain influenced by its usual causes (confounders, prior variables, etc.). This preserves the causal edges into X but changes the conditional distribution structural equation of X.

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Definition 8 Shiit-Scalll intervention). Let M = (I,J, subset I C I. For each j E I choose scale 0z E R and shift by E R. The shift-scale intervention
be an SCM and non-empry
(I, (az : (a)) jet. bz: (b)))Et)
produces a new SCM
Mas := (,, (x aat fi(x,)))) bb, ieī,
(i(xz)),
i $
Perfect or do-intervention recovered as the special case 0j = 0, =
Definition (Shift- Scale counterfactual distribution). Let M be the twin SCM of M. Apply the shift-scale intervention only to the first copy:
(Mowin))
(Mewia) (I.2)))))))"
If this intervened twin SCM admits measurable solution (X, X)) E X X, call the joint law
P (x.x") the shift-scale counterfactual distribution of M under the intervention ss ss ag, by).
Given a subset a set of coordinates I IIIUT and parameters az by E TR. we want to replace the structural equation
x = f-(-) X, = ag F)( + by,
for each j E I. All other coordinates stay unchanged. Encoding these modifications into a single map
25(2,0',e) a, (', bby jei, j$t
0( (e) == (g)(x, (', jETUTT
(s)
gives the intervened twin update rule. Specifically
j iI = (z) (u,,e) bj
i'j=''ET'
()( av fi(x', ,) bv.
Thus each copy (un-primed and primed) is modified only on the requested coordinates, allowing independent interventions on the worlds.
2.2 Semantics of Cyclic SCMs
One natural interpretation of cyclic structural equations is by assuming an underlying discrete-time dynamical system, where the equations act update rules: the value of each variable at time t+1 is computed from the values at time t. The system is then analyzed in the limit as f -> co, focusing on the fixed points to which the dynamics converge. Mooij al. (2013) demonstrate that alternative, yet natural, interpretation of SCMs emerges when considering systems of ordinary differential equa- tions (ODEs). By examining the equilibrium (steady-state) solutions of such ODEs, one arrives at structural causal model that is time-independent, but still retains meaningful causal semantics with respect to interventions. Specifically, the semantics of interventions and counterfactuals (see also Appendix A.1) remain valid and well-defined in this steady-state context, as rigorously formalized by Mooij al. (2013) and further extended by Bongers al. (2021). These are by no means the only routes to structural causal models; indeed, SCMs may arise through a variety of alternative construc- tions and representations, depending on the nature of the system under consideration. Although many physical processes exhibit inertia, static cyclic structural causal models (SCMs) remain appropriate when we focus on equilibrium behavior sample at temporal resolution coarser than the fastest feedback loop. Examples include segelaegulatory networks in single-cell genomics (Rohbeck et al., 2024); market-equilibrium models (Bongers al., 2021); predator- or reproductive hormone axes exhibit feedback loops (Clarke et al., 2014), etc. Our mathematical framework is agnostic about the interpretation of cycles: we formulate everything directly at the level of structural equations with exogenous noise.
ecological systems; Thyroid

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3 Theory
Theorem 1 (Global E P-contraction = simple SCM). Let = (X, .e) be an SCM whose endogenous index set I is finite. Assume each coordinate domain (X; S R) is non-empry and closed. Fix p E [1, co] and endow every product space with the go-morm |æ>|p ;= Cvez (x;l"))/r (for p = oo take the maximum norm). Suppose there exists E [0, 1) such that
(5(x,e) ell, s x 1| -ul- -llp for all x, y E X, EEE.
(1)
Then M is uniquely solvable with respect every subset S I, and hence M a simple SCM.
Proof. For any subset O S let Q := pa O) O. Because every X, is closed in IR (hence complete) and O is finite, the product No = IIieo X, complete under the 8 metric; the same holds for the full space X.
For each pair (me, (pa(O)) define
hsqiem(os Xo Xo, u+ fo zQ. TIV(OU) ppa()) E7 (App)())
where the "dummy" coordinates ZI\(OUO) and ()pa(() may be chosen arbitrarily because they do not influence fo.
For u, v E Xo define u (u, sQ. (OUO)) 0 (v. zQ zI((OU)) E X. Then || ö|p I|u r|[p because ū, coincide outside O Using (1),
(putoo (u) :x (b)ll, = [ī, e) - fo(ū, S
e)I S xi|l - -llp-
Thus each map " is s-contraction the complete metric space (No, |l llm)). By the Banach fixed-point theorem, for every (æQ (pa())) there exists a unique element (I ppa()) Xo satisfying u² h (u*).
Measurability. Fix any Xo- Defining Picard iterates (0) ū and ((++1) Trr (pato)) (n) one obtains sequence of measurable functions converging pointwise to u². Limits of measurable functions are measurable, hence the map
go(rg, (pa(o)) == uu" (rrg, Spa(o))
measurable.
Equivalence.
(-) By definition, (oa(o)) is the unique fixed point u² of the map hraremio) (u) = folu, xQ, (I(() Cpa(O): (p)))))
Hence, if xo go ),)( then xo hx (xo) fo(æ,
(+) Suppose xo fo( Then zo satisfies zo fo(x, h (HO), it is fixed point of heg By Banach's 's theorem, the fixed point is unique, so
To = u = =00 (pa(o))
Therefore,
xo
xo fo(
establishing the defining equivalence in the notion of unique solvability holds for all x and for Pe-almost every
Because C was arbitrary, the same reasoning applies to every subset, establishing that M is uniquely solvable with respect to all subsets and hence is simple SCM.
5

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The global contraction condition implies unique solvability for every subset; hence the models are simple SCMs in the sense of Bongers al. (2021). This places directly inside the setting where their closure results for do-interventions, marginalization and twin networks apply.
Theorem 2 (Unique Solvability of Twin SCM under Shift-Scale Interventions), Let M X, &, Pε) be an SCM such that the causal mechanism f : X & X satisfies global ( oe-contraction for some constant 0 I < 1. Let MMwiu be the associated twin SCM, and fix subset I CIUI' along with thift-scale coefficients at E &Rl by R satisfying:
0max sup layl S 1.
jEI
Then the shift-scale intervened twin SCM (M ast ag. uniquely solvable with respect to every subset of endogenous variables, and is thus simple SCM. In particular, the associated counterfactual distribution P(x.x1) well-defined.
Proof. Let (z,z') E X X' be the endogenous variables of the twin SCM and the twin map f(x, ()), ef( f(x',()))))))))),))) e)). is global r-contraction on x X:
, 1 de) 1)l () -- f(//////////// ,) // s ==((xxx')-(00 (/pp
Now define the shift-scale intervened map 5(x, by:
[a, + bj, j jEi. (f
(2, e)
i
We now show that "(:,,,e is global contraction in the &--ormm with the same constant s. Let u = (I, x'), = v.. Then for any IUZ'
(f f)( s amax (f, fu, ((,,,e)) jeī,
- e)) -0,(,,))
j $ I.
Therefore,
1/p
lğ(u, e) - pll -
Σ
-(((,e))
jet
Since Qmax S have:
lī e) g(v, e)ll & [If(u, f(e, < K||u lu - vlpp-
Thus g(;e)) e) is also global k-contraction on (X x X, llp), a complete metric space since A' is closed and miies-dimensional.
By Banach's 's fixed-point theorem (as used in Theorem 1), the shift-scale intervened twin SCM has unique solution X') for each e, and the solution map is measurable.
Since the SCM is uniquely solvable with respect all subsets UZ.. the intervened twin SCM is simple. Pushing forward Ps through this solution map defines the shift scale counterfactual distribution IP (x,x).
Proposition 8.2 in Bongers et al. (2021) shows that the class of simple SCMs is closed under (i) marginalization, (ii) perfect interventions (do), and (iii) the twin construction. does not give sufficient condition for simplicity; it presupposes simplicity. Theorem 2 gives sufficient analytic conditions for unique solvability the twin under a class of soft, parametric shift-scale interventions, thereby ensuring well-posed counterfactuals beyond hard do.

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Proposition (Closure under (composed) Shift-Scale Interventions). Let M Pe) be an SCM such that for some pE [o] I.JJN E,
llf(z, e) - fly. e)llp I =
pllp-
LE.YEX,VEEE
with constant [0, 1). Consider finite sequence of shift-scale interventions applied successively to M:
ss (1 (2) b())
ss ss(((m) (m), ()()))
2 1,
where for every r =, i,... and every I(r) one has 1 S
Then the resulting (composed) intervened model equivalent to a single shift-scale in- tervention ss(I, rommp, (comp) with |a S 1 for all j E I, and its structural function g(x, e) acomp ④ [(x e) + bcoup satisfies
l19(z,e) e) gly, e)llp S x xII vllp:
Va,ye X, EE
Consequently the intervened SCM remains k-contractive and hence is a simple SCM.
Proof. (1) Composition reduces to a single affine map. For any coordinate j E I the successive updates act Ijj + a, (...a
(...a, (1)
(2, e) + b(1) ...) + b/m-1)) + f/m) which can be
written a comp [z( e) + b bouup with somp II jeio: aj (r) and a corresponding affine drift 1y bymmp, aj Since each factor satisfies / < have mamm < Hence the final system coincides with g( e) @ f(z, e) + bcomp,
(2) Contraction constant is preserved. Let D doap( denote the diagonal matrix of multiplicative factors. For any u IR RR [D [|u||p because every diagonal entry satisfies comp laj < 1. Hence for all æ, and any
ll((z, e) - gly. elp = -ltplll)_(((ee)--ia___ f(u.))).)))
f(y,e)llo S <|| yllp-
Thus g inherits the same global contraction constant K.
(3) Simplicity after intervention. Because the intervened mechanism is still k-contractive with < 1, Theorem 1 applies verbatim: the intervened model is uniquely solvable with respect to every subset of endogenous variables, i.e, it simple.
Remark (Scale factors exceeding one). The proof of Proposition 1 used the bound layl S 1 to conclude that the diagonal scaling D diag obeys |Dulll < ||ul|p and therefore does not
enlarge the global Lipschitz constant. If some multipliers satisfy lay) > 1, simplicity can still hold, but one must verify additional criterion:
Let be the original contraction constant of f and define
= (max (ayl) s.
Emax
jeī
Because | = max, layl, the intervened mechanism is globally -ipschipp [D] e) DJ(y, <) Kimax| - y1p If Kmax <. the model remains contraction and therefore simple; KKmax 2 1 the contraction proof no longer ensures uniqueness and additional analysis is required.
7

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Proposition 2 Bab-Gaussian Tails under Gaussian Noise for Shift Scalle Counterfactuals) Let the SCM M = (I, J. X, & f,PP) satisfy the global 8 -cootrractoon condition of Theo- rem I with constant K < 1. Apply any shift-scale intervention ss, dz bz) to the twin SCM, with maxzet (ay S and let (X,X)) be the unique endogenous solution (Definition 2) of the intervened twin model. Assume
(a) 1-Lipschitz in noise (in G): For all E X, e1 E
(1 - f(x, e2) S |le - -2112- (b) Gaussian noise: ~ (u, Σ) with E 3 o²1 for some 0²
Then for every --Lisschit,fuunciionn X x X - (W.E.. 12)
P PM(X. X') -EM(,, X') 21 S exp -
12
1>
Hence (XXX) is sub-Gaussian with proxy 2(1 - k) -22
Proof. Let + E + X X denote the unique measurable solution map of the shift-scale intere-itteeeeed twin SCM. That is,
(X, X') +EE),
where E P& is the exogenous random variable. By the global k-contraction in and the 1- 1-Lipschitz property e (both (9) & L-Lipschitz with constant == v2 in e:
v2
( - ((e2) - 1 1- k
lei - 62ll2 for all c1, E.
Take any h X x X' + R that 1-Lipschitz in e Write E = μ 22/2 with ~ (0,
Then g(2) == h (# is L/ppsshhtt in 62, Since Σ 0²1, we have || S o.
By Gaussian concentration for Lipschitz functions (c.g., (Vershynin, 2018, Thm. 5.2.3)),
X')] 2 S exp
= exp
2T220²
4(1 K) 2
This proves the stated concentration inequality.
Remark 2 (L version of Proposition 2). Let all Lipschitz and contraction conditions in Proposition
be measured in E (1 S p co), and keep the Gaussian noise assumption E Ni,, E). Then for every 1-Lipschitz h : (X x X, Il,) + R and all t > O.
P MIXXX X')
EM EX(X,X') X'XXX')) 2)) S exp
2
2 (1 (1-x)-2 ||>) /
||zl|e where ||ll2->p supz40 ||
In particular, if S I and d = dim (E), then
1,
p p2 2,
122
11a-p S 1|1122pp with 114122-p d 1 < 2,
-0² for p 2 and 2 (1 x) 202 2 for p Proof sketch. Write E = p+ 11/2,,2 Z ~ N' I); then = + ( x212+))))) (1-x) 2--
so the proxy equals 2 (1 (1-x)-20-
Lipschitz in l2 and Gaussian Lipschitz concentration applies.
Example
Consider linear cyclic SCM with mutual dependence between consumption (C) and income (!):
C = 0.50 I +1+ Ec.
(Fc, FT) ~ ~N(0, 0.)4
(2)
= 0.40 + 0.50 + Er,
8

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Written in matrix form X = (C,I)" X AX+ + E with
0.50
1
A
b
== (Ec, ET)
0.40
0.5
Because the spectral norm is ||Al|e & |||| = v.25 + 0.16 = 0.6403 < 1, the model is globally
contractive and therefore simple in the sense of Definition 5.
For any 2x2 matrix we have (I A)-1 = 0.50 1.25
[8 6] with ab 1, (I A) 1-ab [b.]. With (a, b) (0.50,0.40)
1.25 0.625 and therefore
]
B[X] (I A) b (1.5625, 1.125),
(3)
Cov(X) (I A) (0.0412) (0.00)() A)-
0.0781 0.0563 0.0725
0.0563
(4)
Thus Xots MM, Σ) with and Σ given in (3)(4). The correlation between C and I is p 0.75.
Shift-Scale intervention on I. Suppose fiscal policy reform dampens the effect of both consump- tion and random shocks on income by a factor = 0.8, and provides a fixed income supplement of 1.0 units:
ss as 9) I + a(0.40 C + 0.50 E,) + 0.8, B = 1.0.
The intervened structural parameters
0
0.50)
(114) 1.4 -1(11) so | S || v0.25 0.1024 = 0.5936 < 1.
A'
8'
0.32
Scaling also affects the exogenous term: E" = (Ec, TEI) Σe = diag(0.04, a2.044
=
diag(0.04, 0.0256). Contractivity preserved, hence unique interventional equilibrium exists:
E [im] - (2.024, 2.048)
0.0658 0.0363 0.0363 0.0421
Cov(Xm) (T -') - A -
Pim R 0.69.
(6)
Consumption rises by ~ 29 % and income by 82 %, while the C-I correlation falls from 0.75 to 0.69.
From intervention to counterfactual via the twin SCM. Equations (5)-6) describe the interven- tional distribution P(Xmt), i.c. what would observe if the shift-scale policy were enacted for the whole population. To answer individual-Ievel counterfactual queries ("what would this household's consumption be had the policy applied?") we follow the twin network construction. Let (c, i) denote the actually observed consumption and income for one household. The twin-SCM duplicates every endogenous variable and shares the same exogenous noise:
C 0.50 / +1 Ec,
C = 0.50 1 Ec,
(Twin SCM)
= 0.40 + 0.50 + F,, I' = 0.8004004 + 0.50 + F2) +1.
where the primed copy encodes the shift-scale intervention ss(L, a=0.8, 3=1.0) and the unprimed copy remains factual. From the factual equations, we can solve
Ec = 0.50
E, = 0.40 0.50.
(7)
Eliminating C' from the primed equations of the twin SCM yields
= 0, Ec.
Substituting gives the counterfactual income i'( 21 21
16 i 1.190476 + 0.761905
Back-substitution furnishes the counterfactual consumption C'(c, = + 0.595238- 0.119048 Above equations give the counterfactual response mapping (c. (C,,). Because the mapping is affine and the factual distribution is Gaussian, the marginal counterfactual (C', I') is again Gaussian with [[((", I') = (2.024, 2.048) and covariance exactly matching (6). Here, the twin-SCM reconciles individual counterfactual semantics (C" i), l'(c, i)) with the
25 -iiio
population-level interventional distribution already reported, while remaining in closed form due to the lincarity and contractiveness of the model.

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Limitations
The k-coontraction condition must hold uniformly over the entire state space; many realistic feedback systems may violate this in certain regimes, even though they still admit unique equilibria. Our con- centration result further relies on Gaussianity of the exogenous variables. For heavy-tailed or merely bounded-moment noise, one typically obtains polynomial rather than exponential concentration, which we do not analyze here. We also invoke Banach's theorem on closed (and hence complete) coordinate domains; models whose natural domains are open or lie on manifolds require additional care. From a practical standpoint, certifying global Lipschitz constants of black-box simulators is challenging; data-driven or local contraction diagnostics may be more feasible in applications. Finally, our closure results explicitly cover only shift-scale maps with bounded gains (lay) < 1); interventions with larger multiplicative factors, stochastic policies, more general functional forms are not yet addressed.
At present, our analysis is restricted to shift-scale interventions. This class, however, already strictly generalizes hard (do-) interventions and provides a principled foundation for broader extensions. The key theoretical principle underpinning is, the global Lipschitz constant of the intervened system remains below one. This property ensures unique solvability of the intervened SCM, even in the presence of eycles, and is abstract enough to accommodate richer intervention types, provided they do not destroy contractivity. Consequently, our results extend directly to any intervention family (including nonlinear, stochastic, more complex parametric changes) that preserves the contraction property. An explicit analysis of broader classes of interventions, and sufficient conditions under which they preserve contractivity, is promising direction for future work
guarantees is the preservation of global contractivity-
6 Conclusion and Future Work
We have established a principled foundation for counterfactual inference in cyclic structural causal models under shift-scale interventions. Leveraging global contraction assumption, proved that such models are simple, ensuring unique solvability even the presence feedback. We further showed that shift-scale interventions preserve solvability, are closed under composition, and admit sub-Gaussian tail bounds for counterfactual functionals under natural regularity assumptions. These results demonstrate that contraction- based SCMs offer mathematically tractable yet expressive class for reasoning about interventions and counterfactuals in cyclic settings. Future research may focus on developing deep generative models for cyclic counterfactuals, leveraging the theoretical foundations established here.
Acknowledgments and Disclosure of Funding
This research was funded part by the Indo-French Centre for the Promotion of Advanced Research (IFCPAR/CEFIPRA) through project number CSRP 6702-2.
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A Supplementary Material
A.1 Equivalence of Twin SCM and Pearl's Abduction-AActoo--rreciicii (AAP) Procedure
Theorem 3. Let M (I,J, & PE) be a structural causal model that is simple (i.e., uniquely solvable with respect every subset of endogenous variables). Assume:
(i) the exogenous vector E Pg has a joint density and mutually independent compo- nents:
(ii) the counterfactual query is defined by a perfect intervention do(XA :- _AA.oor some A I.
Let X M denote the factual world, and define the counterfactual outcome via:
Twin SCM definition: Solve the intervened twin SCM where the do-intervention is
applied to the primed copy, yielding random variables (X, X').
Pearl's AAP (Pearl, 2009) procedure: Sample E ~ Pg conditioned on X = goobs and solve the intervened SCM with this fixed noise to obtain X
Then, the counterfactual distribution derived from the twin SCM coincides with the distribu- tion obtained from Pearl's ceneeciinnn-ctioopeeeieie (AAP) procedure:
Px
gobe
Px
Proof. Since M is simple, by definition there exists unique measurable function g X such that for every E &
g(e) = f(g(e),)), so that X ((E) a.s.
Similarly, after the intervention do( xA == a) the intervened SCM admits a unique measurable solution map gdo E + X' (Proposition 3.8, Bongers et al. (2021)).
In the twin SCM construction, both copies (factual and counterfactual) share the same exogenous vector E, and the intervention is applied only to the first copy. Solving the twin SCM yields:
X. X')
(g
E),
do (E)).
Let obs X' be the observed factual outcome. Pearl's procedure involves:
Abduction: condition on the observation = zbb, which corresponds to conditioning on the set {e E g(e) rbbs];
Action: apply the intervention obtain the new function doo;
Prediction: evaluate xf d0 using the same noise, but now drawn from the posterior
IP (E)))())=
Define the posterior distribution on exogenous noise:
Mg (A) ((E A (E) bbbs),
for all measurable S E.
This measure supported on the set
E g(e)= = x ohs
Equality of counterfactual laws. By construction, X'= = (E) the counterfactual distribution from the twin SCM (given X = zbbs) is:
X (B) = Mgoba (e: le:go (e) B), B
Pearl's method also conditions on (X = obs J. inducing the same posterior Hyabe over exogenous variables. Then, the counterfactual outcome is e = odo (E) with E Mgoba, Thus,
X B)
Myyobe (e (e( E B).
which is the same expression as for the twin SCM.
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The counterfactual law derived from the twin SCM coincides with the one obtained from Pearl's abduction-action cettoo--attioopeiiiiiet prediicion procedure under unique solvability.
Remark 3 (Notation X M). The shorthand X M means that the random vector X is solution of the SCM M.
References
Bongers, S., Forré, P., Peters, J., and Mooij, J. M. (2021). Foundations of structural causal models with cycles and latent variables. The Annals Statistics, 49(5):2885 2915.
Pearl, (2009). Causality, Cambridge University Press, 2 edition.
20


Reviews:
[{"review_id": "aWC1EHS5t7", "reviewer": "Reviewer_vToH", "summary": "This paper proposes a theoretical framework for counterfactual inference in cyclic structural causal models (SCMs) using shift–scale interventions. While most existing counterfactual frameworks assume acyclic structures (DAGs), many real-world systems, specifically dynamical systems, involve feedback loops. By leveraging a global ℓp-contraction condition, the authors ensure unique solvability of cyclic SCMs even after shift–scale interventions. The paper further establishes closure properties under composition and derives sub-Gaussian concentration bounds for counterfactual outcomes under additional Lipschitz regularity assumptions in the exogenous noise.", "questions": "- Could you include a simple, illustrative example of a cyclic SCM to help demonstrate your theoretical results more concretely and make them easier to understand?\n\n- How might your results be used as a foundation to extend the framework to more general types of interventions beyond shift–scale?", "limitations": "It is addressed in the weaknesses.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "## Strengths\n- The paper makes a clear contribution by extending counterfactual reasoning to cyclic causal models.\n- The mathematical arguments are carefully detailed and all definitions, theorems, and proofs are clearly presented.\n\n## Weaknesses\n- The framework relies heavily on a global contraction condition (k < 1), which is difficult to check or guarantee in practical settings.\n- There are no toy examples to help readers understand how the theory plays out in practice.\n- The results are limited to shift–scale interventions and a discussion on how the results for these interventions can be generalized to other interventions is missing.  \n\n## Minor Comments\n- Page 3: The equation between line 83 and 84, 2nd line. ) is not needed.", "was_revised": true, "final_justification": "The authors’ response addressed my concerns, and the provided example improves the paper’s readability. I will maintain my positive score.", "rebuttal": "We are grateful to the reviewer for detailed review and helpful comments. \n\n\n(W1) \n> The global contraction is a sufficient condition that lets us invoke Banach’s fixed-point theorem to guarantee unique solvability. Future works may try to relax it to weaker forms. \nIn practice, one may verify it via power‑iteration, semidefinite‑program and guarantee it with spectral‑norm or Jacobian penalties already standard in deep learning pipelines. \n\n*\"Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks\"*, Fazlyab et al., 2019\n\n*\"Spectral Normalization for Generative Adversarial Networks\"*, Miyato et al., 2018\n\n\n---\n(W2) & (Q1)\n\n> We have added a detailed worked example to the revised manuscript.\n\n## **Example:**\n > Consider a linear cyclic SCM with mutual dependence between consumption ( $C$ ) and income ( $I$ ):\n\\begin{equation}\n\\begin{aligned}\nC = 0.50\\,I + 1 + E_{C}, \\quad\nI = 0.40\\,C + 0.50 + E_{I},\n\\end{aligned}\n\\qquad\n(E_{C},E_{I})^\\top \\sim \\mathcal N(\\mathbf 0,\\,0.04\\,\\mathbf I_2).\n\\end{equation}\nWritten in matrix form  $X =(C,I)^\\top$, $X = A X + b + E$ with  \n\\begin{align*}\nA = \\begin{pmatrix} 0 & 0.50 \\\\\\ 0.40 & 0 \\end{pmatrix}, \\qquad\nb = \\begin{pmatrix} 1 \\\\\\ 0.5 \\end{pmatrix}, \\qquad\nE = (E_C, E_I)^\\top.\n\\end{align*}\nBecause the spectral norm is $\\|A\\|_2 = \\sqrt{0.50\\times0.40}=0.447 < 1$,  the model is globally contractive and therefore simple in the sense of Definition 4. \nFor any $2{\\times}2$ matrix $A=\\bigl[\\begin{smallmatrix}0 & a\\\\\\ b & 0\\end{smallmatrix}\\bigr] $ with \\(ab<1\\),\n$\n(I-A)^{-1}= \\frac{1}{1-ab}\\bigl[\\begin{smallmatrix}1 & a\\\\\\ b & 1\\end{smallmatrix}\\bigr].\n$ With \\\\((a,b)=(0.50,0.40)\\\\) we have\n\\\\((I-A)^{-1}=%\n\\bigl[\\begin{smallmatrix}1.25 & 0.625\\\\\\ 0.50 & 1.25\\end{smallmatrix}\\bigr]\\\\) and therefore\n\\begin{align*}\n\\mu = \\mathbb E[X] = (I-A)^{-1} b \n            = (1.5625,\\;1.125)^\\top, \\\\\\\n\\Sigma = \\operatorname{Cov}(X) = (I-A)^{-1}(0.04\\,\\mathbf I_2)(I-A)^{-\\top}\n            =\\begin{pmatrix}0.0781 & 0.0563\\\\\\ 0.0563 & 0.0725\\end{pmatrix}.\n\\end{align*}     \n\n> Thus  $ X_{\\text{obs}}\\sim\\mathcal N\\ \\bigl(\\mu,\\Sigma\\bigr) $ with $ \\mu $ and $ \\Sigma $ .  The correlation between \\\\(C\\\\) and \\\\(I\\\\) is \\\\(\\rho=0.75\\\\).\n\n> **Shift–scale intervention.**\nSuppose a fiscal policy reform dampens the effect of both consumption and random shocks on income by a factor $\\alpha=0.8$, and provides a fixed income supplement of $\\beta=1.0$ units:\n\\\\[\n\\mathrm{ss}(I,\\alpha,\\beta) :\nI \\leftarrow \\alpha\\bigl(0.40\\,C + 0.50 + E_{I}\\bigr) + \\beta,\n\\quad\n\\alpha=0.8, \\beta=1.0.\n\\\\]\nThe intervened structural parameters  are\n\\begin{align*}A'=\\begin{pmatrix}0 & 0.50 \\\\\\ 0.32 & 0\\end{pmatrix}, \\qquad\nb'=\\begin{pmatrix}1 \\\\\\ 1.4\\end{pmatrix},\n\\quad \\text{so}\\quad\n\\\\|A'\\\\|_2=\\sqrt{0.50\\times0.32}=0.400<1.\n\\end{align*}\n\n\n> Scaling also affects the exogenous term:\n\\\\(\nE'=(E_{C},\\,\\alpha E_{I})^\\top,\n\\Sigma_{E'}=\\operatorname{diag}(0.04,\\alpha^{2} \\\\cdot 0.04)\n           =\\operatorname{diag}(0.04,0.0256).       \n\\\\)       \n\n\n\n> Contractivity is preserved, hence a unique interventional equilibrium exists:\n\\begin{align*}\n\\mathbb E[X_{\\text{int}}] =(I-A')^{-1} b' \n        =(2.024,\\;2.048)^\\top,\\\\\\\n\\operatorname{Cov}(X_{\\text{int}}) =(I-A')^{-1}\\Sigma_{E'}(I-A')^{-\\top}\n  =\\begin{pmatrix} 0.0658 & 0.0363 \\\\\\ 0.0363 & 0.0421\\end{pmatrix},\n\\qquad\n\\rho_{\\text{int}} \\approx 0.69.\n\\end{align*}\nConsumption rises by \\\\(\\sim\\ 29\\\\%\\\\) and income by \\\\(\\sim 82 \\\\%\\\\), while the\n\\\\(C\\\\)–\\\\(I\\\\) correlation falls from \\\\(0.75\\\\) to \\\\(0.69\\\\).\n\n\n> **From intervention to counterfactual via the twin SCM.**\nAbove equations describe the interventional distribution $P(X_{\\text{int}})$, i.e. what we would observe if the shift–scale policy were enacted for the whole population. To answer individual–level counterfactual queries (``what would this household’s consumption be had the policy applied?’’) , we follow the twin network construction:  \nLet $(c,i)$ denote the actually observed consumption and income for one household.  The twin‑SCM duplicates every endogenous variable and shares the same exogenous noise:\n\n\\\\[\n\\boxed{\n\\begin{aligned}\nC &= 0.50 I + 1 + E_{C},              &  C' &= 0.50I' + 1 + E_{C},\\\\\\ \nI &= 0.40C + 0.50 + E_{I},           &  I' &= 0.8\\bigl(0.40C' + 0.50 + E_{I}\\bigr) + 1.\n\\end{aligned}}\n\\\\]\n\n> where the primed copy encodes the shift–scale intervention\n$\\mathrm{ss}(I,\\alpha{=}0.8,\\beta{=}1.0)$ and the unprimed copy remains\nfactual. From the factual equations, we can solve\n\\begin{equation}\nE_{C}=c-0.50i-1,\n\\qquad\nE_{I}=i-0.40c-0.50.\n\\end{equation}\n\n> Eliminating $C'$ from the primed equations of the twin SCM yields\n\\\\(\n0.84I' = 1.72 + 0.8E_{I} + 0.32E_{C}.\n\\\\)\nSubstituting $E_{C}$ and $E_{I}$ gives the counterfactual income \\\\( \nI'(c,i) = \\frac{25}{21} + \\frac{16}{21}i \n        = 1.190476 + 0.761905i.\n\\\\) Back‑substitution furnishes the counterfactual consumption\n\\\\(\nC'(c,i) = c + \\tfrac{25}{42} - \\tfrac{5}{42}i\n        = c + 0.595238 - 0.119048i. \\\\)\n\n\n> Above equations give the counterfactual response mapping $(c,i)\\mapsto(C',I')$. Because the mapping is affine and the factual distribution $X_{\\mathrm{obs}}\\sim\\mathcal N(\\mu,\\Sigma)$ \nis Gaussian, the marginal counterfactual $(C',I')$ is again Gaussian with\n\\\\(\\mathbb E[(C',I')^\\top]=(2.024,2.048)^\\top\\\\) and covariance $\\begin{pmatrix} 0.0658 & 0.0363 \\\\\\ 0.0363 & 0.0421\\end{pmatrix}$.\nHere, the twin‑SCM reconciles individual counterfactual semantics\n\\\\((C'(c,i),I'(c,i))\\\\) with the population‑level interventional\ndistribution already reported, while remaining in closed form due to the linearity and contractiveness of the model.\n\n\n\n\n\n---\n(W3) & (Q2)\n> Our current framework is restricted to shift–scale interventions, but this class already strictly generalizes hard interventions and serves as a foundation for further extensions. The core theoretical principle underpinning our results is the preservation of global contractivity (i.e., the global Lipschitz constant remains below one) after intervention. This property ensures unique solvability of the intervened SCM, even in the presence of cycles, and is abstract enough to accommodate more general intervention types, provided they do not destroy contractivity. Thus, our results can be directly extended to any class of interventions (including nonlinear, stochastic, or more complex parametric changes) that preserve this contraction property. An explicit analysis of broader classes of interventions, and sufficient conditions under which they preserve contractivity, is a promising direction for future work"}, {"review_id": "Ak1GW8uGuH", "reviewer": "Reviewer_Na7U", "summary": "Structural causal models are the classical framework in counterfactual inference. However, these models typically require an directed acyclic graph, which excludes the possibility of feedback loops. The article introduces a notion of counterfactuals that generalizes beyond acyclic graphs and can handle cycles. The underlying mathematics is based on contraction mappings and Banach fixed point theory.", "questions": "see above.", "limitations": "yes.", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths:\n- The manuscript is well-written and clear. I enjoyed reading it.\n- The manuscript could provide a foundation for follow-up works, as well-posedness of structural causal models with cycles and well-posedness of a corresponding set of interventions is thorougly discussed.\n\nWeaknesses:\n- While the mathematical exposition is on point, the main results e.g. Thm.1 and Thm.2 follow in a relatively straightforward manner from Banach fixed point theory. The technical contributions are therefore somewhat limited.\n- The authors could expand the discussion on the meaning and interpretations (including a philosophical level) of structural causal models with cyclic graphs. In many physical process there are no feedback effects that are immediate as modelled by the authors. I am thinking of thermodynamic systems, mechanical systems, fluid dynamical systems, etc. that have some sort of \"inertia\", which means that there is a clear temporal sequence that characterizes how the variables affect each other over time. In such a case modelling with a directed acyclic graph is maybe more appropriate... Can the authors provide simple examples where cyclic graphs are known to occur?\n\nOn a similar note, the field of control theory (i.e. the science of feedback) has dealt with feedback loops for a long time although the focus is maybe more on physical and engineering systems, wheras in causal inference the focus lies more on social and economical systems. In control theory there is the notion of well-posedness that has been studied extensively, which is very related to the content of the manuscript (although approached from a different angle). Here is for example a survey on well-posedness for a control-theoretic lens: \"Well-posed systems-The LTI case and beyond\" by Marius Tucsnak, George Weiss, Automatica, 2014.", "was_revised": true, "final_justification": "I keep my score in view of rebuttal and discussion.", "rebuttal": "We thank the reviewer for careful reading and thoughtful comments. We are glad to hear that you enjoyed reading the manuscript.\n\n---\n(W1)\n>  We agree that the Banach Fixed Point (BFP) theorem underlies our proofs; however, our contribution is not merely a restatement of BFP for cyclic SCMs. The novelty lies in how we specialise it to cyclic structural causal models (SCMs) and in the additional consequences.\n---\n(W2) \n> -  The term “cyclic” in structural causal models (SCMs) could also be understood as referring to simultaneous structural equations that capture the steady-state behavior of an underlying dynamical system, rather than its transient, time-dependent evolution. One way to interpret these cyclic structural equations is by assuming an underlying discrete-time dynamical system, where the equations act as update rules: the value of each variable at time $t+1$ is computed from the values at time $t$ . The system is then analyzed in the limit as $t\\rightarrow \\infty$ , focusing on the fixed points to which the dynamics converge.\nMooij et al. (2013) demonstrate that an alternative, yet natural, interpretation of SCMs emerges when considering systems of ordinary differential equations (ODEs). By examining the equilibrium (steady-state) solutions of such ODEs, one arrives at a structural causal model that is time-independent, but still retains meaningful causal semantics with respect to interventions. Specifically, the semantics of interventions and counterfactuals remain valid and well-defined in this steady-state context, as rigorously formalized by Mooij et al. (2013) and further extended by Bongers et al. (2021). This is by no means the sole pathway to structural causal models; indeed, SCMs may arise through a variety of alternative constructions and representations, depending on the nature of the system under consideration. \n\n*\"From Ordinary Differential Equations to Structural Causal Models: the deterministic case\", Joris M. Mooij, Dominik Janzing, Bernhard Scholkopf, 2013.*\n\n> **Although many physical processes exhibit inertia, static cyclic structural causal models (SCMs) remain appropriate when we focus on equilibrium behaviour or sample at a temporal resolution coarser than the fastest feedback loop.\n Examples include gene‑regulatory networks in single‑cell genomics (Rohbeck et al., 2023); market‑equilibrium models (Bongers et al., 2021); predator–prey ecological systems;  Thyroid or reproductive hormone axes exhibit feedback loops (Clarke et al., 2012), etc.**\n\n*\"Bicycle: Intervention-Based Causal Discovery with Cycles\", Rohbeck et al., 2024.*\n\n*\"Modelling mechanisms with causal cycles\", Brendan Clarke, Bert Leuridan & Jon Williamson, 2013.*\n\n> - With our limited background in control theory, we note, as you have rightly pointed out, informally that control theorists refer to a feedback interconnection as “well posed” when there exists a unique, stable mapping from inputs to outputs (Tucsnak & Weiss, 2014).  In the context of SCMs, the unique solvability of the structural equations serves an analogous role, ensuring that interventions (do) yield well-defined outcomes."}, {"review_id": "KyS82mFTc1", "reviewer": "Reviewer_vCep", "summary": "This paper addresses the challenge of counterfactual inference in cyclic structural causal models (SCMs), which better reflect the feedback loops and cyclical dependencies found in many real-world systems such as biological networks. While traditional counterfactual methods assume acyclic models (DAGs), this work departs from that norm and explores shift-scale interventions within cyclic SCMs. The authors propose a new framework for conducting counterfactual analysis under these conditions and demonstrate its theoretical soundness and practical utility, laying groundwork for more realistic causal reasoning in complex systems.", "questions": "1. Which part of the paper explains how the proposed method handles cyclic structures?\n2. Could the authors elaborate where the twin-SCM construction interacts with the cyclic structure?\n3. What is the computational cost of solving the (shift-scale) twin SCM in high dimensions, and are there efficient algorithms?\n4. Can the authors provide simulations or real-data case studies to demonstrate (a) cycles in action, (b) contraction diagnostics, and (c) counterfactual estimation under shift-scale interventions?", "limitations": "Yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 1, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n- Addresses a major gap by extending counterfactual inference to cyclic SCMs, which better model feedback loops in real systems (e.g., biology).\n\nWeaknesses\n- The paper never clearly highlights which sections or results are specifically responsible for handling cyclic structures, making it harder for readers to identify where cycles are addressed.\n- The paper lacks empirical validation. No simulations or real-data experiments are provided to illustrate practical performance.\n- Computational aspects (e.g., how to verify contraction or solve high-dimensional twin SCMs) are not discussed, which could hinder adoption in large-scale systems.", "was_revised": true, "final_justification": "The authors have adequately addressed my concerns. I will raise my score to accept.", "rebuttal": "We are grateful to the reviewer for careful reading and constructive comments offered.\n\n(W1) & (Q1,Q2)\n>  We recognize that the literature on counterfactual inference has predominantly focused on acyclic structural causal models (SCMs). It is therefore essential to clarify how our work generalizes these results to the cyclic setting. Our paper does not assume acyclicity. Rather, a major motivation and contribution of our work is to establish a rigorous theoretical framework for counterfactual inference in cyclic SCMs. All our theoretical results, proofs, and constructions are formulated for general SCMs\nthat may be cyclic. The acyclic (DAG) case is simply a special instance.\n\n---\n(W2) & (Q2, Q4)\n> We appreciate the reviewer’s concern. We have added a detailed worked example to the revised manuscript\n\n## **Example:**\n > Consider a linear cyclic SCM with mutual dependence between consumption ( $C$ ) and income ( $I$ ):\n\\begin{equation}\n\\begin{aligned}\nC = 0.50\\,I + 1 + E_{C}, \\quad\nI = 0.40\\,C + 0.50 + E_{I},\n\\end{aligned}\n\\qquad\n(E_{C},E_{I})^\\top \\sim \\mathcal N(\\mathbf 0,\\,0.04\\,\\mathbf I_2).\n\\end{equation}\nWritten in matrix form  $X =(C,I)^\\top$, $X = A X + b + E$ with  \n\\begin{align*}\nA = \\begin{pmatrix} 0 & 0.50 \\\\\\ 0.40 & 0 \\end{pmatrix}, \\qquad\nb = \\begin{pmatrix} 1 \\\\\\ 0.5 \\end{pmatrix}, \\qquad\nE = (E_C, E_I)^\\top.\n\\end{align*}\nBecause the spectral norm is $\\|A\\|_2 = \\sqrt{0.50\\times0.40}=0.447 < 1$,  the model is globally contractive and therefore simple in the sense of Definition 4. \nFor any $2{\\times}2$ matrix $A=\\bigl[\\begin{smallmatrix}0 & a\\\\\\ b & 0\\end{smallmatrix}\\bigr] $ with \\(ab<1\\),\n$\n(I-A)^{-1}= \\frac{1}{1-ab}\\bigl[\\begin{smallmatrix}1 & a\\\\\\ b & 1\\end{smallmatrix}\\bigr].\n$ With \\\\((a,b)=(0.50,0.40)\\\\) we have\n\\\\((I-A)^{-1}=%\n\\bigl[\\begin{smallmatrix}1.25 & 0.625\\\\\\ 0.50 & 1.25\\end{smallmatrix}\\bigr]\\\\) and therefore\n\\begin{align*}\n\\mu = \\mathbb E[X] = (I-A)^{-1} b \n            = (1.5625,\\;1.125)^\\top, \\\\\\\n\\Sigma = \\operatorname{Cov}(X) = (I-A)^{-1}(0.04\\,\\mathbf I_2)(I-A)^{-\\top}\n            =\\begin{pmatrix}0.0781 & 0.0563\\\\\\ 0.0563 & 0.0725\\end{pmatrix}.\n\\end{align*}     \n\n> Thus  $ X_{\\text{obs}}\\sim\\mathcal N\\ \\bigl(\\mu,\\Sigma\\bigr) $ with $ \\mu $ and $ \\Sigma $ .  The correlation between \\\\(C\\\\) and \\\\(I\\\\) is \\\\(\\rho=0.75\\\\).\n\n> **Shift–scale intervention.**\nSuppose a fiscal policy reform dampens the effect of both consumption and random shocks on income by a factor $\\alpha=0.8$, and provides a fixed income supplement of $\\beta=1.0$ units:\n\\\\[\n\\mathrm{ss}(I,\\alpha,\\beta) :\nI \\leftarrow \\alpha\\bigl(0.40\\,C + 0.50 + E_{I}\\bigr) + \\beta,\n\\quad\n\\alpha=0.8, \\beta=1.0.\n\\\\]\nThe intervened structural parameters  are\n\\begin{align*}A'=\\begin{pmatrix}0 & 0.50 \\\\\\ 0.32 & 0\\end{pmatrix}, \\qquad\nb'=\\begin{pmatrix}1 \\\\\\ 1.4\\end{pmatrix},\n\\quad \\text{so}\\quad\n\\\\|A'\\\\|_2=\\sqrt{0.50\\times0.32}=0.400<1.\n\\end{align*}\n\n\n> Scaling also affects the exogenous term:\n\\\\(\nE'=(E_{C},\\,\\alpha E_{I})^\\top,\n\\Sigma_{E'}=\\operatorname{diag}(0.04,\\alpha^{2} \\\\cdot 0.04)\n           =\\operatorname{diag}(0.04,0.0256).       \n\\\\)       \n\n\n\n> Contractivity is preserved, hence a unique interventional equilibrium exists:\n\\begin{align*}\n\\mathbb E[X_{\\text{int}}] =(I-A')^{-1} b' \n        =(2.024,\\;2.048)^\\top,\\\\\\\n\\operatorname{Cov}(X_{\\text{int}}) =(I-A')^{-1}\\Sigma_{E'}(I-A')^{-\\top}\n  =\\begin{pmatrix} 0.0658 & 0.0363 \\\\\\ 0.0363 & 0.0421\\end{pmatrix},\n\\qquad\n\\rho_{\\text{int}} \\approx 0.69.\n\\end{align*}\nConsumption rises by \\\\(\\sim\\ 29\\\\%\\\\) and income by \\\\(\\sim 82 \\\\%\\\\), while the\n\\\\(C\\\\)–\\\\(I\\\\) correlation falls from \\\\(0.75\\\\) to \\\\(0.69\\\\).\n\n\n> **From intervention to counterfactual via the twin SCM.**\nAbove equations describe the interventional distribution $P(X_{\\text{int}})$, i.e. what we would observe if the shift–scale policy were enacted for the whole population. To answer individual–level counterfactual queries (``what would this household’s consumption be had the policy applied?’’) , we follow the twin network construction:  \nLet $(c,i)$ denote the actually observed consumption and income for one household.  The twin‑SCM duplicates every endogenous variable and shares the same exogenous noise:\n\n\\\\[\n\\boxed{\n\\begin{aligned}\nC &= 0.50 I + 1 + E_{C},              &  C' &= 0.50I' + 1 + E_{C},\\\\\\ \nI &= 0.40C + 0.50 + E_{I},           &  I' &= 0.8\\bigl(0.40C' + 0.50 + E_{I}\\bigr) + 1.\n\\end{aligned}}\n\\\\]\n\n> where the primed copy encodes the shift–scale intervention\n$\\mathrm{ss}(I,\\alpha{=}0.8,\\beta{=}1.0)$ and the unprimed copy remains\nfactual. From the factual equations, we can solve\n\\begin{equation}\nE_{C}=c-0.50i-1,\n\\qquad\nE_{I}=i-0.40c-0.50.\n\\end{equation}\n\n> Eliminating $C'$ from the primed equations of the twin SCM yields\n\\\\(\n0.84I' = 1.72 + 0.8E_{I} + 0.32E_{C}.\n\\\\)\nSubstituting $E_{C}$ and $E_{I}$ gives the counterfactual income \\\\( \nI'(c,i) = \\frac{25}{21} + \\frac{16}{21}i \n        = 1.190476 + 0.761905i.\n\\\\) Back‑substitution furnishes the counterfactual consumption\n\\\\(\nC'(c,i) = c + \\tfrac{25}{42} - \\tfrac{5}{42}i\n        = c + 0.595238 - 0.119048i. \\\\)\n\n\n> Above equations give the counterfactual response mapping $(c,i)\\mapsto(C',I')$. Because the mapping is affine and the factual distribution $X_{\\mathrm{obs}}\\sim\\mathcal N(\\mu,\\Sigma)$ \nis Gaussian, the marginal counterfactual $(C',I')$ is again Gaussian with\n\\\\(\\mathbb E[(C',I')^\\top]=(2.024,2.048)^\\top\\\\) and covariance $\\begin{pmatrix} 0.0658 & 0.0363 \\\\\\ 0.0363 & 0.0421\\end{pmatrix}$.\nHere, the twin‑SCM reconciles individual counterfactual semantics\n\\\\((C'(c,i),I'(c,i))\\\\) with the population‑level interventional\ndistribution already reported, while remaining in closed form due to the linearity and contractiveness of the model.\n\n\n\n---\n(W3) & (Q3)\n> Our work is intended as a foundational contribution establishing well-posedness and unique solvability for counterfactual queries in cyclic SCMs under shift–scale interventions, rather than as a recipe for scalable counterfactual estimation. We do not provide a general-purpose algorithm for estimating counterfactuals or for solving high-dimensional twin SCMs. \nThe solution to the twin SCM (and thus the counterfactuals) can always, in principle, be obtained by iterating the contraction map (Banach fixed-point iteration), which is well known to converge geometrically fast for contraction mappings. \n\n>In practice, one may verify contraction via power‑iteration, semidefinite‑program and guarantee it with spectral‑norm or Jacobian penalties already standard in deep learning pipelines. For further details, please refer to: \n*\"Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks\"*, Fazlyab et al., 2019;\n*\"Spectral Normalization for Generative Adversarial Networks\"*, Miyato et al., 2018.\nHowever,  scalability to extremely large systems is outside our current scope and an important direction for future work.\n\n\n\n\nWe hope that the additional example addresses the reviewer’s concerns. We kindly request that you reconsider your score in light of these clarifications."}, {"review_id": "nAgzLZMFeX", "reviewer": "Reviewer_Ucnx", "summary": "The paper focuses on cyclic SCMs and more precisely on their behavior under a type of soft interventions, the shift-scale interventions do(aX+b). A number of theoretical results is proved, including the unique solvability of twin SCMs, used for modelling counterfactual distributions, under ss interventions. The latter implies that counterfactual distributions associated with such interventions are well-defined for cyclic SCMs.", "questions": "Q1. What are the key differences between Theorem 2 from your paper and Proposition 8.2 from Bogers et.al.?\n\nQ2. Could I kindly ask the authors to explain what P-almost surely means in Definition 2? It could be helpful if you add an explanation in the paper for completeness.\n\nQ3. Could you elaborate on the last sentence from the paragraph in Section 5?", "limitations": "Limitations are mentioned. Maybe some more discussion on the challenges that other types of soft interventions pose would be helpful.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 2, "originality": 3, "strengths_and_weaknesses": "+ The paper is clearly written, and results are well explained.\n+ I read the proofs of the two theorems only. Their exposition was nice, and I found them easy to follow. The proofs seemed correct.\n+ The field of cyclic SCMs is indeed under explored and important. Thus, efforts to shed more light in this domain have value.\n\n- I found the connection of this paper to prior work poor. I strongly advise the authors to: (1) mention previous results from Bongers et al in cyclic SCMs, and especially their results related to hard interventions; (2) highlight any potential overlap in methodology and draw a connection to this paper and your explicitly mentioning the extra challenges you had to face in order to prove your results. See my questions for some examples\n\n- The contributions of the paper are rather limited. It would be interested to see a similar study for other subfamilies of soft interventions and/or the case with more than one simultaneous soft interventions. The latter produces counterfactual distributions which might include more than two counterfactual worlds and hence cannot be modeled by twin SCMs.", "was_revised": true, "final_justification": "The authors promised to include the comparison I asked.", "rebuttal": "We sincerely thank the reviewer for their careful reading and thoughtful, constructive feedback.\n\n\n\n\nThe contributions of the paper are rather .... two counterfactual worlds and hence cannot be modeled by twin SCMs.\n> Our current work is restricted to shift–scale interventions, but this class already strictly generalizes hard interventions and serves as a foundation for further extensions. The core theoretical principle underpinning our results is the preservation of global contractivity (i.e., the global Lipschitz constant remains below one) after intervention. This property ensures unique solvability of the intervened SCM, even in the presence of cycles, and is abstract enough to accommodate more general intervention types, provided they do not destroy contractivity. Thus, our results can be directly extended to any class of interventions (including nonlinear, stochastic, or more complex parametric changes) that preserve this contraction property. An explicit analysis of broader classes of interventions, and sufficient conditions under which they preserve contractivity, is a promising direction for future work\n\n>  Regarding multiple simultaneous soft interventions, as you have rightly pointed out that classical twin SCMs are limited to two worlds. Extending our fixed-point and contractivity analysis to multi-world counterfactuals is a natural and promising avenue for future research. \n\n---\n\nI found the connection of this paper to prior work poor. I strongly ... See my questions for some examples.\n\n> Thank you for this valuable suggestion. We will include a discussion of prior results by Bongers et al., particularly their findings on hard interventions. This will help clarify how our work extends these foundational contributions.\n\n> Our responses to your questions provide further clarification on these aspects.\n---\n\n(Q1) \n> Theorem 2 provides sufficient conditions for unique solvability of the twin SCM under shift–scale interventions. Proposition 8.2 is a closure property: it states that if an SCM is simple, then its twin SCM, marginalizations, and perfect interventions will also be simple. It does not provide a sufficient condition to ensure simplicity (e.g., contraction); rather, it relies on the SCM being already simple.\n\n---\n\n(Q2) \n> It means that the structural equations hold with probability 1 under the probability measure $\\mathbb{P}$ defined on the underlying probability space. In other words, the set of  elements of the sample space $\\omega \\in \\Omega$ for which the equation does not hold has probability zero.\n\"Almost surely\" (abbreviated as a.s.) is a standard concept in probability theory.\n\n\n---\n\n(Q3)\n> We have rephrased the sentence as: \"Future research may focus on developing deep generative models for cyclic counterfactuals, leveraging the theoretical foundations established here\". \nNote that the majority of counterfactual inference approaches using deep generative models restrict the causal structure to directed acyclic graphs (DAGs).\n\n\nWe hope our responses effectively address the reviewer’s concerns and respectfully ask that you consider raising  your score based on these clarifications."}]

Author rebuttal:

Meta-review: The paper develops a formalism for structural causal models that can include cycles along with a class of interventions under which variables and distributions are still well-defined, including counterfactual distributions.

Reviewers agreed the paper made a strong foundational contribution for an important problem.

The abstract should provide some simple description of shift-scale interventions.

One or more simple examples should help motivate the theoretical framework. Authors gave one in the discussion, this should be polished and included in the paper. Authors should consider whether any figures could help increase understanding for some readers.

There is some important discussion not currently in the paper that should also be added. What are the **semantics** of interventions and counterfactuals? The mathematical framework seems agnostic about whether cycles represent e.g. time dynamics or a time-independent steady state solution, so perhaps the answer to the previous question requires first making additional assumptions about which setting is under consideration. The semantics of counterfactuals in a DAG are well-known: "The value variable $x_i$ would have taken had some counterfactual condition held for other causally-related variables." Does the present framework allow any meaningful semantics, or is an intuitive/linguistic understanding precluded by e.g. infinite recursion?
Decision: Accept (poster)