A Measure-Theoretic Axiomatisation of Causality
Authors: Junhyung Park
Simon Buchholz
Bernhard Schölkopf
Krikamol Muandet
Venue: NeurIPS 2023 oral (2023)
Keywords: Causality
probability theory
causal models
TL;DR: We propose causal spaces, a measure-theoretic axiomatisation of causality.

Abstract: Causality is a central concept in a wide range of research areas, yet there is still no universally agreed axiomatisation of causality. We view causality both as an extension of probability theory and as a study of what happens when one intervenes on a system, and argue in favour of taking Kolmogorov's measure-theoretic axiomatisation of probability as the starting point towards an axiomatisation of causality. To that end, we propose the notion of a causal space, consisting of a probability space along with a collection of transition probability kernels, called causal kernels, that encode the causal information of the space. Our proposed framework is not only rigorously grounded in measure theory, but it also sheds light on long-standing limitations of existing frameworks including, for example, cycles, latent variables and stochastic processes.

A Measure-Theoretic Axiomatisation of Causality
Junhyung Park
Simon Buchholz
Empirical Inference Department MPI for Intelligent Systems 72076 Tübingen, Germany
Empirical Inference Department
MPI for Intelligent Systems 72076 Tübingen, Germany
junhyung parketuebingen.mpp de
simon, wuwchhol.@ttuueeneen..n.o de
Bernhard Schölkopf
Krikamol Muandet CISPA Helmholtz Center for Information Security
Empirical Inference Department MPI for Intelligent Systems 72076 Tübingen, Germany bs bs@tuebingen.mmp
66123 Saarbrücken, Germany
muadett@cispa.dd
Abstract
Causality is central concept in wide range of research areas, yet there is still no universally agreed axiomatisation of causality. We view causality both as an extension of probability theory and as a study of what happens when inter- venes on a system, and argue in favour of taking Kolmogorov's measure-theoretic axiomatisation of probability causality. To that end, we propose the notion of a causal space, consisting of a probability space along with collection of transition probability kernels, called causal kernels, that encode the causal information of the space. Our proposed framework is only rigorously grounded in measure theory, but also sheds light on long-standing limitations of existing frameworks including, for example, cycles, latent variables and stochastic processes.
the starting point towards axiomatisation of
1
Introduction
Causal reasoning has been recognised as hallmark of human and machine intelligence, and in the recent years, the machine learning community has taken up rapidly growing interest in the subject
46 53
in particular in representation learning [55] 41 622 57 I1 40 and natural language
processing 35 19]. Causality has also been extensively studied in wide range of other research
domains, including. but not limited to, philosophy [39 64 16), psychology [60], statistics [45] [66
including social, biological and medical sciences [51 31 29 26), mechanics and law (6).
The field of causality was born from the observation that probability theory and statistics (Figure Tab cannot encode the notion of causality, and so we need additional mathematical tools to support the enhanced view of the world involving causality (Figure lb) Our goal in this paper to give an axiomatic framework of the forwards direction of Figure Tb which currently consists of many competing models (see Related Works), As a starting point, we observe that the forwards direction of Figure Ta i.e. probability theory, has set of axioms based are widely accepted and used"] and hence argue that it is natural to take the primitive objects of this framework the basic building blocks. Despite the fact that all of the existing mathematical frameworks of causality recognise the crucial role that probability plays/ should play in any causal
measure theory (Axioms 2.1 that
Kolmogorov's axiomatisation is without doubt the standard in probability theory. However, we are aware of other, less popular frameworks, for example, that is more amenable to Bayesian probability 34 one based game theory [99] and imprecise probabilities 61
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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Data Generating Process
Probability Theory
Causal Data
Causal Reasoning
Data
Generating Process
Data
Statistics
Causal Discovery
(a) Statistics (or machine learnin)) is inverse problem (b) Causal discovery is inverse problem of causal of probability theory.
reasoning.
Figure 1: Data generating processes and data.
theory, it is surprising that few of them try to build directly upon the axioms of probability theory, and those that do fall short in different ways (see Related Works).
On the other hand, we place manipulations at the heart of our approach to causality; in other words, we make changes to some parts of system, and we are interested in what happens to the rest of this system. This manipulative philosophy towards causality is shared by many philosophers [64,, and is the essence behind almost all causal frameworks proposed and adopted in the statistics/machine learning community that we are aware of.
To this end, we propose the notion of causal spaces (Definition 2.2 constructed by adding causal objects, called causal kernels, directly to probability spaces. We show that causal spaces strictly generalise (the interventional aspects of) existing frameworks, i.e. given any configuration of, for example, a structural causal model or potential outcomes framework, we can construct causal space that can carry the same (interventional) information. Further, we show that causal spaces can seamlessly support situations where existing frameworks struggle, for example those with hidden confounders, cyclic causal relationships or continuous-time stochastic processes.
We stress that our paper should not be understood as a criticism of the existing frameworks, or that somehow our goal is to replace them. On the contrary, we foresee that they will continue to thrive in whatever domains they have been used in, and will continue to find novel application areas.
Related Works
Most prominently, there are the structural causal models (SCMs) 44 46). based most often on directed acyclic graphs (DAGs). Here, the theory of causality is built around variables and structural equations, and probability only enters the picture through distribution on the exogeneous variables
33 Efforts have been made to axiomatise causality based on this framework [21] 23 28 but models based on structural equations or graphs inevitably rely on assumptions even for the definitions themselves, such as being confined to finite number of variables, the issue of solvability in the case of non-recursive (or cyclic) cases, that all common causes (whether latent or observed) are modeled, or that the variables in the model do not causally affect anything outside the model. Hence, these cannot be said to be an "axiomatic definition" in the strictest sense. In a series of examples in Section 4 we highlight cases for which causal spaces have natural representation but SCMs do not, including common causes, cycles and continuous time stochastic processes.
The potential outcomes framework is a competing model, most often used in economics, social sciences or medicine research, in which we have a designated treatment variable, whose causal effect we are interested in, and for each value of the treatment variable, we have separate, potential outcome variable [31 26]. There are other, perhaps lesser-known approaches to model causality,
such as that based on decision theory [1 52 on category theory [32 20). on agent explicitly performing actions that transform the state space [5],, or settable systems (63]
Perhaps the works that are the most relevant to this paper are those that have already recognised the need for an axiomatisation of causality based on theeoreet probability theory. Ortega [42 uses a particular form of a tree to define a causal space, and in so doing, uses an alternative, Bayesian set-up of probability theory [4] has an obvious drawback that it only considers countable sets of "realisations", clearly ruling out many interesting and commonly-occuriig cases, and also does not seem accommodate cycles. Heymann et al. (27 define the information dependency model based on measurable spaces to encode causal information. We find this to be highly interesting and relevant approach, but the issue of cycles and solvability arises, and again, only countable sets of outcomes are considered, with the authors admitting that the results are likely not to hold with uncountable sets. Moreover, probabilities and interventions require additional work be taken care of. Lastly, Cabreros and Storey [12] attempt to provide a measure- measure-theoreeie grounding to the potential

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outcomes framework, but thereby confine attention to the setting of a finite number of variables, and even restrict the random variables to be discrete.
Finally, we mention the distinction between type causality and actual causality. The former is a theory about general causality, involving statements such as "in general, smoking makes lung cancer more likely". Type causality is what we will be concerned with in this paper. Actual causality, on the other hand, is interested in whether particular event was caused by particular action, dealing with statements such as "Bob got lung cancer because he smoked for 30 years". . It is an extremely interesting area of research that has far-reaching implications for concepts such as responsibility. blame, law, harm 3) model explanation 7 and algorithmic recourse [36]. Many definitions of actual causality have been proposed [25] [2 24], but the question of how to define actual causality is still not settled (. The current definitions of actual causality are all grounded on (variants) of SCMs,
and though out of the scope of this paper, it will be an interesting future research direction to consider how actual causality can be incorporated into our proposed framework.
2 Causal Spaces and Interventions
Familiarity with measure- probability theory is necessary, and we succinctly collect the most essential definitions and results in Appendix [A] Most important to note is the definition of a transition
probability kernel (also given at the end of Appendix A.I):: for measurable spaces (E, E) and (F,T),
a mapping K E x F > [0, oo] is called a transition probability kernel from (E, E) into (F, F) if the mapping + K (1z, B) is measurable for every set B E I and the mapping B + K K(x,, B) a probability measure on (F.F) for every æ E E. Also worthy of mention is the definition of
measurable reclangle: if ( (E, E) and (F,3) are measurable spaces and A E & and B E F, then the
measurable rectangle A x B is the set of all pairs (z, y) with x E A and y E B. All proofs are deferred to Appendix F. We start by recalling the axioms of probability theory, which we will use as the starting point of our work.
Axioms 2.1 (Kolmogorov
38 A probability space is a triple (S, P), where:
(i) Z is a set of outcomes;
(ii) H is collection of events forming a o-algebra, i.e. collection of subsets of S such that
(a) Ω E HC; (b) if A E K, then Ω A E H.; (c) if A1, A2,.. E H, then Un An E H;
(iii) IP is probability measure on (S) i.e. a function P. P:K + [0. 1] satisfying (a) P(0) = 0: E.. F(An) for any disjoint sequence (An) in K (c) IP)) 1.
(b) P(UnAA)
In the development of probability theory. one starts by assuming the existence of a probability space (2, H. P). However, the actual construction of probability spaces that can carry random variables corresponding to desired random experiments is done through (repeated applications of) two main results- those of conescu-Tulces and Kolmogorov 14, p. p.160, Chapter IV, Section 4]; the former constructs a probability space that can carry finite or countably infinite chain of trials, and the latter shows the existence of a probability space that can earry a process with an arbitrary index set. In both cases, the measurable space (H) is constructed as product space:
(i) for a finite set of trials, each taking place in some measurable space Ee), t= 1,..., n, we have (H, / (E,, E,);
(ii) for countably infinite set of trials, each taking place in some measurable space (Fe, E N, we have (52, H() = 0teN EE,, E
(iii) for a process (X, : t E T) with an arbitrary index set T, we assume that all the X, live in the same standard measurable space (E, E), and let (2, = (E,E)T eIeT (E, E).
In the construction of a causal space, we will take as our starting point probability space (Ω, H... where the measure P is defined on a product measurable space (S2, H) Ster (E,, Es) with the (E,, &,) being the same standard measurable space if T is uncountable. Denote by P(T) the power set of T, and for S E P(T), we denote by H.s the sub-o-algebra of K = G generated by measurable rectangles XIeT At, where A, E &, differs from E, only for ( E S. In particular, Ha = H] is the trivial olagebrra of = X4 Also, we denote by 2s the subspace xses F. of Ω Xter E,, and for T 2 S 2 U. we let TSU denote the natural projection from Us onto Ly.
3

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Definition 2.2. A causal space is defined as the quadruple (St, PKK, where (9, P) Ex &tere, PP) a probability space and K {Ks S E P(T)), called the causal mecha- nism, is collection of transition probability kernels Ks from (9,Hs) into (R,H)), called the causal kernel on Hys that satisfy the following axioms:
(i for all A E H and w E 92
Ko(w, -)= IP(A):
(ii) for all E Ω A E Ky and H, Ks(w, B) - 1a (W) B)
B); in particular, for A E Hs, Ks A) 1 [w K Q) Lا(w) = (A).
Here, the probability measure should be viewed as the "observational measure", and the causal mechanism IK, consisting of causal kernels Ks for S E (TT), contains the "causal information" of the space, by directly specifying the interventional distributions. We write la(w) when viewed as a function in for a fixed A, and (A) when viewed measure for fixed w Ω. Note that K cannot be determined "independently" of the probability measure I., since, for example, K6 is clearly dependent on by ii)
Before we discuss the meaning of the two axioms, we immediately give the definition of an interven- tion. An intervention is carried out on sub-o-algebra of the form Hy for some U E (T)). In the fol- lowing, for S E P(T), we denote Wy = (w)) Then note that Ω = Q2 x lTT s and for any w E Ω, we can decompose it into components as w = (ws,wry, wr,s). Then Ks(w, A) = Ks( wrys), ) A) for any A E H only depends on the first we component of = (ws, sT). As slight abuse of notation, we will sometimes write Ks (Ws, A) for conciseness.
Definition 2.3. Let (9, H. IP, K) = (XteT E,, &IETE,, P. K) be causal space, and U E P(T). Q probability measure on (S), (G)) and L {Lv V E P(U)) a causal mechanism on (0,H6, Hy. Q). An intervention on He via (Q, LL) new causal space (Ω, H, pdo(U.2), (o((U, where the intervention measure pdo(U.() is probability measure on H() defined, for A H. by
pdb(U..0) (A) I Q(dw)
and (K (K d.).. :S E P(T)) is the intervention causal mechanism whose intervention causal kernels are
KS L..L). (w. = Lsnu dwly)) Suu (Wsr,,
The intuition behind these definitions is as follows. Starting from the probability space (2,, J. P), we choose a "subspace" on which to intervene, namely a sub-o-algebba Her of H. The intervention is the process of placing any desired measure Q on this "subspace" (S2, Hy,), along with an internal causal mechanism on this "subspace" The causal kernel Ky corresponding to the "subspace" Hy. which is encoded in the original causal space, determines what the intervention measure on the whole space H will be, via equation (I.. For the causal kernels after intervention, the causal effect first takes place within HG via the internal causal mechanism L, then propagates to the rest of HC via equation (2).
The definition of intervening on a o-algebra of the form Hy given in Definition 2.3 sheds light on the two axioms of causal spaces given Definition 2.2
Remark 2.4. Trivial Intervention Axiom (i of causal mechanisms in Definition 2.2 ensures that intervening on the trivial o-algebra (i.e. not intervening all) leaves the probability measure
intact, i.e. writing Q for the trivial probability measure on (0, Ω). have pdo(0.C) P.
Interventional Determinism Axiom (ii of Definition ensures that for any A E Ky, we have (po((..0) (A) = Q(A), which means that if we intervene on the causal space by giving Hy a particular probability measure Q, then Hy indeed has that measure with respect to the intervention probability measure.
*Choosing Q to have measure on a single element would correspond to what is known as a "hard intervention" in the SCM literature. Letting Q and IL be arbitrary would allow obtain any "soft intervention".

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The following example should serve as further clarification of the concepts.
Example 2.5. Let Ei = Es = R, and E be Lebesgue o-algebras on Ei and Eg- Each e1 E1 and e2 E E2 respectively represent the altitude in metres and temperature in Celsius of random location. For simplicity, we
Alt
Tem
Figure 2: Altitude and Temperature.
assume jointly Gaussian measure IP on (9,)C)
=EE E2, &1 2 E2), say
1000
300
1 ) For each
15
with mean vector
and covariance matrix
10
-15
E En and A E E2, we let K (e1, A) be the conditional measure of given
e1, i.e. Gaussian with mean 1200-6x and variance 3. This represents the fact that, if we intervene by fixing the altitude of a location, then the temperature of that location will be causally affected. However, if we intervene by fixing temperature of a location, say by manually heating up cooling down a place, then we expect that this has no causal effect on the altitude of the place. This can be represented by the causal kernel K2(e2, B) = -((B) for each B E E i.e. Gaussian measure with mean 1000 and variance 300 regardlexs of the value of ez. The corresponding "causal graph" " we
20
would be Figure 2 If we intervene on & with measure 01000. i.e. we fix the altitude at 1000m, then the intervention measure (do((16uooo) on (F1z, &2) would be Gaussian with mean 10 and variance If we intervene on E2 with any measure Q the intervention measure pdo(2, on (Ei, &1) would still be Gaussian with mean 1000 and variance 300.
The following theorem proves that the intervention measure and causal mechanism are indeed valid.
Theorem 2.6. From Definition 2.3
pdo(U.2) is indeed a measure on (H,HH), and ( is indeed
valid causal mechanism on (2, H. pdo(U.D)), i.e. they satisfy the axioms of Definition 2.2
To end this section, we make a couple of further remarks on the definition of causal spaces.
Remark 2.7.
(i) We require causal spaces to be built on top of product probability spaces, as opposed to general probability spaces, and causal kernels are defined on sub-o-algehras HC of the form Hyg for S E PTT), as opposed to general sub-o-algebras of H. This is because, for two events that are not in separate components of product space, one can always intervene on one of those events in such a way that the measure on the other event will have to change, meaning the causal kernel cannot be decoupled from the intervention itself. For example, in a dice-roll with outcomes 2,, 2, 3, 4, 5, 6} each with probability 1 if we intervene give measure
6* roll then the other outcomes are forced to have measure O. Only if we consider separate components of product measurable spaces can we set meaningful causal relationships that are decoupled from the act of intervention itself.
(ii) We do not distinguish between interventions that are practically possible and those that are not. For example, the "causal effect of sunlight on the moon's 's temperature" cannot be measured realistically, as would require covering up the sun, the information encoded in the causal kernel would still correspond to what would happen when we cover up the sun.
3 Comparison with Existing Frameworks
In this section, we show how causal spaces can encode the interventional aspects of the two most widely-used frameworks of causality, i.c. structural causal models and the potential outcomes.
3.1 Structural Causal Models (SCMs)
Consider an SCM in its most basic form, given in the following definition.
Definition 3.1 (146) p.83, Definition 6.2]). structural causal model e = (S, P) consists of
a collection S of (structural) assignments X, := fPA, Nz).j = 1...., d, where PA, C (X1,-., Xa) {X,) are called the parents of X, and Nj are the noise variables; and distribution over the noise variables such that they are jointly independent.
The graph G of an SCM is obtained by creating one vertex for each X, and drawing directed edges from each parent in PA, to Xy. This graph is assumed be acyelic.
Below, we show that unique causal space that corresponds to such an SCM can be constructed.
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First, we let the variables Xz.j = 1.... take values in measurable spaces (E,, respectively. and let (92, = (E,,
Ez). An SCM e entails a unique distribution P' over the variables X (X1, Xa) by the propagation of the noise distribution P through the structural equations ʃj 46 p.84, Proposition 6.3), and we take this IP as the observational measure of the causal space. More precisely, assuming (1.....() df is a topologiaal ordering, we have, for A, E Ej.J = ....,,.
P(A1 Eg ... x Ea) = P((n1 f11) E A1))
P(A, A2 Es x x Fa) = P([((11, n2): : (f
A1 x A2))
P(A, x x Aa) =
na) (f. falfi(a))).).... (.. E A x ... x Aat).
Finally, for each S E P((], d}) and for each E Zt define 1,5 13 if j & S and 1 = w if
j E S. Then we have
Ks(w,Ap x1....x x Ad) = TP (111,,.... na): S (F." (m1); ( E A1.
(1 n1))
Aat).
This uniquely specifies the causal space (S, P. K) that corresponds to the SCM C. While this shows that causal spaces strictly generalise (interventional aspects of) SCMs, there are fundamental philosophical differences between the two approaches, as highlighted in the following remark.
Remark 3.2.
(i) The "system" in an SCM can be viewed as the collection of all variables Xi.....Xd. and the "subsystems" the individual variables or the groups of variables. Each structural equation f, encodes how the whole system, when intervened on, affects a subsys- tem X,, i.e. how the collection of all other variables affects the individual variables (even though, in the end, the equations only depend on the parents). This way of encoding causal effects seems somewhat inconsistent with the philosophy laid out in the Introduction, that we are interested in what happens to the "system" when we intervene on a "subsystem". It also seems inconsistent with the actual action taken, which is to intervene on subsystems, not the whole system, or the parents of particular variable.
In contrast, the causal kernels encode exactly what happens to the whole system, i.e. what measure we get on the whole measurable space (S2, H(), when we intervene on a "subsystem", i.e. put a desired measure on sub-o sub-o-algebra of 7
(ii) The primitive objects of SCMs are the variables Xj. the structural equations fy and the distribution PN over the noise variables. The observational distribution, as well as the interventional distributions, are derived from these objects. It turns out that unique existence of observational and interventional distributions are not guaranteed, and can only be shown under the acyclicity assumption or rather stringent and hard-to-verify conditions on the struc- tural equations and the noise distributions [10). Moreover, it means that the observational and interventional distributions are not decoupled, and rather are linked through the struc- tural equations fj. and as result, it is not possible encode the full range of observational and interventional distributions using just the variables of interest (see Example 4.1.
In contrast, in causal spaces, the observational distribution P. as well as the interventional distributions (via the causal kernels), are the primitive objects. Not only does this automati- cally guarantee their unique existence, but it also allows the interventional distributions (i.e. the causal information) to completely decoupled from the observational distribution.
(iii)
Galles and Pearl (1 Section 3) propose three axioms of counterfactuals based on SCMs (called causal models in that paper), namely, composition, effectiveness and reversibility. Even though these three concepts can carried over to causal spaces, the mathematics through which they are represented needs to be adapted, since the tools that are used in causal spaces are different from those used in causal models of Galles and Pearl [2T]. In particular, work directly with measures the primitive objects, whereas Galles and Pearl (21) use the structural equations as the primitive objects, and the probabilities only enter through measure on the exogenous variables. Thus, the three properties can be phrased in the causal space language as follows:
In this sense, some philosophy is shared with generalised structural equation models (GSEMs) 48

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Composition For S, S T, denote by Q the measure on HSUR obtained by restricting [do(S Then (do(S pdo(sUR.Q') In words, intervening on Hy via the measure Q is the same as intervening on HSUR via the measure that it would have we intervened on His via Q
This is not in general true. A counterexample can be demonstrated with simple SCM, where X1, X2 and X3 causally affect Y, in a way that depends not only on the marginal distributions of X1, X2 and X3 but their joint distribution, and X1, X2 and X3 have no causal relationships among them. Then intervening on X1 with some measure Q cannot be the same as intervening on X1 and X2 with Q IP, since such an intervention would change the joint distribution of 1, X2 and X3. even if we give them the same marginal distributions.
Effectiveness For S S R T, if we intervene on H via measure Q, then His has measure Q restricted to Hy.
This is indeed guaranteed by interventional determinism (Definition 2.2 so effec- tiveness continues to hold in causal spaces.
Reversibility For S, R. S T, let Q be some measure on Hys and Q1 and Q2 be measures
on Hsuu and H Huur respectively such that they coincide with Q when restricted to Hy.
Then if pao(SURt,1)) (BB) = Q2 for all B E Hy and if pdb( = QC)))
for all C E R. then pd)(s,0) (A) = (A) for all A E HR.
This does not hold in general in causal spaces; in fact, Example 4.2 is a counterexample of this, with S = 0.
3.2 Potential Outcomes (PO) Framework
In the PO framework, the treatment and outcome variables of interest are fixed in advance. Although much of the literature begins with individual units, these units are in the end i.i.d. copies of random variables under the stable unit treatment value assumption (SUTVA), and that is how we begin.
Denote by (ś, Kt IP) the underlying probability space. Let : â > Z be the "treatment" variable, taking values in a measurable space (2,3). Then for each value z of the treatment, there is separate random variable Y, Ω - V, called the "potential outcome given Z =" taking values in ; we also have the "observed outcome", which is the potential outcome
measurable space (), ()):: consistent with the treatment, i.e. Y Yz. The researcher is interested in quantities such as the "average treatment effect", E[Y,, Y=2), where E is the expectation with respect P. measure the causal effect of the treatment. Often, there are other, "pre-treatment variables" or "covariates", which we denote by X Ω + X, taking values in measurable space (X', X). Given these, another object of interest is the "conditional average treatment effect", defined as EYY., - Yzu X].
It is relatively straightforward to construct causal space that carry this framework. We define Ω = z x y x X' and D( = 3 & D @ X. We also define P, for each A E 3. B E 2) and C e X, as P(A x x C*) = P(Z E A, B, E C). As for causal kernels, we are essentially only interested in K2(2 B) for B E D) and we define these to be K2(2, B) = PYYY, E B).
4 Examples
In this section, we give few more concrete constructions of causal spaces. In particular they are designed to highlight cases which are hard to represent with existing frameworks, but which have natural representations in terms of causal spaces. Comparisons are made particularly with SCMs.
4.1
Confounders
The following example highlights the fact that, with graphical models, there is no way to encode correlation but no causation between two variables, using just the variables of interest.
Example 4.1. Consider the popular example of monthly ice cream sales and shark attacks in the US (Figure Baj. that shows that correlation does not imply causation. This cannot be encoded by an SCM with just two variables as Figure 3b since no causation means no arrows between the variables, which in turn also means dependence. One needs to add the common causes into the
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T
Jan
Mar
May
Jul
Sep
Nov
Month
S
S
Ice cream sales
Shark attacks
(h)
Figure 3: Correlation but no causation between ice-cream sales and shark attacks. S stands for the number shark attacks, for ice cream sales, for temperature and for economy.
model (whether observed or latent). the most obvious one being the temperature (high temperatures make people desire ice cream more, as well as go to the beach more), as seen Figure 3el Now we have a model in which both dependence and no causation are captured. But can stop here? There are probably other factors that affect both variables, such as the economy (the better the economic situation, the more likely people are buy ice cream, and to take beach holidays) Not only is the model starting to lose parsimony, but as soon as we stop adding variables to the model. we are making assumption that there are further confounding variables out there in the world B
Figure 3d
In contrast, causal spaces allow us model any observational and causal relationships with just the variables that we were interested in, without any restrictions or the need to add more variables. In this particular example, we would take as our causal space (Eạ E2, &1 @ Eg, P. P,),,K), where F1 = Es TR. with values in En and E2 corresponding to ice cream sales and shark attacks respectively, and ε = E being Lebesgue a-algebras. Then we can let be a measure that has strong dependence between any A E ε1 and B E &2, but let the causal kernels be K₃ K1(x. B) P(B) for any x E₁ and B E E2, and likewise K2(1, A) = P(A) for any x E E2 and E &.
Nancy Cartwright argued against the completeness of causal Markov condition, using an example of two factories [3] p. 108], which there may not even be any confounders between dependent variables, not even unobserved one. If we accept her position, then there are situations which SCMs would not able to represent, whereas causal spaces would have no problems at all.
4.2 Cycles
As mentioned before, cycles in SCMs cause serious problems, namely that observational and inter- ventional distributions that are consistent with the given structural equations and noise distribution may not exist, and when they do, they may not exist uniquely. This is an artefact of the fact that these distributions are derived from the structural equations rather than taken as the primitive objects. In the vast majority of the cases, cycles are excluded from consideration from the beginning and only directed acyclic graphs (DAGs) are considered. Some works study the solvability of cyclic SCMs 23 10, where the authors investigate under what conditions on the structural equations and the noise variables there exist random variables and distributions that solve the given structural equations, and if so, when that happens uniquely. Other works have allowed cycles to exist, but restricted the definition of an SCM only to those that have a unique solution 23
49].
Of course, cyclic causal relationships abound in the real world. In our proposed causal space (Definition 2.2), given two sub-o-algebras His and Hyr H. nothing stops both of them from having causal effect on the other (see Definition B. for precise definition of causal effects), but we are still guaranteed to have a unique causal space, both before intervention and after intervention on either HCs or Hy. The following is an example of a situation with "cyclic" causal relationship.
Example 4.2. We want to model the relationship between the amount of rice in the market and its price per kg. Take as the probability space (E1 x Eg, &1 @ P), where E, E2 R with values in E₁ and Eg representing the amount of rice in the market in million tonnes and the price
*One solution could be to add a single "variable" that collects all of the confounders into one, but then the numerical value of this "variable", as well as its distribution and the structural equations from this "variable" into and I, would be completely meaningless.

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Price
Price
I
2
R
Rice
Rice
Rice
(b)
(c)
(d)
Figure 4: Rice in the market in million tonnes and price per kg in KRW.
of rice per kg in KRW respectively, & Eg are Lebesgue a-algebras and TP is for simplicity taken to be jointly Gaussian. Without any intervention, the higher the yield, the more rice there is in the market and lower the price, as in Figure [4b] If the government intervenes on the market by buying up extra rice or releasing rice into the market from its stock, with the goal of stabilising supply at 3 million tonnes, then the price will stabilise accordingly. say with Gaussian distribution with mean 4.5 and standard deviation 0.5, as in Figure 4c The corresponding causal kernel will be On the other hand, if the government fixes the price of rice at a price, say at 6,000 KRW per kg, then the farmers will be incentivised to produce more, say with Gaussian distribution with mean 4 and standard deviation 0.5, as in Figure 4d The corresponding causal
K (3,3x)
2 _ =
V22
kernel will be (2 y)
1( 4-5
Causal spaces treat causal effects really what happens after intervention takes place, and with this approach, cycles can be rather naturally encoded, as shown above. We do not view cyclic causal relationships an equilibrium of dynamical system, or require be viewed an acyelic stochastic process, as done by some authors [46 p.85, Remark 6.5].
4.3 Continuous-time Stochastic Processes, and Parents
A very well established sub-field of probability theory is the field of stochastic processes, in which the index set representing (most often) time can be either discrete or continuous, and in both cases, infinite. However, most causal models start by assuming finite number of variables, which immediately rules out considering stochastic processes, and efforts to extend to infinite number of variables usually consider only discrete time steps [46, Chapter 10] or dynamical systems ( 47 50). Since probability spaces have proven to accommodate continuous time stochastic processes in a natural way, it is natural believe that causal spaces, being built up from probability spaces, should be able to enable the incorporation of the concept of causality in the theory of stochastic processes.
Let W be todelly set, in particular W N {0, I,..... W Z - {.., -2 1,, 1, ...). W = R = [0, oo) or W R = ( (o0, oo) considered as the time set. Then, we consider causal spaces of the form (2, H. IP. K) (XLET E,, @ieT E PK,,, K). where the index set T can be written other index set T. The following notion captures the intuition that causation can only go forwards in time.
T W T for
Definition 4.3. Let HH, K)) = (x, &,, P, P.) be causal space, where the index set can be written as T W x T, with W representing time. Then we say that the causal mechanism K respects time, or that K is a time-ree resprerspeting causal mechanism, if, for all Wt. , W2
W with
Wi < W2, we have that H. w x has no causal effect (in the sense of Definition B. on H 1o xf
In a causal space where the index set T has time component, the fact that causal mechanism K respects time means that the past can affect the future, but the future cannot affect the past. This already distinguishes itself from the concept of conditioning conditioning on the future does have implications for past events. We illustrate this point in the example of Brownian motion.
Example 4.4. We consider I-dimensional Brownian motion. Take (XLER) Et, SLER P,P,,K, where, for each t E R,,, Eg = R and & is the Lebesgue o o-algebra, and IP is the Wiener measure. For we have causal kernels Kz(x,y) [2)(tt=a
I(c-o) z(C-)(1- (v=x)² and Ke(x,v)
/25s
The former says that, if we intervene by setting the value of the process to x at time 8, then the process starts again from x, whereas the latter says that if we intervene at time I, the past values at time are not affected. On the left-hand plot of Figure 5 we set the value of the process at time to O. The past values of the process are not affected, and there is discontinuity at time 1 where the process starts

---

2
1
1
A dn mr
Mah
0
0
2
0
Figure 5: 1-dimensional Brownian motion, intervened and conditioned have value 0 time 1.
again from O. Contrast this to the right-hand plot, where we condition on the process having value 0 at time 1. This does affect past values, and creates Brownian bridge from time 0 to time 1.
Note, Brownian motion is not differentiable, so no approach based on dynamical systems is applicable.
Remark 4.5. The concept of purents is central in SCMs the structural equations are defined on the parents of each variable. However, continuous time dense, so given two distinct points in time, there is always time point in between. Suppose we have one-dimensional continuous time Markov process (Xe) SER [14 p. p.169], and a point to in time. Then for any t < to. Xg has a causal effect on Xto. but there always exists some t' with t < t' < to such that conditioned on X X, does not have a causal effect on Xt.. meaning X, cannot be parent of Xt. In such a case, X cannot be said to have any parents, and hence no corresponding SCM can be defined.
5 Conclusion
In this work, we discussed the lack of a universally agreed axiomatisation of causality, and some arguments as why measure-theoretic probability theory can provide a good foundation on which to build such an axiomatisation. We proposed causal spuces, by enriching probability spaces with causal kernels that encode information about what happens after an intervention. We showed how the interventional aspects of existing frameworks can be captured by causal spaces, and finally we gave some explicit constructions, highlighting cases which existing frameworks fall short.
Even if causal spaces prove with time to be the correct approach to axiomatise causality, there is much work to be done in fact, all the more in that case. Most conspicuously, we only discussed the interventional aspects of the theory of causality, but the notion of counterfactuals also seen as a key part of the theory, both interventional counterfactualls advocated by Pearl's ladder of causation [44] Figure 1.2] and backtracking counterfactuals [58). We provide some discussions and possible first steps towards this goal in Appendix [ but mostly leave it as essential future work. Only then will we be able to provide a full comparison with the counterfactual aspects of SCMs and the potential outcomes. As discussed in Section [] the notion of actual causality is also important, and would be interesting to investigate how this notion can be embedded into causal spaces. Many important definitions, including causal effects, conditional causal effects, hard interventions and sources, as well as technical results, were deferred to the appendix purely due to space constraints, but we foresee that there would be many more interesting results be proved, inspired both from theory and practice. In particular, the theory of causal abstraction 49] 2 3 should benefit from our proposal, through extensions of homomorphisms of probability spaces to causal spaces
As a final note, we stress again that our goal should not be understood as replacing existing frameworks. Indeed, causal spaces cannot compete in terms of interpretability, and in the vast majority of situations in which SCMs, potential outcomes or any of the other existing frameworks are suitable, we expect them to be much more useful. In particular, assumptions are unavoidable for identifiability from observational data, and those assumptions are much better captured by existing framework." However, just as measure- thheorect probability theory has its value despite not being useful for practitioners in applied statistics, we believe that it is a worthy endeavour to formally axiomatise causality.
$ "In similar vein, Keurti et al. (37 consider homomorphisms between groups of interventions. 'Researchers from the potential outcomes community and the graphical model community are arguing as to We do not take part in this debate.
which framework is better for which situations (30 43
10

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Acknowledgments and Disclosure of Funding
We express sincere gratitude to Robin Evans the University of Oxford, Michel de Lara École des Ponts ParisTech and Wojeiech Niemiro at the University of Warsaw for fruitful discussions and providing valuable feedback on earlier drafts. We also thank anonymous reviewers for their suggestions for improvements.
This work was supported by the Tübingen Al Center.
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A Mathematical Preliminaries
In this section, we recall some basic facts about measure and probability theory that we need for the development in the main body of the paper. We follow Çinlar [14]
A.1 Measure Theory
Suppose that E is set. We first define the notion of o-algebra. non-empty collection of E is called o-algebra on E if it is closed under complements and countable unions, that is, if
(1 A = EAAEE
(ii A1, A2... E = U A
14, p.2]. We call {0, E} the trivial a-algebra of E. If is an arbitrary collection subsets of E, then the smallest o-algebra that contains C, or equivalently, the intersection of all o-algebras that contain C, is called the a-algebra generated by C, and is denoted oC.
A measurable space is pair (E, E), where E is set and is o-algebra on E 14 p.4].
Suppose (E, E) and (F,3) are measurable spaces. For A and B E I, we define the measurable rectangle A x B as the set of all pairs (z, y) with æ E A and E B. We define the product o-algebra I E F the o-algebra generated by the collection of all measurable rectangles. The measurable space (E x F, @ F) is the product of (E, E) and (F, (F [14. p.4]. More generally, if ( (E,, En) are measurable spaces, their product is
(E,
E
E.).
where E₁ x... En is the set of all n-tuples (I1,..., Inn with 2; in E, for L...., n and & &..... En is the o-algebra generated by the measurable rectangles A1 x... x An with A, in & for = L..,, 14 p.44]. If I' is arbitrary (countable uncountable) index set and (E,, E,) is measurable space for each t E T, the product space {Eç : t E T is the set X Eg of all collections (xt)er with x4 E E, for each & T. A rectangle in Xier F4 is a subset of the form
X A, = (x= tE
X E ot A₁ for each in T} tET
where A, differs from F, for only a finite number of t. is said to be measurable if A, E & for every & (for which A, differs from F2). The o-algebra Xier F, generated by the collection of all measurable rectangles is called the product o-algebra and is denoted by X 1ier & 14 p.45].
collection of subsets of is called a p-system if is closed under intersections [14] p.2]. two measures μ and generating E, then and v are identical [14 p. 1616, Proposition 3.7].
measurable space (E, E) with ({E) v(E) co agree a p-system
Let (E, E) and (F,F) be measurable spaces. A mapping f F -> F Fis measurable if f- B e E for every B E [14 p.6].
Let E) and (F.3) be measurable spaces. Let f be bijection between and F, and let f denote its functional inverse. Then, f an isomorphism f is measurable relative to and F, and is measurable with respect to 3 and E. The measurable spaces (E, E) and (F, 3) are isomorphic if there exists an isomorphism between them [1] p.1]..
A measurable space (E, E) is standard measurable space if it is isomorphic to (F, BF) for some Borel subset F of R. Polish spaces with their Borel o-algebra are standard measurable spaces [1 1.I
Let A C F. Its indicator, denoted by 1 is the function defined by
i r E
1x
if $
15

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14 p.8]. Obviously, 14 is E-measurable if and only if A E & A function f:E. - I. is said to be simple if is of the form
/ = Σ
04
==1
for some n E N, d1,..., On E R and A1. An E & 14 p.8]. The A1.... A, E & can be chosen measurable partition of E, and is then called the canonical form of the simple function f. A positive function on E is E-measurable if and only if it is the limit of an increasing sequence of positive simple functions 14 p. 10, Theorem 2.17].
measure on a measurable space (F, E) is mapping jr: & + [0, oc] such that
(i) p())
(ii) (( En= (AA))) for every disjoint sequence (An) in
14, p.14]. A measure space is a triplet (E, LE,C,H) H., where (E, E) is measurable space and j is measure
measurable set is said to be negligible if (a = 0, and an arbitrary subset of E is said to be negligible if it is contained in measurable negligible set. The measure space is said to be complete every negligible set is measurable [4] p. 17].
Next, we review the notion of integration of real-valued function f E - R with respect to μ [1 p.20, Definition 4.3].
(a) Let f: - |0, oo| be simple. If its canonical form is f Ei= a, a; with as E R, then we define
[dp Σ a.)(A.).
(b) Suppose f: + J0, oc] measurable. Then by above, we have a sequence (fr) of positive simple functions such that fn + f pointwise. Then we define
f [du = lim
/ fndd,
n-xo
where f fn is defined for each n by (a).
(c) Suppose f: E + 1-o0, oo] is measurable. Then f = max{f. f. 0} and f== min 0} are measurable and positive, so we can define I f dp and f du in (b). Then we define
J fdw= =
/
dp f
f
provided that least one term on the right be positive. Otherwise, fdu undefined. If I f+ du and J f dp oc, then we say that is integrable.
Finally, we review the notion of transition kernels, which are crucial in the consideration of conditional distributions. Let (E, E) and (F,3) be measurable spaces. Let K be a mapping J [0, so). Then, K is called transition kernel from (E, E) into (F.3) if
(a) the mapping æ + K (x, B) is measurable for every set J; and
(b) the mapping B + K (z, B) measure on (F,F) for every x E.
A transition kernel from (F,E) into (F,F) is called probability transition kernel if K(x, F) = for all æ E E. A probability transition kernel from (E, E) into (E, is called a Markov kernel on (E, 8) 14 p.37.39,40J.
A.2 Probability Theory
Now we translate the above measure-theoretie notions into the language of probability theory, and introduce some additional concepts. A probability space is a measure space (2,,C, such that
16

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P(Z) [14] p.49]. We call Ω the sample space, and each element E St an outcome. We call H collection of events, and for any A E H. we read ((A) as the probability that the event A occurs [4 p.50].
random variable taking values in measurable space (F,, E) function x E., measurable with respect to H and E. The distribution of X is the measure i on (E, é) defined by ((A) P(X A) [14 p.51]. For arbitrary T, let X₁ be random variable taking values in (E, 8) for each t E T. Then the collection {X,:t e ] is called stochastic process with state space (E, 8) and parameter set 14 p.53].
Henceforth, random variables are defined on (2, and take values |-oc, oo). We define the expectation of random variable X 52 + [-00, oo] as [[X] lg X ddI [4 p.57-58). We also define the conditional expectation 14 p. p.140, Definition 1.3]. Suppose F is a sub-o-algebra H.
(a) Suppose X is a positive random variable. Then the conditional expectation of X given is any positive random variable E satisfying
[V X] = E X
for all V Ω + [0, co] measurable with respect to F.
(b) Suppose X : Ω - [-o0, oo] is random variable. If [[X] exists, then define
Eg X = - E
where X X and Eg X are defined in (a).
Next, define conditional probabilities, and regular versions thereof [14 pp. 149-151]. Suppose H E H. and let I' be sub-o-algebra of H. Then the conditional probability of given I is defined
Py = EglH.
Let Q(H) be a version of P3 for every H J. Then Q (w, + (I) is said be regular version of the conditional probability Py provided that Q be probability transition kernel from (12, F) into (H, Regular versions exist (2,H) is a standard measurable space [4 p. 1515 Theorem 2.7].
The conditional distribution of random variable X given F is any transition probability kernel L: (w, B) + (B) from (), into (E, E) such that
P B) L(B)
for all B E.
(E, is standard measurable space, then version of the conditional distribution of X given F exists [14] p.151].
Suppose that T is totally ordered set, i.e. whenever r, 8, T with r < and and for any è E T. exactly one of < t, s = f and & < holds [88 p.62]. For each t E T. let Fe be sub-o-algebra of H. The family T (3, : E T} is called filtration provided that F, I for < t [14] p.79]. A filtered probability space (Ω, H., F P) is a probability space (H, endowed with filtration J.
have <
Finally, we review the notion of independence and conditional independence. For a fixed integer 2, let F,, be sub-o-algebin of H. Then (F1,..., ..., F..,,n) is called an independency if
P P(H .... Hx) = (H1) ......
for all H1 E FT,... H, E F,, Let 7 be arbitrary index set. Let F, be sub-o-algebra of K for each / E T. The collection (3, T) called an independency its every finite subset is an independency [14] p.8
Moreover, 3 FI,..., Fn are said be conditional independent given F if
P3 H1 n... n HA) = Pp (H1)..PP ...P. (H,)
for all Hi E F F1, Hn Tn 14 p.8]
17

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B Causal Effect
In this section, define what it means for sub-o-algebra of the form Hg to have causal effect on an event A E H..
Definition B.1. Let (2,,,, (XIETE, &, K) be a causal space. U @PT)), H an event and 'f a sub-o-algebra of DC (not necessarily of the form Hy for some S E P(T)).
(i If Ks(w, A) Ksyu(w (w, A) for all e P(T) and all w E Ω then say that My has no causal effect on A, or that Hy is non-causal to A.
We say that Hy has no causal effect on F. or that Hy is non-causal to F. if, for all A E F, Hy has no causal effect on A.
(ii) If there exists w E such that (w(w, A) + P(A), then we say that Ky has an active causal effect on A, or that Hy is actively causal to A.
We say that Hy has an active causal effect F, or Hy is actively causal to F, if Ky has an active causal effect on some A E F.
(iii) Otherwise, we say that Hy has a dormant causal effect on A, or that Hy is dormantly causal A.
We say that Hy has dormant causal effect on I, or that Hy is dormantly causal F, Hy does not have an active causal effect on any event in F and there exists A E F on which Hy has a dormant causal effect.
Sometimes, we will say that Hy has a causal effect on A to mean that Hyr has either an active or a dormant causal effeet on A.
The intuition is as follows. For any S E P(T) and any fixed event E H, consider the function has no causal effect on A, then it means that the causal kernel does not depend on the went component of wg. Since this has to hold for all S E P(T). it means that it is possible to have, for example, Ky A) = F(A) for all w E and yet for Hy to have causal effect on A. This would be precisely the case where Ky has a dormant causal effect on A, and it means that, for some P(T), wg + Ks((wsne wsyu, A) does depend on the wsnu component.
A). If HC We + ()
We collect some straightforward but important special cases in the following remark. Remark B.2. (a) If Hy has no causal effect on A, then letting S = U in Definition B. 1 and applying Definition 2.2 we can see that, for all w Ω
Ko A) Kuyo (w, A) = Ko A) = P(A).
In particular, this means that Fly cannot have both no causal effect and active causal effect
A.
(b) It is immediate that the trivial o-algebra Hgg {0, S2}} has causal effect on any event E H. Conversely, it is also clear that Hy for any U E P(T)) has no causal effect on the trivial o-algebra.
(c) Let U E P(T) and I' a sub-o-algebra of H. If If nF + 2). then Hyr has an active causal effect on F, since, for Hy n with A of 0 and A # Ω, Definition 2.2itiioon22211 tells that (=(,, A) = (.(.) & IP))). In particular, Hy has an active causal effect on itself. Further, the full o-algebra H Hy has an active causal effect on all of its sub-o-algebras except the trivial o-algebra, and every Hy, U E P(T)) except the trivial o-algebra has an active causal effect on the full o-algebra HC.
(d) Let U PTT) and F2,F2 be sub-o-algebras of H. If S F2 and Hy has no causal effect on T2. then it is clear that Hy has no causal effect on F1.
(c) If Ke has no causal effect on an event A, then for any V E P(T) with V C U, Hy has no causal effect on A. Indeed, take any S P(T). Then using the fact that Hy has no causal effect on A, see that, for any E Ω
(w, A) = (ws
applying Definition B. with S
18

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Ksy A)
since V CU
= Ks(w,A) A)
applying Definition B. i with S.
Since S E ((T) was arbitrary, we have that Fly has no causal effect on A.
(f) Contrapositively, if U. E P(T) with V _ U and Hy has causal effect on A, then Hy has a causal effect on A.
U E P(T) has no causal effect on A, then for any V E P(T), we have
(v = Aw)r(w,A).
Indeed,
Kuur(w, A) ((ounram since U V has no causal effect on A by A)
since
(h) I V E P(T) and neither Hy nor Hy has a causal effect on A, then Hruv has no causal effect A. Indeed, for any S E P(T') and any w S
(s A) K(sroni(ws A)
V has no causal effeet on U has causal effect A. Since S P(T) was arbitrary, Hyrv has no causal effect on A.
Ksyu(w, A)
K)(w,
(i) Contrapositively, U, V PTT and Huuv has causal effect on then either Hy or Hy has causal effect on A.
Following the definition of no causal effect, we define the notion of trivial causal kernel.
Definition B.3. Let (S, H. IP, K) = (XteT &ier&, F, K) be a causal space, and U E P(T). We say that the causal kernel Ky is trivial if Hy has no causal effect on Hryu.
Note that we can decompose H. H Hy @ Hryv, and so H is generated by events of the form A for A Hy and E Hry.- But if Ky trivial then we have, by Axiom 2.2 ii) Ky (w, B) (P))) for such a rectangle.
We also define a "conditional" version of causal effects.
Definition B.4. Let (2, H. K) =
(Xter Et, &ter & P. IK) a causal space, U, E P(T), A H an event and F sub-o aalebba of H (not necessarily of the form Hy for some S E (T))).
(i) If (suv(w, = (ys A) for all S E P(T) and all w E Ω, then say that Hy has no causal effect on A given Hv. or that Hy non-causal to A given Hy.
We say that Her has no causal effect on F given Hv, or that Hy is non-causal to F given Hy, if, for all A F, Hy has causal effect on A given Hy.
(ii) If there exists w Ω such that Kouv (w, A) + Kv A), then we say that Hy has an active causal effect on A given Hv, or that Hy is actively causal to A given Hy.
We say that Hy has active causal effect F given Hv. that Hy is actively causal I given Hy. Hy has an active causal effect on some A E F.
(iii) Otherwise, we say that Hy has a dormant causal effect on A given Hv, or that Hy is dormantly causal A given Hy.
We say that Hy has dormant causal effect on F given Hy, or that Hy is dormantly causal F given Hy. if Ky does not have active causal effect on any event in F given Hy and there exists E F on which Hy has a dormant causal effect given Hv.
Sometimes, we will say that Hy has causal effect on given Hy mean that Hy has either an active a dormant causal effect A given Hly
19

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The intuition is as follows. For any fixed 5 E P(T) and any fixed event A E H. consider the function wsuv K Suv ( V) () A). Hf has no causal effect A given Hy, then means that the causal kernel does not depend on the WSn(U' v) component of wsuv; in other words, Hy only has an influence on through its component.
We collect some important special cases in the following remark. Remark B.5.
(a) Letting V = U, we always have Ksuu(w, A) = K (suu))(())))) (w; A) = Ksuu (w; A) for all w and H, which means that Hy has no causal effect on any event A E H given itself.
(b) If Ky has causal effect A given Hv, then letting U = S in Definition B.4 that, for all w E Ω
Kuuv (w, A) = (v(w, A).
In particular, this means that Hy cannot have both no causal effect and active causal effect on given Hyy.
(c) The case V reduces Definition B.4 to Definition B.1 i.e. Hy having no causal effect in the sense of Definition B.1 the same as Fy having no causal effect given {0, 22 in the sense of Definition B.4 etc.
(d) It is possible for Ky to be causal to an event A, and for there exist V E P(T) such that Hy has causal effect on A given Hv. However, if Ky has causal effect on A. then for any E P(T), Hyr has no causal effect on given Hy. To see this, note that Remark B tells us that U V also does not have any causal effect on A. Then given any P(T).
Ksuv(w, A) = ((sun)(UN))(
applying Definition B. to SUV. Since S E P(T) was arbitrary, Hy has no causal effect on given Hy.
C Interventions
In this section, we provide few more definitions and results related to the notion of interventions, introduced in Definition 2.3
(UU,U)) behave in some
First, we make few remarks on how the intervention causal kernels K special cases, depending on the relationship between U and S.
Remark C.1.
(a) For S E P(T') with U C S, we have, for all w E Ω and all A E H,
k do(U: (w, S Lu (wo, duly Ks (sl))ss(((ss) (wwssssww))).......
Swn (dw'y) (s((w wb),
by Definition 2.2
Ks KK((Wsyr, ).
A).
This means that, after an intervention on Her subsequent interventions on Hg with Hy C Hy simply overwrite the original intervention. Note that this reminiscent of the "partial ordering on the set of interventions" in [49], but in our setting, this is given by the partial ordering induced by the inclusion structure of aub-o-algebras of H.
(b) For E PP(T) with S U.
K do
(w, A) = I.s (wss Ku (wby A)
for all w E Ω and A H. i.e. KS do( is product of the two kernels Ky and Ls p.39]; in particular, K do(U.Q)))
(w, A) Ls(w, A) for all A Hy.
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(c) For S E P(T) with SNU = =,
Kg k Aw,A) = / Lu (ww, duhr Ks u ((ws, w/y),, A)
/ Q(dwky) K (wws. wh A)
by Definition 2
for all w E Ω and E H. i.e. the effect of intervening on Hy with Q then Hg is the same intervening on Hous with product measure of Q on Ky and whatever measure place on Hy.
We give it name for the special case in which the internal causal kernels are all trivial (see Definition B.3).
Definition C.2. Let (Ω, H,, P,, Ik) = (Xter Fa, &,, P.K) be a causal space, and U E P(T) and Q a probability measure (y). A hard intervention Ky via Q is new causal space (12, H. plo(U:2)) Kdb(U.(,hhd))) where the intervention measure [do(U..0) is probability mea- sure (2, defined in the same way as in Definition 2.3 and the intervention causal mechanism gdo(U;Q,bard) {K [Kdo(U.Q,bard) S E P(T) consists of causal kernels that obtained from the intervention causal kernels in Definition 2.3 in which Lsnu is a trivial causal kernel, i.e. one that has no causal effect on Henss.
From the discussion following Definition B.3 Lsnu A B) = LA (wssn))Q((
we have that, for A E Hsny and B E Hoys
The next result gives an explicit expression for the causal kernels obtained after a hard intervention. Theorem C.3. Lei H.HH..P. IK) = (XtET Ea, &ier&,, IP, K) be a causal space, and U E P(() and Q probability measure on (D,HC)) Then after a hard intervention on Hy via Q the intervention causal kernels Kd hhard are given by
KS kdo(U) (w, A) = Kg K-d(U, (ws, A) = ((del()ss) Sou s),
Intuitively, hard interventions do not encode any internal causal relationships within Hy, after subsequently intervene on Hys the measure Q that we originally imposed on Hy remains on Hyyss.
The following lemma contains couple of results about particular sub-o-algebras having no causal effects on particular events in the intervention causal space, regardless of the measure and causal mechanism that was used for the intervention.
Lemma C.4. Ler (2, H, H.P,K) K) = (XteT E PK be a causal space, and U E PP(T), Q a probability measure on HD) and LL = (Lv V E E(U)) a causal mechanism (R, Q). Suppose we intervene on Hy via IL.
(i) For A Hy and V P(T) with VNU = 0. Hy has no causal effect on A (c.f. Definition kij in the intervention causal space (S2, H., pdo(U.Q) (.d.(U.Q...)... i.e. events in the a- algebra Hy on which intervention took place are not causally affected by o-algebras outside Hy.
(ii) Again, let V P(T) with V VNU causal space, HV had no causal effect on A, then in the intervention causal space, Hy has no causal effect on A either.
=
0. and also let A E H be any event. If. in the original
(iii) Now let V E P(T)), A H any event and suppose that the intervention Hy via Q is hard. Then Hy had no causal effect on A in the original causal space, then Ky has no causal effect on A in the intervention causal space either.
Lemma C.4 and (ii) tell us that, if FCy had no causal effect on A in the original causal space, then by intervening on Her with N = 0 or by any hard intervention, we cannot create a causal effect from H. on A. However, by intervening on a sub-o-algebra that contains both Hy and (a part of) A, and manipulating the internal causal mechanism I. appropriately, it clear that we can create causal effect from Hy.
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The next result tells us that if a sub-o-algeera Hy has dormant causal effect on event A, then there is sub-o-algebra of 36 and hard intervention after which that sub-o-algebra has an active causal effect on A.
Lemma C.5. Let (Q,HC, = Et, Cr, be a causal space, and U E P(T). For an event A E H. Hf has dormant causal effect on A in the original causal space, then there exists a hard intervention and a subset V C U such that in the intervention causal space, Hy has an active causal effect A.
The next result is about what happens to a causal effect of a sub-o-algebre that has no causal effect on an event conditioned another sub-o-algebin after intervening on that sub-o-algebra.
Lemma C.6. Ler (S, HC, P. K) (Xter & P. I)) be a causal space, and U, V E P(T). For an event A E H., suppose that Hy has no causal effect on given Hy (see Definition B.4 Then after an intervention on Hy via any (Q,L), L), Heny has no causal effect on A.
The next result shows that, under hard intervention, time-resp causal mechanism stays time-respecting.
Theorem C.7. Let (S,H,, PP, K) = (XIET En, SIETE,,,,P be causal space, where the index set T can be written as W x T, with W representing time and KK respecting time. Take any U E P(T) and any probability measure Q on Hy. Then the intervention causal mechanism QaalU.. also respects time.
D Sources
In causal spaces, the observational distribution P and the causal mechanism K are completely decoupled. In Section [31]] we give detailed argument to why this is desirable, but of course, there is no doubt that the special case in which the causal kernels coincide with conditional measures with respect to P is worth studying. To that end, introduce the notion of sources.
Definition D.1. Let (92, J. P, K) Ex, &,erE, P.P) K) be a causal space. U E P(T)), A H an event and I sub-o-algeeba of JC. We say that Hy is a (local) source of A (y A) is version of the conditional probability Pxt (A)). We say that HG; is a (local) source of T if He; is a source of all E F. We say that Hy a global source of the causal space if Hy a source of all E H.
Clearly, source allgebrrs are not unique (whether local or global). It is easy to see that Hgg = { !?} and H( = Hr Ster&, are global sources, and axiom (ii) [f of Definition 2.2 implies that any H[g is a local source of any of its sub-o-algebray including itself, since, for any A E Hy, Px, (A) A Also, a sub-o-algebra of a source not necessarily source, nor r-algebra that contains a source necessarily source (whether local or global). In Example 2.5 above, altitude is a source of temperature (and hence a global source), since the causal kernel corresponding to temperature coincides with the conditional measure given altitude, but temperature is not source of altitude.
When we intervene on Ky (via any (Q, L)), Hy becomes global source. This precisely coincides with the "gold standard" that is randomised control trials in causal inference, i.c. the idea that, if we are able to intervene on Fly, then the causal effect of HCy on any event can be obtained by first intervening on Hy, then considering the conditional distribution on Hy. Next is a theorem showing that when one intervenes on Hy, then Hy becomes a source.
Theorem D.2. Suppose have causal space (Ω, PP K) (xieT &I IK),, and let U E P(T).
(i) For any measure Q on Hy and any causal mechanism IL. on Hy, Q). the causal kernel K (db(U.Q.)) Ky is a version of ( which means that Hy is a global source a- Ky
algebra of the intervened causal space (Ω,, (,,
Ht,,H, pdo(UU0)) R.)).
(ii) Suppose V E P(T) with V C U. Suppose that the measure Q on (Ω, Hy: factorises over Ky and Kov,, Le. for any E Hy and B Kyy,, Q(A B) ())
Then after a hard intervention on Hy via Q, the causal kernel KV podolU which means that Ky is a global source o-algebra of the intervened causal space (N,3C, D2,,C,1dolff (do(U.)) gdo(U,QQhard))
gdo(U,Q.arr a version of
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Let A E H be an event, and U E P(T). By the definition of the intervention measure (Definition we always have
( = Q(dw) A),
hence [pdl((:.)) (A) can be written in terms of P and Q if Ku(w,, A) can be written in terms of IP. This can be seen to occur three trivial cases: first, if HGr a local source of A (see Definition D.I.. in which case Kr(w: (w, A) Prcce (w, A); secondly, if Hyr has no causal effect on A (see Definition P(A); and finally, if A E Hy, in which case, by intervention we have (w, A) 1 (w). In the latter case, do not even have
B.Ip. in which case K (w. A)
determinism (Definition 2 i
dependence on F. Can we generalise these results?
Lemma D.3. Let PC, K) = Et, Bter K)) be a causal space. Let E H be an event, and U P(T)). If there exists a sub-o-algebra of (not necessarily of the form Hy for some V E ))()) such that
(i) the conditional probability Pul(U,.) A) can be wrillen in terms of and Q;
(ii) the causal kernel Ku(-, can be written in terms of I. for all B E 9:
then (pdo(U,0) (A) can be writlen in terms of and Q
Remark D.4. The three cases discussed in the paragraph above Lemma D.3 are special cases of the
Lemma with 9 being any sub-o-algebra of H with (0, 22} C9 Hy. In this case, condition (ii) is
trivially satisfied since we have k B) = (.).) by intervention determinism (Definition) 2.2 ii)
have pdo(U.0) (,, A) (- A), which means that the be written in terms of TP.
and for condition (i), by Theorem D.2
problem reduces to checking if Ku
Proof. By law of total expectations, for any V E P(T), we have
Tdd((C:)) (A)
/
pdo(U, govg (w, A)Fdo(U..0) (dw)
/ photur (w: A) f Q(dw") (ul dw).
Here, pdk(UU0) (w, A) can be written in terms of P and Q by condition (i). Moreover, note that it suffices to be able to write the restriction of Ky to Hy: V 9 in terms of P, since the integration of Hy G-measurable function. Since the collection of intersections (Dn B, E Ky, B E () is n--system that generates Hy V S [14 p.5, 1.18), it suffices to check that Ku(w', D B) can be written in terms of IP for all D E Hy and B E 9. But by interventional determinism (Definition 2.2 ii). we have (u'w', D n B) = 1 D Ku (w', .B. Since Kurw', B) can be written in terms of by condition (ii), the restriction of Ku (w', to Hy 9 can be written in terms IP, and hence pdo(U.Q) (A) can written in terms
and Q.
Corollary D.5. Let (2, KKK.K) = (XIET Ex, EIeT P,KKK be a causal space. Let H be an event, and U E P(T)). If there exists V P(T) such that condition (i) of Lemma D.3 is satisfied with 9 = Ky and one of the following conditions is satisfied:
(a) Hy is local source of Hy;
(b) Hy has causal effect on Ky;
(c) V.U.
then (Po(( (A) can be written in terms of P and Q
Proof. Condition (i) Lemma D.3 is satisfied by hypothesis. If one of (a), (b) or (c) is satisfied, then
trivially, condition (ii) of Lemma D.3 is also satisfied. The result now follows from Lemma D.3
The above reminiscent of "valid adjustments" in the context of structural causal models 46 p. 115, Proposition 6.41], and in fact contains the valid adjustments.
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E Counterfactuals
There are various notions of counterfactuals in the literature. The one considered in the SCM literature is the interventional counterfactual, which captures the notion of "what would have happened if we intervened on the space, given some observations (that are possibly contradictory to the intervention we imagine we would have done)". Recently, backtracking counterfactuals have also been integrated into the SCM framework [58]. This captures the notion of "what would have happened if background conditions of the world had been different, given that the causal laws of the system stay the same?" Finally, we note that in the potential outcomes framework, the random variables representing "potential outcomes" that form the primitives of the framework can be directly counterfactual.
Vanilla probability measures have just one argument, i.e. the event. Conditional measures and causal kernels (in the sense of our Definition 2.2) have two arguments, the first being the outcome which we either observe or force the occurrence of, and the second being the event in whose measure we interested. For both of the above concepts of counterfactuals, we need to go one step further and consider three arguments. The first is the outcome which we observe, just like in conditioning, and the last should be the event in whose measure we are interested. For interventional counterfactuals, the second argument should be an outcome which we imagine to have forced the occurrence of given that we observed the outcome of the first argument, and for backtracking counterfactuals, the second argument should be an outcome which we imagine to have observed instead of the outcome in the first argument which we actually observed.
From these principles, we tentatively propose to extend Definition 2.2 to account for interventional counterfactuals follows.
Definition E.1. A causal space is defined as the quadruple PP, K), where P2, = (XteT E,, &teT&,, IP) a probability space and = (K5 S E &PTT), ) J sub-o-algebra of HJ, called the causal mechanism, is a collection of functions Ks,5 : St x Ω x H + [0, 1], called the causal kernel on Hig after observing F. such that
(i) for each fixed E f2 and A E H, Ks n, A) is measurable with respect to Hy:
(ii) for each fixed w E Ω and E H,, (ws A) is measurable with respect to F;
(iii) for each fixed pair (w, E Q Q (s.s(W m, -) is a measure on HC:
(iv) for all A E K and w, 2
Ko.9(w, n1 A) = (T A);
(v) for all A E Hss all B E HC and all w, E Ω
Ks n, A B) = (s n, B); in particular, for A E Hs, (w, n, A) (w(w) (w, 17, 12) La(w;;
(vi) for all A E H. E l and sub-o-algebras F C 9 C H.
Eg Ks,s(w,, A)] K5 (w,,,))
Note that letting F = () trivially recovers the causal space defined in Definition 2.2 Moreover, letting S = 0, we recover the conditional distribution given F.
Recall that the one of the biggest philosophical differences between the SCM framework and our proposed causal spaces (Definition 2.2 was the fact that SCMs start with the variables, the structural equations and the noise distributions as the primitive objects, and the observational and interventional distributions over the endogenous variables are derived from these, whereas causal spaces take the observational and interventional distributions the primitive objects (the latter via causal kernels). Note that, in the above extended definition of causal spaces incorporating interventional counterfactuals (Definition E.I. we applied the same principles, in that we treated the observational distribution (P), interventional distributions (K5,(),13)) ) and the (interventional) counterfactual distributions (,))) as the primitive objects.
This differs significantly from the SCM framework, where again, the (interventional) counterfactual distributions are derived from the structural equations, by first conditioning on the observed values of
24

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the endogenous variables to get a modified (often Dirac) measure on the exogenous variables, then intervening on some of the endogenous variables, deriving the measure on the rest of the endogenous variables by propagating these through the same structural equations. We see the value in this approach in that the (interventional) counterfactual distributions can be neatly derived from the same primitive objects that are used to calculate the observational and interventional distribution. However, we argue that this cannot be an axiomatisation of (interventional) counterfactual distributions in the strictest sense, because it relies on assumptions. In particular, it strongly relies the assumption that the endogenous variables have no causal effect on the exogenous variables, and when this assumption is violated, i.e. when there is hidden mediator, calculation of (interventional) counterfactual distributions is not possible. In contrast, Definition E. treat the (interventional) counterfactual measures the primitive objects, and does not impose any a priori assumptions about the system.
As mentioned in Section of the main body of the paper, we leave further developments of this interventional counterfactual causal space, as well as the definition of backtracking counterfactual causal space, as essential future work.
Proofs
Theorem 2.6 From Definition 2.3
(do(U.0) is indeed measure (2,, and (do(U.C..) is indeed
valid causal mechanism on K, ())( i.e. they satisfy the axioms of Definition 2.2
Proof. That (do(U.Q) a measure on (H.H) follows immediately from the usual construction of measures from measures and transition probability kernels, see e.g. Çınlar [44 p.38, Theorem 6.3). It remains to check that (do(U) is valid causal mechanism in the sense of Definition 2.2
(i) For all A e H and w E S
k (w, A) = g
/ T Taplwo, dult) Ker (Wwo, w()); A)
(dd'') (ullll")lwww.. A)
ppd((U.0) (A),
where applied Axiom 2.2 Lp.
For all A Hys and E H. we have, by Axiom 2.2 using the fact that A Hy S Hssu,
K s
do(U..LL)
(w, AB
Lsnu(wsnu dwyy K (us((((ss((ss
S Linu (unuu (( (WWs),) w/)) B)
/ Lsnu (wsuu du(y)
1 (w / suu (wwwy why,, B).
where, in going from the third line to the fourth, we split the w/ in 1 ((wss w/)) into components (w'sntr w')ss) and notice that since A E Hig. la does not depend on the component winys- Here, the map w'snu + (a((Ss (((((ssssssss))) w'snu)) is meassuaas,,,,,, so can write it the limit of an increasing sequence of positive Hinu-simple functions (see Section A.I. say (fa)ne with fn = Enn b,, 1B., where B,, E Ksnu. Like- wise, the map wy + Ksuu ((ws),.w))) B) is He-measurable, so we can write it as the limit of an increasing sequence of positive Her-simple functions, say (gn)neN with
L E Cj2, Ics,, where Cj Hy. Hence
KS (O..O.L))) (w, AnB) = / Lsnu dw/)) ( lim fnr (w(wss))) ( [im In an(e)))
~0000
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Since, for each wy. both of the limits exist by construction, namely the original measurable functions, have that the product of the limits is the limit of the products:
(w, (n, = Isnu (wanu, duyy) lim
(9 n-000
Here, since fn and were individually sequences of increasing functions, the pointwise products fagn also form an increasing sequence of functions. Hence, we can apply the monotone convergence theorem to see that
(d((.. (w, An
K5
lim
Lsnu (wsnu duly I (W'snu (nn (w(())
n=+00
Mrn
/ LIsn duly ) Bi, (wsnu) (w/)
lim
ben Cin
in Jn
lim
bin Cjn Lisu (wsnu, Bi, n (j.))
ja ===1
lim
bin Sin 1 Bi, (wsn)) Lsnn Cz.)
Mm
lim
Din 1 (wsne)
Σ Jn=1
Cja Lsni (wsnu: Cja)
n +00
Min
lim
bb 1B,, (wsne)
lim
Snn Lsnu (wsnu Cja)
lim fo (owss))) n-
/ Lsnoo dwly) n
lim
Σ
cj (wb))
Jn=1
((Wss, Lsnu (wsnu, dw'y)) lim In (w/)) n-000
LA(ws)) / Lsnu (wsku dw'y)) Ksuu (Wws wb),
= (lAWw)) (( Lo) (ws, B)
where, from the fourth line to the fifth, we used Axiom 2.2 [i} from the sixth line the seventh. we used that limit of the products is the product of the limits again, noting that both of the limits exist by construction; from the eighth line to the ninth, we used monotone convergence theorem again. This is the required result.
Theorem Let (H, P, Et, &ter P, K) be causal space, and U E P(T) and Q probability measure on He). Then after a hard intervention on Hy via Q the intervention hand))) are given by
CR
causal kernels KS
hand) KS
(w, A) haad (ws. =
/ ((dwwnss)K Kou sou ((ws, ((ws, wbys),
Proof. We decompose Hyr as product o-algebra into Hsnu @ Hors Then events of the form BnC with B Hsnu and C' E Hoys generate Hy, so for fixed wsnt, the measure Lsnu (wsnu.)) is completely determined by Lsnu (Wsnv, B CCC) for all B E Hsnu, C' E Hyys.. But we have
(weun B n C) Swnne (B) (wsnu, C)
by Axiom 2.2
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= Swanc (B)Q(C),
since Lsnu is trivial and C' E Hyn.s. So the measure Lisu (wsnU, :) is product measure of ówene and Q. Hence, applying Fubini's theorem,
Kg K Cad (w, A)
/ Lssno (sn (usos (wsr,,))))))))))
/ /
wors), A)) (dw'sno ( (ddw))sss) / (s wby A) A);
/
required.
Theorem D.2 Suppose we have a causal space (22, (2,, = (xier 1ter P,,,K) and let U E ((T).
(i) For any measure Q on He and any causal mechanism L on (2, Q), the causal kernel
K (do(tiQ.L) Ky is version of (ll(U..)) which means that Hyy is global source o-algebra He of the intervened causal space (52, HC, pdo(U.0), (.
(ii) Suppose V E P(T) with V S U. Suppose that the measure ! on (N,HC) factorises over Hy and Hor,, i.c. for any E Hy and B Hey, Q(A n B) (())((B).
Then after hard intervention on Hy via Q, the causal kernel K is version of pp which means that Hy is a global source o-algebra of the intervened causal space (D,H,, (d)(U. kd)(U,)))..
Proof. Suppose that / = Ex b. is -imple function, i.e. with B, E Hy for i = 1, m. m.
Then for any B Hyy
J. (dw)paalt:)) (dws)
Σ 1 B, (ploo( (dw)
b, (do
(B (BnB.)
Σ b, / ((dw))o((,BBBB)
by the definition of (dd((U.Q)
m /
(((w)))nnB)W))
by Axiom
it
(.)
Q(w))
Now, for any He-measurable map g S2 + I. we can write it as a limit of an increasing sequence of positive Hi-simple functions f.. (see Section A.I. so for any E Hy.
/ ((w))rdo(U.0)) (dw) / lim fn (w)) (dw) n
lim / fn (alo)) (dw) n-
by the monotone convergence theorem
27

---

lim / (d(w) B
by above
lim fn fn(w)
by the monotone convergence theorem
no0
(dw))
We use this fact in the proof of both parts of this theorem.
ii First note that we indeed have Kü dd((U. = Kv, by Remark C. a map w + (w, A) is me-measurable, for any B E Hy.
For any A E H, the
J Ku A) (dw) 8
Ko A)Q((w)
by above fact
B
f (B(w) A)Q((ww)
= S Kulw, An Aolw,AADe
(n)wAAn))((da))
by Axiom
= pdo(U..) - (AnB)
= / (Lnn((w) (ans(w))oolC)))
/ (w) (dw)
((U)
So A)) (do(U 96
() is indeed version of the conditional probability (A), which means that Hy is a global source of H, pdo(U()) (.do(U,...))
Ke d.
(ii) For any A E H, the map + Kv Kuu(lu0)) (w, A) is Hy-mmesurable and hence Hyr measurable, so for any E Kv C C.V.
Kv doll (wv, (Ip(((.))(dwv)))
B
Kv KdblU..)
(wv, (dQ(wwv))
by above fact
f K (WV, An ()Q()
by Axiom 2.2
/ / (e (( B)Q(dwv)
I Kulwu, A (dw()
where, in going from the third line to the fourth, we used Theorem C.3 and to go from the fourth line to the fifth, we used the hypothesis that Q factorises over Hy and Huv,, meaning ((dworv)QQdaa
= Q(dwr). So K (du(U. (w,A) A) is indeed a version the conditional probability puu(U:))) (A), which means that Hly is a global source of (0, pdo(U1.0) (do(U.0)).
Lemma C.4 Let P.K. = (XIeT E,, P, PP K) be causal space, and U E P(T), Q probability measure on (9,Hg) and I. (Iv V E P(U)) causal mechanism on (2, Hy, Q). Suppose we intervene on Hy via LQ.
28

---

(i) For E Hy and V E P(T) with V nu = 0, HCy has no causal effect on A (c.f. Definition B. in the intervention causal space (12, JC, pdo(U. ( i.c. events in the o- algebra Hy on which intervention took place are not causally affected by o-algebras outside Hu.
iii Again, let V E P(T)) with VnU = 0. and also let A E H be any event. If, in the original causal space, Hyy had no causal effect on then the intervention causal space, Jy has no causal effect on A either.
(iii) Now let V E P(T), A E H any event and suppose that the intervention on Hy via Q is hard. Then if Ky had no causal effect A in the original causal space, then Hy has causal effect on A in the intervention causal space either.
Proof.
Take any S E P(T). See that
doo (w, A) = / (wnutwsnt (suussrs Kg
/ Lsnu (wsnu: d (A(w())
I Lsnu (wsnu, dwly) (S ((
I Lisivynu (4))) ( K(s (4s ( ssV (w, A)
where, in going from the first line to the second and from the second line to the third, we used the fact that A Hy, and in going from the third line to the fourth, we applied the fact that (S \ V) n U = S n U since V NN @. Since S E P(T) was arbitrary, Hyy has causal effect on A in the intervention causal space.
(ii) Take any P(()) See that
Kg do (w, A) = / Lsstrlwsnn (sur((((sss)
/ Lsnu (wsnu, ( ((ooa
= I
dodot (w,
where, in going from the first line to the second, we used the fact that Hy has no causal effect on in the original causal space, and in going from the second line to the third, we used U n V = 0, which gives us $ n = (S V) n U and (S U) V = (S V) U Since E PTT) was arbitrary, Hy has no causal effect on A in the intervention causal space.
(iii) Take any S E P(T). Apply Theorem C.3 to see that
K Kg homl)) (w, A) (ksuiins
Q(dwly ((sddlrs)))sss)))((((sss
Def. B. B.
/ Q(duly (
f Q(dwin s) K ((
Def. B.
f Reddiinssn
29

---

K do (w, A);
where, in going from the second line to the third, we used that (SUU)\V = ((S))) Since (PT)) was arbitrary, Hy has no causal effect on A in the intervention causal space.
Lemma C.5 Let (2, = = (XIeT SteT &t, IK) be causal space, and U E P(T). For an event A E Ht, if DCr has a dormant causal effect on A in the original causal space, then there exists hard intervention and subset V S U such that in the intervention causal space, Ny has an active causal effect on A.
Proof. That Hy has a dormant causal effect on A tells us that (o(w, AA = P(A) for all w E Ω. but there exists some S E PPTT) and some Wo E 52 such that Ks(wo, A) * Ksy (wo, A). We must have S U A 0, since otherwise S U S and cannot possibly have (s(wo, A) j Kg (wn, A). Then we hard-intervene on Hego with the Dirac measure on wo- Then apply Theorem C.3 to see that
K (.,,S hard) (ww))ns: = / wwo (dw's ) (s((o))rssss w snu
Ks(wo, A) * Ksy (w:: A)
Note that the intervention measure A is equal to Ksyo (wo, A):
pdo(S) (A) = f
A) = Ksyo (wo, A).
Putting these together, we have
K K (oS hard) (wb, A) + pdo(s)Uibno)(A) (A), (o hurd)), Henu has an active
L.c. in the intervention causal space (9, H. (UU K snu causal effect on A.
Lemma C.6. Let (K, = (x SIET P. IK) be causal space, and U, V E P(T). For an event A E K. suppose that Hy has no causal effect on given Hyy (see Definition B.4). Then after an intervention on Hy via any (Q, L), Hyy has causal effect on A.
Proof. Take any probability measure Q on (9,, Hly) and any causal mechanism L on KCy Q). Then see that, for any S E P(T) and all w E Ω
doV.. (w, A) K
Lsnv ( Ksv (
Lisv
=
doV..2 sx(UVV A),
where, in going from the first line to the second, we used the fact that Hy has no causal effect on A given Hy, and in going from the second line to the third, we used identities Snv = V))V
and (S LUV) (U \ V/) = (S), (U \ V)) U.. Since S E P(T) was arbitrary, we have that Hoy has no causal effect on A in the intervention causal space.
0
Theorem C.7
(Xter & P., TK) be a causal space, where the index set T can be written as = W with W representing time and K respecting time, Take any E P(T) and any probability measure Q on Hy. Then the intervention causal mechanism (dd(U.QQhaad) also respects time.
Let (2, H. P. K)
30

---

Proof. Take any W1, wh E W with Wi < w2. Since K. respects time, we have that H x has no causal effect on D[ wy xf in the original causal space. To show that H w2x has no causal effect on H w1 x after hard intervention on Hy via Q, take any S E P(2)) and any event A E H w1 xf" Then using Theorem
d (w, A)
= Q(dw') (WWs, A)
/ Q(dw') ((dw") (SLUU) w2xt (Wws) csuor xx)))
(Wws) x = / (((sd))(((SLt)))))))) (1dd
( (x0) A)
/ Q(dw') (SV x Jue (WSr ( x
K (5(00wgx3
RdolU, Q.ard)) (w, A)
where, from the second line to the third, we used the fact that H w2xf has no causal effect on A, from the third line the fourth we used the fact that n K wy xf has no causal effect on
A (by Remark B.2e) and Remark B.2 and from the fourth line to the fifth, we used that
(ISUU){3 HC
WP x (U nH. w)x
xT) and (U (SUH, wa UUU wy wT))
(S H oit 1.
Since S P(T) was arbitrary, we have that H my x has no causal effect on A
(Definition
Since H ur x was arbitrary, H u2 x has no causal effect on H
10 xf and so
[gdoUU.Q.hnn respects time.
31


Reviews:
[{"review_id": "NhBv6qIvVb", "reviewer": "Reviewer_TfjY", "summary": "(Preamble warning: given the amount and time I had to review Neurips papers, and given other duties, I focused solely on the main content of the paper, not consulting the appendices that were actually longer than the paper.) \n\nThe paper proposes and discusses a new framework, based on measure-theoretic probabilities, to encode causal systems, both on the observational and interventional level. \n\nThis is done through the use of probability kernels and probability measures defined over Cartesian products of spaces. The framework generalizes a number previously proposed frameworks,", "questions": "Questions:\n\n*  P6, 240-243: what is meant by unique existence? Is it meant that a given observational/interventional system can only be represented by a unique set of measures and kernels? The exact meaning of uniqueness is unclear to me here.\n\n* In example 4.2., I agree that one variable can influence the other (in a causal way), but I would argue that we are also faced here with a dynamical process (rice production can only be impacted by price for the next period of rice harvesting, and vice-versa). Would it be possible to provide an example of cyclicity where we are not faced with time-evolving processes?\n\n* P8, L318: could you give some references to identify \"by some authors\"?\n\n* Comutability: one question that arises from the presented framework (but is probably to be adressed in future work) is about the computability of the presented framework? Maybe it would be good to mention a few cases where this is achievable, if authors have already identified such cases. \n\n* P9, L359-360: it is not entirely clear to me how the proposed axiomatization and the measure-theoretic view of probabilities are linked in the proposed framework. To be more precise, my feeling is that the two proposed axioms of no action-no causation and of agreement witht he causal mechanisms are not especially linked to a measure-theoretic view, which offers the technical tools to derive it. For example: to which extent do we really need $\\sigma$-additive measures rather than finitely additive ones to express these two axioms? Of course the first option is technically/pragmatically more simple to deal with, but I am not completely convinced that this is at the heart of the expressed axioms.  \n\nSuggestions:\n\n* Figure 1: suggesting that probability theory is confined to modelling data-generating process is a bit misleading, as a Bayesian setting allows one to put probabilities on any quantity of interested (including those not generated by data). \n\n* Given this absence of discussions about the philosohical concepts of causality and how these axioms encode them, speaking of \"Axiomatization of causality\" seems a bit strong to me. Why not referring in the title to \"Generalised causal model through measure-theory\"? \n\n* P3, L108-113: the vocabulary used here as well as the various differentiation made is a bit ambiguous: why speaking of \"chains of trials\" that make one think of repeated experiments when in all further examples all measurable spaces belong to distinct variables? \n\n* P5, L195-196: an underlying assumption here is that the causal kernels are 1. perfectly specified and 2. encode causal relationship. In practice, nothing prevents those two points to be false, and I think it should be mentioned somewhere that those assumptions are made. \n\n", "limitations": "Limitations were adequately adressed.", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "* A unifying framework relying on measure-theoretic probabilities that is well-justified from a mathematical point of view, and can be summed up by two technical constraints on the system. \n\n* A quite well-written paper, with illustrative examples showing how the framework can be applied. ", "weaknesses": "* The authors make bold claims on probability theory: while it is quite okay to rely on measure-theoretic probabilities to derive a framework, I would argue that claiming that this view of probability is \"near-undisputed and universally accepted\" (P1, L24-25) and that \"In ... probability theory, one starts by assuming the existence of a probability space\" (P3, L101-102) is not true in the modern world. For instance and only to mention a few works, De Finetti betting accounts of probabilities allow for finitely additive probabilities, and does start by defining a probability space on a $\\sigma$-algebra. Similarly, Shafer/Vovk account of probabilities from a game-theoretic point of view does not start from a measure-theoretic view point. Finally, one could also argue that causality could be seen through the lense of other uncertainty theories, and that the unversality of probability theory could be challenged itself (see works of Peter Walley on imprecise probabilities extending De Finetti's betting argument, and works of followers). So even if I think that the adopted viewpoint here is reasonable, I would avoid suggesting that it is the only one that can be considered, or that it can hardly be questioned. \n\n* Sufficiency of the proposed axioms: the axioms are given as mathematical constraints over the system, and makes sense in a fully probabilistic setting (it is less clear if ones consider more expressive languages than probabilities). However, such kinds of axioms (unchange with respect to no new information + conformity with imposed constraints on the system) can also be found in other settings such as, e.g., Jeffrey's updating rule in the presence of uncertain information. I somehow miss a discussion of what makes those axioms peculiar to the situation of causalilty? Or does it follows from the assumptions that the probabilistic kernels DO model an exisiting cause rather than something else? And in this later case, one could imagine a situation where specified kernels do not encode actual causality relationships, while still following the proposed axioms. My feeling is that the axioms concern more mathematical properties ensuring that the system will behave well, rather than axioms ensuring that the encoded concepts and systems will indeed be causal in nature.\n\n* Not clear whether more generality gives a better means to identify causation: In Example 4.1. I appreciate the fact that the proposed model is much more flexible than SCM, however this example raises the question of knowing whether the system is not to general to be able to identify causalities? More precisely, assume we observe the S-I correlation, as well as measure the temperature. If we focus on the SI, we then have two models that perfectly explains the data: one with no causation and observed correlation, and one with causation (where one would have to specify the causation mechanisms). In contrast, SCM would not be able to account for S-I correlation without assuming causaition, and would incitate the modeller to add potential explaining causations. So it is unclear whether the provided generality comes with better capacities of identifying potential causations in practical modelling situations? \n ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for your time and effort in reviewing our submission; we are very grateful for your overall positive evaluation of the paper and your valuable suggestions. We will do our best below to address your concerns and queries. \n\n**Weaknesses**\n\nThank you for this point. We are aware that other theories of probability exist other than Kolmogorov's. Of those you mentioned, we are aware of de Finetti's approach, and are vaguely aware of Shafer/Vovk's game theoretic framework. We were not aware of Walley's work - thank you for pointing it out. There is also an approach more amenable to Bayesian probability ([Jaynes and Bretthorst, 2003], or https://en.wikipedia.org/wiki/Cox%27s_theorem). We acknowledge that these works are valuable, and we propose to make more of an effort to acknowledge them. We also propose to water down our language from \"near-undisputed and universally accepted\". However, we believe it is equally unreasonable to treat Kolmogorov's framework as simply one out of many on the same level of significance, given that the vast majority of research in probability theory, as well as the vast majority of textbooks and lecture courses on probability theory, is built on it. \n\nWe agree that the two axioms we provide are a priori mathematical properties that are not in themselves causal in nature. However, this is also true for probability spaces - they are simply measure spaces where the whole set happens to have measure 1. Same for densities - they are simply functions that happen to integrate to 1. Yet, we give these objects a probabilistic interpretation. Our axioms of causal kernels should be viewed in the same light.\n\nThank you for raising this point. We agree that in general identification is difficult, if not impossible, with such generality. However, we would like to point out that if measuring temperature is possible, and if it is in our interest to do so (for identification or any other reason), then our causal space framework accommodates that, so in this sense, nothing is \"lost\" compared to the SCM framework. We maintain that this is advantageous in the modelling perspective over SCMs, where the researcher is \"forced\" to include other variables that explain the correlation. For identification from data in our framework, Section D in the Appendix is dedicated to it, via a concept which we call \"sources\".\n\n**Questions**\n\nP6, 240-243: Uniqueness here is meant simply to contrast with SCMs, where, without the acyclicity assumption, there may be many distributions over the endogeneous variables that satisfy the structural equations and the noise variables (see, for example, [Bongers et al., 2021, Foundations of Structural Causal Models with Cycles and Latent Variables]). In causal spaces, the observational and interventional distributions are always uniquely defined.\n\nExample 4.2: Thank you for raising this point. Please see author rebuttal to all authors. \n\nP8, 318: For example, [Peters et al., 2017, Elements of Causal Inference, Remark 6.5]. We agree that we should give some citation here, and we propose to do so in future versions of this work. \n\nComputability: This is not a question that we had thought to address in this submission at all, but we agree that this is a crucial and interesting research direction. We are grateful to you for pointing it out.\n\nP9, 359-360: We agree that the ideas behind the two axioms we give are not inherently measure-theoretic, and that it would probably be possible to endow other frameworks (e.g. graphical models) with these concepts. However, as you point out, the way these axioms represent these concepts are measure-theoretic, since the transition kernels are very much measure-theoretic tools. \n\n**Suggestions**\n\nFigure 1: In terms of what is included in the outcome space, the parameters in Bayesian setting could also be viewed as \"data\". But we see the point you are trying to raise. What would be your suggestion for the left-hand box?\n\nThe focus of this work was not the philosophical discussions around causality on the level of a philosophy paper, which we hope you understand. However, we maintain that calling our proposal an \"axiomatisation\" is justified in the mathematical sense, in that we lay out what we assume to be true about causal spaces, and we build subsequent definitions and results on these axioms. This is precisely in the form of other \"axiomatic frameworks\" in mathematics, not least the probability axioms of Kolmogorov. Moreover, throughout the text we clearly state what philosophical approach behind causality we take, namely that we view it as a study of what happens when we intervene, and in Remark 2.4, we discuss precisely what our axioms encode, so we do not fully agree with your judgment that there is an \"absence of discussions about the philosohical concepts of causality and how these axioms encode them\". \n\nP3, L108-113: Thank you for this suggestion, yes, we agree that \"chain of trials\" is somewhat misleading. The terminology is taken from [Cinlar, 2011, Probability and Stochastics, page 161]. Of course, the mathematics $(E_t,\\mathcal{E}_t)$ allows for distinct variables even here. We propose to change it to \"set of trials\", would you agree that this is better?\n\nP5, L195-196: We would again like to draw parallels with probability spaces, where, of course, it is equally impossible to perfectly specify the probability measure in practice for any given problem, but we still do so in the modelling process. In the same vein, we feel it is justified to assume that for the causal kernels. We will make this point more explicit.\n\nWe thank you again for your positive review, and raising many interesting points. If accepted, your suggestions are sure to improve our paper. If any of our answers were unclear or we misunderstood your point, please let us know. We would be happy to engage in further discussions. "}, {"review_id": "HAsnkVmkZu", "reviewer": "Reviewer_xRcV", "summary": "This paper proposes a measure-theoretical axiomatization of causalality with the notion of causal space and with a collection of causal kernels encoding the causal information. ", "questions": "(1) I am confused by Definition 2.2 (Causal space).    For a given S, the causal mechanism seems undecided: there are many causal mechanisms satisfying the two axioms.  For two subsets S1 and S2 of the T, What is the relationship among K_{S1}, K_{S2} and K_{S1\\cup S2}?\n(2) It seems that all varaibles are independent in Definition 2.2.   How can one formulate the relationship that variable X causes another variable Y in the causal spapce?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "The paper offers an axiomatization of causality which is based on the measure-theoretical foundation of probabaility theory. ", "weaknesses": "(1) It seems that Definiiton 2.2 is not well-defined because, given a subset S of T, we cannot decide the causal mechanism K_S. Also see my questions below. \n(2)  I really could not understand the claim (Line 147) that \"intervention is the process of placing any desired measure,... along with an internal causal mechanism ...\".   Then it seems that intervention can also be treated as conditioning. ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for reviewing our submission. We regret that you have a rather negative view of the paper, and we hope that our clarifications below, as well as the other reviews, can give you a more positive view. \n\n**Definition 2.2**\n\nThis is the main definition of the paper, containing the two axioms of causal kernels. We regret that you were not convinced by these axioms of causal spaces. We'll do our best to address your concerns. \n\nFor a given $S$, of course there are many possible causal kernels $K_S$ that satisfy the two axioms, but in the end the causal space will have one $K_S$ specified. This is the same in SCMs, where, given a random variable (or a node), there are many possible functions that can act as the structural equation for this variable, but in the end the SCM will specify one function. Or even just in probability spaces, where of course there are many possible probability measures that satisfy the axioms of a probability measure, but a given probability space will have one measure specified. \n\nFor two subsets $S_1,S_2$ of $T$, there is a priori no relationship between $K_{S_1}$, $K_{S_2}$ and $K_{S_1\\cup S_2}$, and for full generality, this is desirable, since, even in the SCM framework, we can easily construct examples where specifying $K_{S_1}$ and $K_{S_2}$ does not fully determine $K_{S_1\\cup S_2}$. \n\nWe are not sure what you mean by the variables being independent in causal spaces, but it is easy to formulate a two-variable situation in the causal space framework with $X$ causing $Y$. This is precisely what was illustrated in Example 2.5. \n\n**Line 147**\n\nThe fact that we take this view on intervention is not a \"claim\", but the philosophy behind our definition of intervention, and it agrees with most other definitions of intervention in the literature, including the SCM case. We pick a subset of the variables on which we would like to intervene (most often a single variable), and we place a desired measure on those (that) variable(s) (for hard interventions, a Dirac measure). The causal components in the frameworks (the structural equations in SCMs, the causal kernels in our causal spaces) then determine what happens to what we did not intervene on. \n\nIn our causal space framework (and in any other framework), intervention is categorically not the same as conditioning - in fact this is the whole point of having any theory of causality. In examples throughout our submission, there are many instances in which intervention is not the same as conditioning, showing that our causal space framework accommodates this. For example, in Example 2.5, conditioning on temperature is not the same as intervening on temperature, and in Example 4.4 and Figure 5, conditioning on the Brownian motion to have a particular value at a particular time point is not the same as intervening on the Brownian motion to have a particular value at a particular time point. \n\nWe thank you again for your review, and we hope that our rebuttal, along with the opinions of the other reviewers, are sufficient to convince you at least of the validity of causal spaces, and if possible, their merits. Please let us know if you have further questions, or if any of our clarifications were unclear. "}, {"review_id": "7v2pv2WZDg", "reviewer": "Reviewer_aMnb", "summary": "The paper develops new foundations for causality based on the measure-theoretic foundations of probability pioneered by Kolmogorov. Interventions are formalized as transition kernels and it's shown that the new approach is more general than existing ones in a number of respects.", "questions": "- Line 49: Why is there no citation to Ibeling & Icard's 2020 \"Probabilistic Reasoning across the Causal Hierarchy\"? This paper was (one of the, if not the) first to axiomatize SCMs in a truly *probabilistic* setting. The references [18, 20] there are still only quasi-probabilistic. (Admittedly, the paper I suggest does enforce an acyclicity constraint, but it is no better or worse than the other two references in this regard.)\n- Line 296: Citation 20 probably should also be thrown in here.\n- Can the rice example be modeled within the space of SCMs that always have unique solutions? I can't see why not. In this particular case, although the framework here may be more natural, it may not add much that is substantive.\n- Example 4.2: notations K_1(3, x) and K_2(6, x) seem inconsistent with each other since the interventions are on different variables.\n- Can the authors comment on whether the framework given invalidates quintessential principles of SCMs, e.g., effectiveness, composition, and reversibility? I am guessing the answers are no (by interventional determinism, line 158), yes, yes. The broader point is that not only do SCMs require additional assumptions to be well-defined, but that these assumptions may also reverberate in causal reasoning.\n- Line 328: Z = {..., 2, 1, 0, 1, 2, ...} should read Z = {..., -2, -1, 0, 1, 2, ...}", "limitations": "Yes, they have.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "The paper is clearly laid out, sound, and represents a noteworthy (and as far as I know, novel) attempt to do causality in terms of mathematical probability theory. I believe the most promising direction in terms of utility is in being able to handle continuous-time causal stochastic processes, which have so far resisted formalization; the approach here does not even rely on dynamical systems.", "weaknesses": "I raise some edits and additional citations below.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the time and effort spent in reviewing our submission, and for the overall positive evaluation. We also thank you for raising interesting points and suggestions for improvements, which we will take into account in future versions of this work. Below, we will try our best to address your concerns and queries. \n\n**Line 49**\n\nThis is a very reasonable citation recommendation. We had of course read the paper during the project, but inexplicably left out its citation in our submission. We will make sure to include it in future versions of this work. Thank you very much for pointing it out. \n\n**Line 296**\n\nAgreed - thank you very much! \n\n**Rice Example**\n\nThank you for making this point - please see author rebuttal to all reviewers. \n\n**Example 4.2**\n\nThank you for pointing this out. We will change one of the $x$'s into a $y$ for better notation. \n\n**Principles of SCMs**\n\nBy effectiveness, composition and reversibility, we assume that the reviewer is referring to the notions introduced under these names in [Galles and Pearl, 1998, An Axiomatic Characterisation of Causal Counterfactuals, Section 3]. This is a very interesting point to consider, and we are very grateful to you for raising it. \n\nFirstly, even though the concepts of effectiveness, composition and reversibility can be carried over to causal spaces, the mathematics through which they are represented needs to be adapted, since the tools that are used in causal spaces are different from those used in causal models of [Galles and Pearl, 1998]. In particular, we work directly with *measures* as the primitive objects, whereas [Galles and Pearl, 1998] use the structural equations as the primitive objects, and the probabilities only enter through a measure on the exogenous variables. Thus, the three properties can be phrased in the causal space language as follows:\n\nEffectiveness: For $S\\subseteq R\\subseteq T$, if we intervene on $\\mathscr{H}_R$ via a measure $\\mathbb{Q}$, then $\\mathscr{H}_S$ has measure $\\mathbb{Q}$ restricted to $\\mathscr{H}_S$. This is indeed guaranteed by interventional determinism (Definition 2.2(ii)), as you said. \n\nComposition: For $S,R\\subseteq T$, denote by $\\mathbb{Q}'$ the measure on $\\mathscr{H}_{S\\cup R}$ obtained by restricting $\\mathbb{P}^{\\text{do}(S,\\mathbb{Q})}$. Then $\\mathbb{P}^{\\text{do}(S,\\mathbb{Q})}=\\mathbb{P}^{\\text{do}(S\\cup R,\\mathbb{Q}')}$. In words, intervening on $\\mathscr{H}_S$ via the measure $\\mathbb{Q}$ is the same as intervening on $S\\cup R$ via the measure that it would have if we intervened on $\\mathscr{H}_S$ via $\\mathbb{Q}$. This is not in general true, as you said. A counterexample can be demonstrated with a simple SCM, where $X_1$, $X_2$ and $X_3$ causally affect $Y$, in a way that depends not only on the marginal distributions of $X_1$, $X_2$ and $X_3$ but their joint distribution, and $X_1$, $X_2$ and $X_3$ have no causal relationships among them. Then intervening on $X_1$ with some measure $\\mathbb{Q}$ cannot be the same as intervening on $X_1$ and $X_2$ with $\\mathbb{Q}\\otimes\\mathbb{P}$, since such an intervention would change the joint distribution of $X_1$, $X_2$ and $X_3$ even if we give them the same marginal distributions. \n\nReversibility: For $S,R,U\\subseteq T$, let $\\mathbb{Q}$ be some measure on $\\mathscr{H}_S$, and $\\mathbb{Q}_1$ and $\\mathbb{Q}_2$ be measures on $S\\cup R$ and $S\\cup U$ respectively such that they coincide with $\\mathbb{Q}$ when restricted to $\\mathscr{H}_S$. Then if $\\mathbb{P}^{\\text{do}(S\\cup R,\\mathbb{Q}_1)}(B)=\\mathbb{Q}_2(B)$ for all $B\\in\\mathscr{H}_U$ and if $\\mathbb{P}^{\\text{do}(S\\cup U,\\mathbb{Q}_2)}(C)=\\mathbb{Q}_1(C)$ for all $C\\in\\mathscr{H}_R$, then $\\mathbb{P}^{\\text{do}(S,\\mathbb{Q})}(A)=\\mathbb{Q}_1(A)$ for all $A\\in\\mathscr{H}_R$. We again agree with you that this does not hold in general in causal spaces. In fact, Example 4.2 in our paper is a counterexample of this, with $S=\\emptyset$. \n\nSo all in all, we agree with your answers of no, yes, yes, as to whether the causal space framework challenges these fundamental properties of SCMs. Composition is an interesting one, because even in the SCM setting, it is very clear that it does not hold if we are interested in the *measure* of the target variable, not just the pointwise evaluation of the structural equations. This is a very interesting discussion, and we propose to include it in future versions of this work. Thank you very much again for raising this point! \n\n**Line 328**\n\nThis is a terrible typo. Thank you very much for pointing it out. \n\nWe thank you again for the review, positive evaluation and valuable suggestions for improvements. We hope to have clarified your questions, and should you have any further questions, or if we misunderstood any of your points, we would be happy to engage in the author-reviewer discussion. "}, {"review_id": "ICF4ScB8ap", "reviewer": "Reviewer_pcib", "summary": "This paper presents a completely novel mathematical description of causal systems, which generalizes the most well-known causal formalisms such as SCMs and potential outcomes. It does so by taking the probability distributions as the foundation, and use this to construct a measure-theoretic approach to causal systems. The result is a very expressive framework, that definitely has its place within the landscape of causal formalisms. \n", "questions": "I do have some conceptual comments about the motivation and supposed superiority of the framework. \n\n1: In your comparison with SCMs, you only consider hard interventions. What about soft interventions? \n\n2: Remark 3.2: \nI find this quite strange. Usually one thinks of an intervention not as an intervention on the system that affects a subsystem, but simply as an intervention on the subsystem itself, and this of course affects the system, just as changes to a part are also changes to a whole. I’m not saying the idea of a “holistic” intervention on an entire system is uninteresting, but merely that this is not usually how causal interventions are understood. In fact, the independent mechanism assumption made by Woodward and taken over by others captures a fundamental modularity that is assumed to hold for causal relations. \n\nSimilarly, the idea of coupling the observational and the interventional distributions is a feature, not a bug: it encodes an assumption about how we take the world to work. \n\nIn sum, although I agree that it is interesting to decouple these distributions and to view interventions as “system-wide”, as your formalism does, I am not sure when and why we would ever want to do so: at heart, the motivation for your framework seems to come with a philosophical theory that is lurking in the background which appears in conflict with the standard interventionist theory of Woodward that causal modellers usually align with. I realize that it would be too much for one paper to introduce both such a rich framework and dive into a philosophical analysis, but my suggestion would then be to here remain more neutral regarding these commitments. \n\n3: footnote 1: do you think your framework is strictly more expressive than GSEMS? They don’t use probabilities, but perhaps you could reconstruct GSEMS by giving positive support to all values, and reasoning about P > 0?\n\n4: Confounders. \nI’m not sure if the authors are aware, but the philosopher Nancy Cartwright has famously argued against the Causal Markov Condition by using a very similar example, the so-called chemical factory. In her case though, she goes even further, stipulating that there is no common cause, not even an unobserved one. If one accepts her stipulation, then CBNs seem indeed unable to express such situations, whereas your framework would have no trouble with it at all. So it might be useful to refer to this discussion. For what it’s worth, the usual reply is similar to the move that you here discuss: simply add a “fictional” exogenous variable that functions as a latent common cause. \n\nThis brings me to footnote 2: I do not agree that such a variable is “completely meaningless”. It simply captures our ignorance. As long as one is committed to the Causal Markov Condition, one is committed to the belief that such a latent variable must exist. Whether we call it one variable or several variables collapsed into one seems merely a matter of terminology.\n\n5: “Of course, cycling relationships abound in the real world.” \nReally? That is quite a controversial claim: do the authors then subscribe to backwards causation? I’d say that cyclic models are useful abstractions of what in the end are non-cyclic systems. Your example is a case in point: it’s not the current price which causes the current amount of rice, and vice versa, it’s the price right now that determines the future amount. \n\nMoreover, although I’m sure the authors are right that SCMs have trouble in representing certain cyclic cases, is this really one of them? It doesn’t seem so hard to model this with an SCM. \n", "limitations": "yes", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "The paper is clearly written, but some background in probability theory is required to understand the technical definitions. There is substantial discussion of related work and open questions, illustrations of the framework, and comparisons to SCMs. Overall this is a great paper. \n", "weaknesses": "See questions.\n\nTypos:\n\n58: “we have designated” -> “we have a designated”\n\n81: “implications concepts” -> “implications for concepts”\n\n107: “the measurable space are” -> “the measurable space is”\n  ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer very much for the time and effort spent in reviewing our submission, and for the positive evaluation. Thank you also for suggestions for improvements; we will reflect them in future versions of this work. We will do our best below to address your concerns in the Questions section. \n\n1. If we understand correctly, you are referring to Section 3.1, lines 214-215. This is not referring exclusively to hard interventions, it is simply constructing the causal kernels. Once these causal kernels are constructed, by placing a non-Dirac measure on the sub-$\\sigma$-algebra on which we want to intervene, and also by utilising the internal causal mechanisms $L$, we can then formulate soft interventions. For future versions of this work, we propose to spell this out more explicitly. \n\n2. Actually the point we were trying to make was precisely in agreement with your view - that we are most often interested in intervening on subsystems, rather than the whole system. Our causal kernels capture precisely this information, i.e. the effect of intervening on subsystems on the whole system. The point we were trying to make was that in the SCM formulation, the mathematics seems to go in the reverse direction to this philosophy, because each structural equation $f_j$ encodes information about the effect of intervening on the whole system on the subsystem $X_j$. Our causal space formalism does *not* view interventions system-wide - each causal kernel encodes information about intervening on the corresponding subsystem (sub-$\\sigma$-algebra). In this sense, we are in complete agreement in terms of the philosophy (with Woodward, and that behind e.g. the SCM framework), we just differ in how to encode this mathematically. \n\nAs for the decoupling of observational and interventional measures, we maintain that it is an advantage to be able to do this for the sake of generality. In order to be able to couple them, as it is done in SCMs, we have to impose the assumption that all of the variables have been included in the model that are required to do this (see Example 4.1). We do treat the case in which observational measure and the interventional measure (at least partially) match as an important special case - see Appendix D on the concept of sources that we introduce there. \n\n3. Some philosophy behind causal spaces is shared with GSEMs in the sense that GSEMs also encode information about what happens to the whole system when intervening on a subsystem. However, there are some technical hurdles to be overcome if one wants to actually reconstruct a GSEM from a causal space. First of all, they consider power sets of the range rather than $\\sigma$-algebras, which, if the variables are allowed to take uncountably many values, causes measure-theoretic problems. Moreover, as you rightly point out, for a given probability measure on the exogenous variables and an intervention, there seems to be no way of propagating this measure on the exogenous variables to sets of values of the endogenous variables. \n\nIn the sense that GSEMs cannot do probabilities and causal spaces are designed explicitly with probabilities in mind, we believe that causal spaces strictly generalise GSEMs. We also do not believe that they worked with power sets rather than $\\sigma$-algebras for any deep reason, and if we wrote out the GSEM formulation with $\\sigma$-algebras, then indeed, as you suggest, we believe it would be possible to reconstruct GSEMs from causal spaces by letting the function $\\mathbf{F}$ map to the support of each intervention. However, for reasons laid out above, there would be no way of uniquely reconstructing causal spaces from GSEMs. \n\n4. This would be very valuable addition to our work, and we are very grateful to you for this comment. We absolutely propose to add a discussion on this example in future versions of this work. \n\nAs for footnote 2, we agree with you that the variable itself is not meaningless. The variable of course can have meaning as a way of capturing our ignorance, precisely as you said. Our point was that such a variable cannot have numerical values and distributions that represent something meaningful. What does it mean for a variable that we name \"everything in the world that we do not know about\" to have value 1? What does it mean for it to be normally distributed? Properly defined random variables should be more concrete, e.g. height of a student in cm, temperature in Celsius, crop yield in tonnes, etc. \n\n5. Thank you for raising this point. Please see author rebuttal to all reviewers. \n\nWe thank you again for your positive review and suggestions, and should you have any more questions or if we misunderstood any of your points, we would be delighted to engage in the author-reviewer discussions. "}, {"review_id": "rbsM2yNzlU", "reviewer": "Reviewer_kUcy", "summary": "This paper builds a rigorous foundation for causal reasoning. It does so by defining causal space, an extension of the concept of probability space by adding stochastic kernels. This richer structure allows the authors to define notions like interventions and causal effects, and generalize the standard frameworks like structural causal models.", "questions": "While understanding the definition of causal space, a natural question of what are some sufficient conditions for its existence arises (similar in spirit to the sufficient conditions for the existence of regular conditional distributions). It would be nice to explore this question.", "limitations": "Not relevant.", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "It is easy to take for granted the advantages provided by the measure-theoretic foundations of probability theory, but it is undeniable that Kolmogorov's insight to do so put probability theory in a respectable position and more importantly allowed for rapid progress on otherwise impossibly difficult problems related to stochastic process by Doob and others. No doubt this axiomatization did not happen without resistance, and many early probabilists considered it an offense to probabilistic intuition.\n\nThis paper is a valiant attempt at axiomatization of causal reasoning, and in my opinion it has a done a fairly good job. The idea of using stochastic kernels to enrich the underlying probabilistic structure for the much more difficult (than plain statistics) problem of causal reasoning is very natural while at the same time difficult to effectuate. The authors do a great job defining all the relevant terms and being rigorous in their treatment.", "weaknesses": "The authors do a lackluster job in providing enough motivation and examples, and discussing potential alternative axiomatizations. A first read of the paper gives an impression of pulling the definitions \"out of the hat\". This is unfortunate since it is important to encourage as many readers as possible to think in this direction and incorporate it in their research.\n\nOn lines 137-138, it is stated that \"$K_S(\\omega, A) = K_S((\\omega_S, \\omega_{T \\setminus S}), A)$ for any $A \\in \\mathcal{H}$ only depends on the first $\\omega_S$ component of $\\omega$\". I don't think it is true (if it is indeed true, please provide a proof). For example, let $T = \\\\{1,2\\\\}, S = \\\\{1\\\\}, E_1 = E_2 = \\\\{0,1\\\\}$ and $\\mathcal{E}_1 = \\mathcal{E}_2 = \\mathcal{P}(\\\\{0,1\\\\})$. Then $\\Omega = E_1 \\times E_2$ and $\\mathcal{H} = \\mathcal{E}_1 \\otimes \\mathcal{E}_2 = \\mathcal{P}(\\Omega)$. Now we could define $K_S$ in such a manner that it depends on the second component of $\\omega \\in \\Omega$. Say:\n$$\nK(\\omega, A) = \\begin{cases}\n0 &\\text{if } \\omega = (0,0), A = \\\\{(0,0)\\\\}\\\\\\\\\n1 &\\text{if } \\omega = (0,1), A = \\\\{(0,0)\\\\}\\\\\\\\\n\\vdots\n\\end{cases}\n$$\n\nA few minor nitpickings:\n1. In section 2, where $\\mathcal{H}_S$ is defined it will be clearer to remind what measurable rectangles are, since it is implicitly being used that for measurable rectangles $A_t$ differs from $\\mathcal{E}_t$ for only finite $t$'s.\n2. In definition 2.2, it is misleading to say \"(product) probability space\". The word \"product\" when talking about measurable spaces (as opposed to _measure_ spaces) has a connotation of product measures, which is not the case here.\n3. I find the notation in equation (2) confusing. It might be helpful to also say that the kernel $K_S^{\\text{do}(U, \\mathbb Q, \\mathbb L)}$ is simply the product $L_{S \\cap U} K_{S \\cup U}$ of stochastic kernels (as defined by Çinlar on page 39).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer very much for the time and effort spent in reviewing our submission, and for the positive evaluation. Your comments make it clear that you understood and agreed with our motivation behind this project, for which we are humbled and extremely grateful. We are also very grateful for the suggestions for improvements, which we will take into account for future versions of this paper. We will do our best to address your concerns below. \n\n**Motivation**\n\nWe fully agree that, as currently written, the axioms somewhat seem to be pulled out of the hat. In fact, these axioms were a result of over a year of contemplating different ways of axiomatising causality. Given more space, you are absolutely right that more discussion of the motivation and potential alternative axiomatisations would certainly benefit the paper, and we propose to do so in the future versions of this work. \n\n**Lines 137-138**\n\nThe causal kernel $K_S$ is a transition probability kernel from $(\\Omega,\\mathscr{H}_S)$ into $(\\Omega,\\mathscr{H})$, so for a given $A\\in\\mathscr{H}$, the map $\\omega\\mapsto K_S(\\omega,A)$ has to be measurable with respect to $\\mathscr{H}_S$ (see definition of transition kernels in the appendix, lines 568-575). If $K_S$ depended on the $T\\setminus S$ components, it would violate this measurability condition. We hope we understood your point correctly, and that this addresses your concern. \n\n**Minor comments**\n\n1. We give the definition of measurable rectangles in the appendix on line 523. In future versions of this paper, we propose to move this definition to the main body in the camera-ready version. \n\n2. You are right. We propose to change this, so that we make it clear that only the $\\sigma$-algebra is in the form of a product, not the measure. Thank you very much for pointing this out - this is a great spot. \n\n3. Actually (2) is not a product of kernels, because $K_{S\\cup U}$ has the $\\omega_{S\\setminus U}$ component that does not get integrated with respect to the measure $L_{S\\cap U}(\\omega_{S\\cap U},\\cdot)$. In the appendix, Remark C.1(b) treats the special case in which the interventional kernel $K^{\\text{do}(U,\\mathbb{Q},\\mathbb{L})}_S$ is a product of kernels. \n\n**Questions**\n\nI'm guessing you mean something like [Cinlar, 2011, page 151, Theorem IV.2.7]? If so, yes, we agree that this is definitely an interesting question to answer in future research. Thank you very much for the suggestion, we really appreciate it! "}]

Author rebuttal:
We would like to thank all reviewers for their time and effort in reviewing our paper, and for making many valuable suggestions that are sure to improve the draft. We are very grateful for your time, and do not take it for granted. We are also very grateful for the mostly positive reviews, and kind words. We answer points raised by individual reviewers separately, except those relating to cycles and Example 4.2, which we answer jointly here. 

**Do cyclic causal relationships truly exist, or is everything acyclic if we include a time component?**

As challenged by reviewer TfjY, it is difficult to think of truly instantaneous cyclic causal relationships (unless they are random variables that do not live in separate components of a product $\sigma$-algebra; see Remark 2.7(i). For example, if $X$ and $Y$ represent the weight of a box in kg and stones respectively, then one could argue that $X$ "causally" affects $Y$ and vice versa, and that this causal effect is instantaneous. However, in our opinion, it makes no sense to talk about the causal effect between random variables that do not live in separate components of a product $\sigma$-algebra. In SCMs, or in any other causal model based on graphs, random variables represented by each node should live in separate components of a product $\sigma$-algebra. Note that this is different to when two random variables do live in separate components of a product $\sigma$-algebra but a measure *happens* to be supported on some kind of a diagonal). 

We believe that even in non-cyclic situations, most causal relationships have a hidden time component, although one could argue that some causal effects really are instantaneous, for example, altitude causally affecting temperature, or the picture of a cow causally affecting its label to be "cow". Our stance is more aligned with reviewer pcib, in that modelling cyclic causal relationships are useful abstractions of what are in the end non-cyclic causal relationships. We also believe that it is extremely difficult to argue against the necessity of such an abstraction, if researchers are happy to model non-cyclic, time-dependent causal relationships without explicitly modelling the time component (as it is in fact very often done) and if we believe in the necessity of being able to do so. It is in this sense that we wrote "cyclic causal relationships abound in the real world", although, as pointed out by reviewer pcib, this sentence may be somewhat misleading. We propose to replace it with something that better reflects the discussion in this paragraph. We also do not want to completely rule out the possibility that one might come across a truly instantaneous and cyclic causal relationship, although we are leaning more towards the belief that such a situation may not exist. 

On the other hand, if one were to insist on modelling the time component explicitly every time it is present (which is not our stance), then we argue that the current tools to model causality within stochastic processes fall short, and we believe that causal spaces make significant contribution in this regard too, as argued in Section 4.3 and as agreed by reviewer aMnb. In answer to reviewer pcib, no, we do not subscribe to causation that goes backwards in time, and it is for this reason that we felt it important to introduce the concept of *time-respecting causal mechanism* in Definition 4.3. 

**Example 4.2**

Reviewers aMnb and pcib suggested that it would not be difficult to model the situation in Example 4.2 with a solvable cyclic SCM [Bongers et al., 2021]. To us, this was difficult, and we didn't manage to do so, both when we were writing the paper and during this rebuttal period. One of the reasons is that the observational measure and the interventional measures are decoupled, and the interventional measure (after intervening on either variable) does not equal the corresponding conditional measure, which means that we must include a common hidden confounding variable. But even apart from this, it seems like a difficult task (at least for us; perhaps we lack the skills or experience for it) to set up the noise variables and their distributions, and to come up with structural equations that yield given desired observational and interventional measures. The example given in [Bongers et al., 2021, Example 3.5] seems like an extremely simple case. We admit that it could still be possible and that we just lack the skills to do it; equally, we did not show that this was *not* possible, and we would be curious to know if the reviewers were able to do this. 

But it should at least serve to illustrate one point - that in SCMs, if we start with noise distributions and structural equations that involve cycles, then it is highly non-trivial to show the existence and uniqueness of a solution, and if we start with the observational and interventional distributions of the endogenous variables, then it is (at least for us) very difficult to find the noise distributions and the structural equations that yield them as a (unique) solution. On the other hand, we argue that causal spaces facilitate a much more natural expression of cyclic causal relationships. 

We would like to thank all reviewers again for their time, and if you have any further questions, or if any of our explanations were unclear, or if we misunderstood any of your points, we would be very grateful if you could let us know. We look forward to engaging in further discussions in the author-reviewer discussion period. 

Meta-review: **This is a very strong paper on the measure theoretic foundation of causality and highly important contribution to the field. Deserves special attention, such as an oral and/or an award (in the theory segment, if there are different categories).**

There was one negative reviewer (xRcV), but the review was short and rather uninformative. The reviewer also disappeared and did not react, neither to requests from the authors nor the AC. The review should therefore be disregarded.
Decision: Accept (oral)