Access to this full-text is provided by De Gruyter.
Content available from Journal of Causal Inference
This content is subject to copyright. Terms and conditions apply.
Open Access. ©2020 T. C. Fraser, published by De Gruyter. This work is licensed under the Creative Commons
Attribution 4.0 License
J. Causal Infer. 2020; 8:22–53
Research Article
Thomas C. Fraser*
A Combinatorial Solution to Causal
Compatibility
https://doi.org/10.1515/jci-2019-0013
Received May 20, 2019; accepted Mar 10, 2020
Abstract: Within the eld of causal inference, it is desirable to learn the structure of causal relationships
holding between a system of variables from the correlations that these variables exhibit; a sub-problem of
which is to certify whether or not a given causal hypothesis is compatible with the observed correlations. A
particularly challenging setting for assessing causal compatibility is in the presence of partial information;
i.e. when some of the variables are hidden/latent. This paper introduces the possible worlds framework as a
method for deciding causal compatibility in this dicult setting. We dene a graphical object called a possible
worlds diagram, which compactly depicts the set of all possible observations. From this construction, we
demonstrate explicitly, using several examples, how to prove causal incompatibility. In fact, we use these
constructions to prove causal incompatibility where no other techniques have been able to. Moreover, we
prove that the possible worlds framework can be adapted to provide a complete solution to the possibilistic
causal compatibility problem. Even more, we also discuss how to exploit graphical symmetries and cross-
world consistency constraints in order to implement a hierarchy of necessary compatibility tests that we prove
converges to suciency.
Keywords: causal inference, causal compatibility, quantum non-classicality
1Introduction
A theory of causation species the eects of actions with absolute necessity. On the other hand, a probabilis-
tic theory encodes degrees of belief and makes predictions based on limited information. A common fallacy
is to interpret correlation as causation; opening an umbrella has never caused it to rain, although the two
are strongly correlated. Numerous paradoxical and catastrophic consequences are unavoidable when prob-
abilistic theories and theories of causation are confused. Nonetheless, Reichenbach’s principle asserts that
correlations must admit causal explanation; after all, the fear of getting wet causes one to open an umbrella.
In recent decades, a concerted eort has been put into developing a formal theory for probabilistic cau-
sation [43, 53]. Integral to this formalism is the concept of a causal structure. A causal structure is a directed
acyclic graph, or DAG, which encodes hypotheses about the causal relationships among a set of random
variables. A causal model is a causal structure when equipped with an explicit description of the parame-
ters which govern the causal relationships. Given a multivariate probability distribution for a set of variables
and a proposed causal structure, the causal compatibility problem aims to determine the existence or non-
existence of a causal model for the given causal structure which can explain the correlations exhibited by the
variables. More generally, the objective of causal discovery is to enumerate all causal structure(s) compati-
ble with an observed distribution. Perhaps unsurprisingly, causal inference has applications in a variety of
academic disciplines including economics, risk analysis, epidemiology, bioinformatics, and machine learn-
ing [29, 42, 43, 48, 62].
*Corresponding Author: Thomas C. Fraser: Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada, N2L 2Y5,
University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1; Email: tfraser@perimeterinstitute.ca
A Combinatorial Solution to Causal Compatibility |23
For physicists, a consideration of causal inuence is commonplace; the theory of special/general rela-
tivity strictly prohibits causal inuences between space-like separated regions of space-time [57]. Famously,
in response to Einstein, Podolsky, and Rosen’s [19] critique on the completeness of quantum theory, Bell [7]
derived an observational constraint, known as Bell’s inequality, which must be satised by all hidden vari-
able models which respect the causal hypothesis of relativity. Moreover, Bell demonstrated the existence of
quantum-realizable correlations which violate Bell’s inequality [7]. Recently, it has been appreciated that
Bell’s theorem can be understood as an instance of causal inference [61]. Contemporary quantum founda-
tions maintains two closely related causal inference research programs. The rst is to develop a theory of
quantum causal models in order to facilitate a causal description of quantum theory and to better under-
stand the limitations of quantum resources [3, 6, 13, 17, 25, 30, 36, 38, 44, 47, 60]. The second is the con-
tinued study of classical causal inference with the purpose of distinguishing genuinely quantum behaviors
from those which admit classical explanations [1, 2, 11, 23–25, 50, 58, 60]. In particular, the results of [30]
suggest that causal structures which support quantum non-classicality are uncommon and typically large in
size; therefore, systematically nding new examples of such causal structures will require the development
of new algorithmic strategies. As a consequence, quantum foundations research has relied upon, and con-
tributed to, the techniques and tools used within the eld of causal inference [13, 30, 50, 60]. The results of
this paper are concerned exclusively with the latter research program of classical causal inference, but does
not rule out the possibility of a generalization to quantum causal inference.
When all variables in a probabilistic system are observed, checking the compatibility status between a
joint distribution and a causal structure is relatively easy; compatibility holds if and only if all conditional
independence constraints implied by graphical d-separation relations hold [39, 43]. Unfortunately, in more
realistic situations there are ethical, economic, or fundamental barriers preventing access to certain statis-
tically relevant variables, and it becomes necessary to hypothesize the existence of latent/hidden variables
in order to adequately explain the correlations expressed by the visible/observed variables [21, 43, 60]. In the
presence of latent variables, and in the absence of interventional data, the causal compatibility problem, and
by extension the subject of causal inference as a whole, becomes considerably more dicult.
In order to overcome these diculties, numerous simplications have be invoked by various authors in
order to make partial progress. A particularly popular simplication strategy has been to consider alternative
classes of graphical causal models which can act as surrogates for DAG causal models; e.g. MC-graphs [34],
summary graphs [59], or maximal ancestral graphs (MAGs) [46, 63]. While these approaches are certainly
attractive from a practical perspective (ecient algorithms such as FCI [53] or RFCI [16] exist for assessing
causal compatibility with MAGs, for instance), they nevertheless fail to fully capture all constraints implied
by DAG causal models with latent variables [22].¹The forthcoming formalism is concerned with assessing the
causal compatibility of DAG causal structures directly, therefore avoiding these shortcomings.
Nevertheless, when considering DAG causal structures directly (henceforth just causal structures), mak-
ing assumptions about the nature of the latent variables and the parameters which govern them can sim-
plify the problem [28, 54, 56]. For instance, when the latent variables are assumed to have a known and
nite cardinality², it becomes possible to articulate the causal compatibility problem as a nite system of
polynomial equality and inequality equations with a nite list of unknowns for which non-linear quanti-
er elimination methods, such as Cylindrical Algebraic Decomposition [31], can provide a complete solution.
Unfortunately, these techniques are only computationally tractable in the simplest of situations. Other tech-
niques from algebraic geometry have been used in simple scenarios to approach the causal compatibility
problem as well [27, 28, 35]. When no assumptions about the nature of the latent variables are made, there
are a plethora of methods for deriving novel equality [21, 45] and inequality [2, 4, 8, 11, 20, 23, 26, 30, 55, 58, 60]
constraints that must be satised by any compatible distribution. The majority of these methods are unsat-
1For concrete and relevant example of this weakness, note that there are observable distributions incompatible with the DAG
causal structure in Figure 11 (which admits of no observable d-separation relations), whereas its associated MAG is compatible
with all observed distributions. An analogous statement happens to be true of the DAG causal structure in Figure 13.
2The cardinality of a random variable is the size of its sample space.
24 |T. C. Fraser
isfactory on the basis that the derived constraints are necessary, but not sucient. A notable exception is
the Ination Technique [60], which produces a hierarchy of linear programs (solvable using ecient algo-
rithms [9, 18, 32, 33, 51]) which are necessary and sucient [37] for determining compatibility.
In contrast with the aforementioned algebraic techniques, the purpose of this paper is to present the
possible worlds framework, which oers a combinatorial solution to the causal compatibility problem in the
presence of latent variables. Importantly, this framework can only be applied when the cardinalities of the
visible variables are known to be nite.³This framework is inspired by the twin networks of Pearl [43], parallel
worlds of Shpitser [52], and by some original drafts of the Ination Technique paper [60]. The possible worlds
framework accomplishes three things. First, we prove its conceptual advantages by revealing that a number
of disparate instances of causal incompatibility become unied under the same premise. Second, we provide
a closed-form algorithm for completely solving the possibilistic causal compatibility problem. To demonstrate
the utility of this method, we provide a solution to an unsolved problem originally reported [22]. Third, we
show that the possible worlds framework provides a hierarchy of tests, much like the Ination Technique,
which solves completely the probabilistic causal compatibility problem.
Unfortunately, the computational complexity of the proposed probabilistic solution is prohibitively large
in many practical situations. Therefore, the contributions of this work are primarily conceptual. Nevertheless,
it is possible that these complexity issues are intrinsic to the problem being considered. Notably, the hierar-
chy of tests presented here has an asymptotic rate of convergence commensurate to the only other complete
solution to the probabilistic compatibility problem, namely the hierarchy of tests provided in [37]. Moreover,
unlike the Ination Technique, if a distribution is compatible with a causal structure, then the hierarchy of
tests provided here has the advantage of returning a causal model which generates that distribution.
This paper is organized as follows: Section 2 begins with a review of the mathematical formalism behind
causal modeling, including a formal denition of the causal compatibility problem, and also introduces the
notations to be used throughout the paper. Afterwards, Section 3 introduces the possible worlds framework
and denes its central object of study: a possible worlds diagram. Section 4 applies the possible worlds frame-
work to prove possibilistic incompatibility between several distributions and corresponding causal struc-
tures, culminating in an algorithm for exactly solving the possibilistic causal compatibility problem. Finally,
Section 5 establishes a hierarchy of tests which completely solve the probabilistic causal compatibility prob-
lem. Moreover, Section 5.1 articulates how to utilize internal symmetries in order to alleviate the aforemen-
tioned computational complexity issues. Section 6 concludes.
Appendix A summarizes relevant results from [22] needed in Section 2. Appendix B generalizes the results
of [50], placing new upper bounds on the maximum cardinality of the latent variables, required for Sections 2
and 5.
2A Review of Causal Modeling
This review section is segmented into three portions. First, Section 2.1 denes directed graphs and their prop-
erties. Second, Section 2.2 introduces the notation and terminology regarding probability distributions to be
used throughout the remainder of this article. Finally, Section 2.3 denes the notion of a causal model and
formally introduces the causal compatibility problem.
2.1 Directed Graphs
Denition 1. Adirected graph Gis an ordered pair G=(Q,E)where Qis a nite set of vertices and Eis a set
edges, i.e. ordered pairs of vertices E⊆Q×Q. If (q,u)∈Eis an edge, denoted as q→u, then uis a child
3Regarding the latent variables, Appendix B.2 demonstrates that the latent variables can be assumed to have nite cardinality
without loss of generality whenever the visible variables have nite cardinality.
A Combinatorial Solution to Causal Compatibility |25
of qand qis a parent of u. A directed path of length kis a sequence of vertices q(1) →q(2) →· · · →q(k)
connected by directed edges. For a given vertex q,paG(q)denotes its parents and chG(q)its children. If there
is a directed path from qto uthen qis an ancestor of uand uis a descendant of q; the set of all ancestors
of qis denoted anG(q)and the set of all descendants is denoted desG(q). The denition for parents, children,
ancestors and descendants of a single vertex qare applied disjunctively to sets of vertices Q⊆Q:
chG(Q) =
q∈Q
chG(q),paG(Q) =
q∈Q
paG(q),(1)
anG(Q) =
q∈Q
anG(q),desG(Q) =
q∈Q
desG(q).(2)
A directed graph is acyclic if there is no directed path of length k>1from qback to qfor any q∈Qand
cyclic otherwise. For example, Figure 1 depicts the dierence between cyclic and acyclic directed graphs.
Denition 2. The subgraph of G=(Q,E)induced by W⊂Q, denoted subG(W), is given by,
subG(W) = (W,E∩(W×W)) ,(3)
i.e. the graph obtained by taking all edges from Ewhich connect members of W.
1
2
3 4
5
(a) A directed cyclic graph.
1
2
3 4
5
(b) A directed acyclic graph.
Figure 1: The dierence between a directed cyclic graph and a directed acyclic graph.
2.2 Probability Theory
Denition 3 (Probability Theory).Aprobability space is a triple (Ω,Ξ,P)where the state space Ωis the set
of all possible outcomes,Ξ⊆2Ωis the set of events forming a σ-algebra over Ω, and Pis a σ-additive function
from events to probabilities such that P(Ω) = 1.
Denition 4 (Probability Notation).For a collection of random variables XI={X1,X2,. . . ,Xk}indexed by
i∈I={1,2,. . . ,k}where each Xitakes values from Ωi, a joint distribution PI=P12...k assigns probabilities
to outcomes from ΩI=i∈IΩi. The event that each Xitakes value xi, referred to as a valuation of XI⁴, is
denoted as,
PI(xI) = P12...k (x1x2. . . xk)=P(X1=x1,X2=x2,. . . Xk=xk).(4)
A point distribution PI(yI) = 1 for a particular event yI∈ΩIis expressed using square brackets,
PI(yI) = 1 ⇔PI(xI) = [yI](xI) = δ(yI,xI) =
i∈I
δ(yi,xi).(5)
4A valuation is a particular type of event in Ξwhere the random variables take on denite values.
26 |T. C. Fraser
The set of all probability distributions over ΩIis denoted as PI. Let kidenote the cardinality or size of Ωi. If
Xiis discrete, then ki=|Ωi|, otherwise Xiis continuous and ki=∞.
2.3 Causal Models and Causal Compatibility
Acausal model represents a complete description of the causal mechanisms underlying a probabilistic pro-
cess. Formally, a causal model is a pair of objects (G,P), which will be dened in turn. First, Gis a directed
acyclic graph (Q,E), whose vertices q∈Qrepresent random variables XQ={Xq|q∈Q}. The purpose
of a causal structure is to graphically encode the causal relationships between the variables. Explicitly, if
q→u∈Eis an edge of the causal structure, Xqis said to have causal inuence on Xu⁵. Consequently, the
causal structure predicts that given complete knowledge of a valuation of the parental variables XpaG(u)=
Xq|q∈paG(u), the random variable Xushould become independent of its non-descendants⁶[43]. With
this observation as motivation, the causal parameters Pof a causal model are a family of conditional prob-
ability distributions Pq|paG(q)for each q∈Q. In the case that qhas no parents in G, the distribution is simply
unconditioned. The purpose of the causal parameters are to predict a joint distribution PQover the congu-
rations ΩQof a causal structure,
∀xQ∈ΩQ,PQ(xQ) =
q∈Q
Pq|paG(q)
(xq|xpaG(q)).(6)
If the hypotheses encoded within a causal structure Gare correct, then the observed distribution over ΩQ
should factorize according to Equation 6. Unfortunately, as discussed in Section 1, there are often ethical,
economic, or fundamental obstacles preventing access to all variables of a system. In such cases, it is cus-
tomary to partition the vertices of causal structure into two disjoint sets; the visible (observed) vertices V,
and the latent (unobserved) vertices L(for example, see Figure 2). Additionally, we denote visible parents
of any vertex q∈V∪Las vpaG(q) = V∩paG(q)and analogously for the latent parents lpaG(q) = L∩paG(q).
v1v2v3
`1`2
v4v5
`3
Figure 2: The causal structure G2in this gure encodes a causal hypothesis about the causal relationships between the visible
variables V={v1,v2,v3,v4,v5}and the latent variables L={ℓ1,ℓ2,ℓ3}; e.g. v2experiences a direct causal influence from
each of its parents, both visible vpaG2(v2) = {v1,v4}and latent lpaG2(v2) = {ℓ1,ℓ2}. Throughout this paper, visible variables
and edges connecting them are colored blue whereas all latent variables and all other edges are colored red.
In the presence of latent variables, Equation 6 stills makes a prediction about the joint distribution
PV∪L(xV,λL)⁷over the visible and latent variables, albeit an experimenter attempting to verify or discredit
5It is seldom necessary to make the distinction between the random variable Xqand the index/vertex q; this paper henceforth
treats them as synonymous.
6This is known as the local Markov property.
7This paper adopts the notational convenient of using λℓ∈Ωℓfor valuations of latent variables ℓ∈Lto dierentiate them from
valuations xv∈Ωvof observed variables v∈V.
A Combinatorial Solution to Causal Compatibility |27
a causal hypothesis only has access to the marginal distribution PV(xV). If ΩLis continuous,
∀xV∈ΩV,PV(xV) =
λL∈ΩL
dPV∪L(xV,λL)(7)
If ΩLis discrete,
∀xV∈ΩV,PV(xV) =
λL∈ΩL
PV∪L(xV,λL).(8)
A natural question arises; in the absence of information about the latent variables L, how can one determine
whether or not their causal hypotheses are correct? The principle purpose of this paper is to provide the reader
with methods for answering this question.
In general, other than being a directed acyclic graph, there are no restrictions placed on a causal struc-
ture with latent variables. Nonetheless, [22] demonstrates that every causal structure Gcan be converted into
a standard form that is observationally equivalent to Gwhere the latent variables are exogenous (have no
parents) and whose children sets are isomorphic to the facets of a simplicial complex over V⁸. Appendix A
summarizes the relevant results from [22] necessary for making this claim. Additionally, Appendix B demon-
strates that any nite distribution PVwhich satises the causal hypotheses (i.e. Equation 7) can be generated
using deterministic causal parameters for the visible variables and moreover, the cardinalities of the latent
variables can be assumed nite⁹. Altogether, Appendices A and B suggest that without loss of generality, we
can simplify the causal compatibility problem as follows:
Denition 5 (Functional Causal Model).A(nite) functional causal model for a causal structure G=
(V∪L,E)is a triple (G,FV,PL)where
FV={fv:ΩpaG(v)→Ωv|v∈V}(9)
are deterministic functions for the visible variables Vin G, and
PL={Pℓ:Ωℓ→[0,1]|ℓ∈L}(10)
are nite probability distributions for the latent variables Lin G. A functional causal model denes a proba-
bility distribution PV:ΩV→[0,1],
∀xV∈ΩV,PV(xV) =
ℓ∈L
λℓ∈Ωℓ
Pℓ(λℓ)
v∈L
δ(xv,fv(xvpaG(v),λlpaG(v))).(11)
Denition 6 (The Causal Compatibility Problem).Given a causal structure G=(V∪L,E)and a distribution
PVover the visible variables V,the causal compatibility problem is to determine if there exists a functional
causal model (G,FV,PL)(dened in Denition 5) such that Equation 11 reproduces PV. If such a functional
causal model exists, then PVis said to be compatible with G; otherwise PVis incompatible with G. The set
of all compatible distributions on Vfor a causal structure Gis denoted MV(G).
3The Possible Worlds Framework
Consider the causal structure in Figure 3a denoted G3a. For the sake of concreteness, suppose one is promised
the latent variables are sampled from a binary sample space, i.e. kµ=kν= 2. Let zµ=Pµ(0µ)and zν=Pν(0ν).
8Appendix A.1 briey discusses what it means for two causal structures to be observationally equivalent.
9We prove this result in Appendix B by generalizing the proof techniques used in [50].
28 |T. C. Fraser
The causal hypothesis G3apredicts (via Equation 11) that observable events (xa,xb,xc)∈Ωa×Ωb×Ωcwill
be distributed according to,
Pabc =zµzν[obsabc (0µ0ν)] + zµ(1 −zν)[obsab c(0µ1ν)]+
+ (1 −zµ)zν[obsab c(1µ0ν)] + (1 −zµ)(1 −zν)[obsabc (1µ1ν)],(12)
where obsabc (λµλν)∈Ωa×Ωb×Ωcis shorthand for the observed event generated by the autonomous functions
fa,fb,fcfor each (λµ,λν)∈Ωµ×Ων. In the case of G3a,
obsab c(λµλν) = (fa(λµ),fb(fa(λµ),λν),fc(fb(fa(λµ),λν),λν)).(13)
For each distinct realization (λµ,λν)∈Ωµ×Ωνof the latent variables, one can consider a possible world
wherein the values λµ,λνare not sampled according to the respective distributions Pµ,Pν, but instead take
on denite values. From the perspective of counterfactual reasoning, each world is modelling a distinct coun-
terfactual assignment of the latent variables, but not the visible variables.¹⁰ In this particular example, there
are kµ×kν= 2 ×2 = 4 distinct, possible worlds. Figure 3b represents, and uniquely colors, these possible
worlds. Note that the denite valuations of the latent variables in Figure 3b are depicted using squares¹¹.
Critically, regardless of the deterministic functional relationships fa,fb,fc, there are identiable consistency
constraints that must hold between these worlds. For example, ais determined by a function fa:Ωµ→Ωa
and thus the observed value for ain the yellow (0µ0ν)-world must be exactly the same as the observed value
for ain the green (0µ1ν)-world. This cross-world consistency constraint is illustrated in Figure 3c by embed-
ding each possible world into a larger diagram with overlapping λµ→asubgraphs. It is important to remark
that not all cross-world consistency constraints are captured by this diagram; the value of bin the yellow
(0µ0ν)-world must match the value of bin the orange (1µ0ν)-world if the value of ain both possible worlds is
the same.
For comparison, in the original causal structure G3a, the vertices represented random variables sampled
from distributions associated with causal parameters; whereas in the possible worlds diagram of Figure 3c,
every valuation, including the latent valuations are predetermined by the functional dependences fa,fb,fc.
For example, Figure 3d populates Figure 3c with the observable events generated by the following functional
dependences,
fa(0µ) = 0afa(1µ) = 1a,
fb(0a0ν) = 3bfb(0a1ν) = 1bfb(1a0ν) = 2bfb(1a1ν) = 0b,
fc(3b0µ0ν) = 0cfc(1b0µ1ν) = 1cfc(2b1µ0ν) = 2cfc(0b1µ1ν) = 3c.
(14)
The utility of Figure 3d is in its simultaneous accounts of Equation 14, the causal structure G3aand the
cross-world consistency constraints that G3ainduces. Nonetheless, Figure 3d fails to specify the probabilities
zµ,zνassociated with the latent events. In Section 4, we utilize diagrams analogous to Figure 3d to tackle
the causal compatibility problem. Before doing so, this paper needs to formally dene the possible worlds
framework.
Denition 7 (The Possible Worlds Framework).Let G=(V∪L,E), be a causal structure with visible vari-
ables Vand latent variables L. Let FVbe a set of functional parameters for Vdened exactly as in Equation 9.
The possible worlds diagram for the pair (G,FV)is a directed acyclic graph Dsatisfying thefollowing prop-
erties:
1. (Valuation Vertices) Each vertex in Dconsists of three pieces (consult Figure 4 for clarity):
10 It is conceivable that this framework, and its associated diagrammatic notation, could be extended to accommodate counter-
factual assignments to the visible variables as well. Such an extension could be useful for assessing compatibility with interven-
tional data, in addition to the purely observational data being considered here.
11 This diagrammatic convention is imminently explained in more depth by Denition 7 and associated Figure 4.
A Combinatorial Solution to Causal Compatibility |29
(a) a subscript q∈V∪Lcorresponding to a vertex in G(indicated inside a small circle in the bottom-
right corner),
(b) an integer ωcorresponding to a possible valuation/outcome ωqof qwhere ωq∈ {0q,1q,. . .}=
Ωq(indicated inside the square of each vertex),
(c) and a decoration in the form of colored outlines¹² indicating which worlds (dened below) the
vertex is a member of¹³.
abc
µν
(a) An example causal
structure G3a.
abc
µ
0ν
0
abc
µ
0ν
1
abc
µ
1ν
0
abc
µ
1ν
1
(b) The possible worlds picture for G3a.
c
c c
c
a
a
b
b b
b
µ
0ν
0
µ
1
ν
1
(c) Identifying consistency constraints among
possible worlds for G3a.
c
0
c
1c
2
c
3
a
0
a
1
b
3
b
1b
2
b
0
µ
0ν
0
µ
1
ν
1
(d) Populating a possible worlds diagram
with the deterministic functions fa,fb,fcin
Equation 14.
Figure 3: A causal structure G3aand the creation of the possible worlds diagram when kµ=kν= 2.
v
ω
valuation ωv∈Ωv
original variable v∈ V
colors indicate world membership
Figure 4: A vertex of a possible worlds diagram dissected.
12 The order of the colored outlines is arbitrary.
13 Every valuation vertex belongs to at least one world.
30 |T. C. Fraser
2. (Ancestral Isomorphism)¹⁴ For every valuation vertex ωqin D, the ancestral subgraph of ωqin Dis
isomorphic to the ancestral subgraph of qin Gunder the map ωq↦→ q.
subD(anD(ωq)) ≃subG(anG(q)) (15)
3. (Consistency) Each valuation vertex xvof a visible variable v∈Vis consistent with the output of the
functional parameter fv∈FVwhen applied to the valuation vertices paD(xv),
xv=fv(paD(xv)) (16)
4. (Uniqueness) For each latent variable ℓ∈L, and for every valuation λℓ∈Ωℓthere exists a unique
valuation vertex in Dcorresponding to λℓ. Unlike latent valuation vertices, the valuations of visible
variables xv∈Ωvmay be repeated (or absent) from Ddepending on the form of FV. In such cases,
duplicated xv’s are always uniquely distinguished by world membership (colored outline).
5. (Worlds) A world is a subgraph of Dthat is isomorphic to Gunder the map ωq↦→ q. Let wor(λL)⊆D
denote the world containing the valuation λL∈ΩL¹⁵. Furthermore, for any subset V⊆Vof visible
variables, let obsV(λL)∈ΩVdenote the observed event supported by wor(λL).
6. (Completeness) For every valuation of the latent variables λL∈ΩL, there exists a subgraph corre-
sponding to wor(λL).¹⁶
It is important to remark that although a possible worlds diagram Dcan be constructed from the pair (G,FV),
the two mathematical objects are not equivalent; the functional parameters FVcan contain superuous in-
formation that never appears in D. We return to this subtle but crucial observation in Section 5.1.
The essential purpose of the possible worlds construction is as a diagrammatic tool for calculating the
observational predictions of a functional causal model. Lemma 1 captures this essence.
Lemma 1. Given a functional causal model (G=(V∪L,E),FV,PL)(see Denition 5), let Dbe the possible
worlds diagram for (G,FV). The causal compatibility criterion (Equation 11) for Gis equivalent to a probabilistic
sum over worlds in D:
PV=
λL∈ΩL
ℓ∈L
Pℓ(λℓ)[obsV(λL)].(17)
The remainder of this paper explores the consequences of adopting the possible worlds framework as a
method for tackling the causal compatibility problem.
4A Complete Possibilistic Solution
Section 3 introduced the possible worlds framework as a technique for calculating the observable predictions
of a functional causal model by means of Lemma 1. In this section, we use the possible worlds framework to
develop a combinatorial algorithm for completely solving the possibilistic causal compatibility problem.
Denition 8. Given a probability distribution PV:ΩV→[0,1], its support σ(PV)is dened as the subset
of events which are possible,
σ(PV) = xV∈ΩV|PV(xV)>0.(18)
14 Readers who are familiar with the Ination technique [60] will recognize this ancestral isomorphism property from the deni-
tion of an Ination of a causal structure. The critical dierence between a possible worlds diagram and an Ination is that vertices
in the former represent valuations of variables whereas vertices in the latter represent independent copies of the variables.
15 The uniqueness property guarantees that each world wor(λL)is uniquely determined by λL.
16 Sometimes it is useful to construct an incomplete possible worlds diagram; for example, Figure 10.
A Combinatorial Solution to Causal Compatibility |31
An observed distribution PVis said to be possibilistically compatible with Gif there exists a functional causal
model (G,FV,PL)for which Equation 11 produces a distribution with the same support as PV. The possi-
bilistic variant of the causal compatibility problem is naturally related to the probabilistic causal compati-
bility problem dened in Denition 6; if a distribution is possibilistically incompatible with G, then it is also
probabilistically incompatible. We now proceed to apply the possible worlds framework to prove possibilistic
incompatibility between a number of distribution/causal structure pairs.
4.1 A Simple Example Causal Structure
Consider the causal structure G5depicted in Figure 5. For G5, the causal compatibility criteria (Equation 11)
takes the form,
Pabc (xaxbxc) =
λµ∈Ωµ
λν∈Ων
Pµ(λµ)Pν(λν)δ(xa,fa(λµ))δ(xb,fb(λµ,λν))δ(xc,fc(λν)).(19)
The following family of distributions for arbitrary xb,yb∈Ωb,
P(20)
abc =z[0axb1c] + (1 −z)[1ayb0c]),0<z<1,(20)
are incompatible with G5. Traditionally, distributions like P(20)
abc are proven incompatible on the basis that they
violate an independence constraint that is implied by G5[43], namely,
∀Pabc ∈M(G5),Pa c(xaxc) = Pa(xa)Pc(xc).(21)
Intuitively, G5provides nolatent mechanism by which aand ccan attempt to correlate (or anti-correlate). We
now prove the possibilistic incompatibility of the support σ(P(20)
abc )with G5using the possible worlds frame-
work.
Proof. Proof by contradiction; assume that a functional causal model FV={fa,fb,fc}for G5exists such that
Equation 19 produces P(20)
abc . Since there are two distinct valuations of the joint variables abc in P(20)
abc , namely
0axb1cand 1ayb0c, consider each as being sampled from two possible worlds. Without loss of generality¹⁷,
let 0µ0ν∈Ωµ×Ωνdenote any valuation of the latent variables such that obsabc(0µ0ν)= 0axb1c. Similarly,
let 1µ1ν∈Ωµ×Ωνdenote any valuation of the latent variables such that obsabc(1µ1ν) = 1ayb0c. Using these
observations, initialize a possible worlds diagram using wor(0µ0ν), colored green, and wor(1µ1ν), colored
violet, as seen in Figure 6a. In order to complete Figure 6a, one simply needs to specify the behavior of b
in two of the “o-diagonal” worlds, namely wor(0µ1ν), colored orange, and wor(1µ0ν), colored yellow (see
Figure 6b). Regardless of this choice, the observed event obsac(0µ1ν) = 0a0cin the orange world wor(0µ1ν)
predicts Pac (0a0c)>0¹⁸ which contradicts P(20)
abc . Therefore, because the proof technique did not rely on the
value of 0<z<1,P(20)
abc is possibilistically incompatible with G5.
abc
µν
Figure 5: A causal structure G5with three visible vertices V={a,b,c}and two latent vertices L={µ,ν}.
17 There is no loss of generality in choosing 0µ0νand 1µ1ν(instead of 0µ1νand 1µ0ν) as the valuations for the worldsbecause the
valuation “labels” associated with latent events are arbitrary. The valuations can not be 0µ1νand 1µ1νbecause of the cross-world
consistency constraint obsc(0µ1ν) = obsc(1µ1ν) = fc(1ν).
18 The probabilities associated to each world by Lemma 1 can always be assumed positive, because otherwise, those valuations
would be excluded from the latent sample space ΩL.
32 |T. C. Fraser
a
0b
xc
1
µ
0ν
0
µ
1ν
1
a
1b
yc
0
(a) An incomplete possible worlds diagram
for G5initialized by P(20)
abc . The worlds are
colored: wor(0µ0ν)green, wor(1µ1ν)violet.
a
0b
xc
1
µ
0ν
0
µ
1ν
1
b
?b
?
a
1b
yc
0
(b) Considering possible worlds produces a
contradiction with P(20)
abc . The additional worlds
are colored: wor(0µ1ν)orange, wor(1µ0ν)
yellow.
Figure 6: The possible worlds diagram for G5(Figure 5) is incompatible with P(20)
abc (Equation 20).
4.2 The Instrumental Structure
The causal structure G7depicted in Figure 7 is known as the Instrumental Scenario [8, 40, 41]. For G7, Equa-
tion 11 takes the form,
Pabc (xaxbxc)=
λµ∈Ωµ
λν∈Ων
Pµ(λµ)Pν(λν)δ(xa,fa(λµ))δ(xb,fb(a,λν))δ(xc,fc(b,λν)).(22)
The following family of distributions,
P(23)
abc =z[0a0b0c]+ (1 −z)[1a0b1c],0<z<1,(23)
are possibilistically incompatible with G7. The Instrumental scenario G7is dierent from G5in that there
are no observable conditional independence constraints which can prove the possibilistic incompatibility of
P(23)
abc . Instead, the possibilistic incompatibility of P(23)
abc is tr aditionally witnessed by an Instrumental inequality
originally derived in [41],
∀Pabc ∈M(G7),Pb c|a(0b0c|0a) + Pbc|a(0b1c|1a)≤1.(24)
Independently of Equation 24, we now prove possibilistic incompatibility of P(23)
abc with G7using the possible
worlds framework.
Proof. Proof by contradiction; assume that a functional model FV={fa,fb,fc}for G7exists such that Equa-
tion 22 produces P(23)
abc (Equation 23). Analogously to the proof in Section 4.1, there are only two distinct
valuations of the joint variables abc, namely 0a0b0cand 1a0b1c. Therefore, dene two worlds one where
obsab c(0µ0ν) = 0a0b0cand another where obsabc (1µ1ν) = 1a0b1c. Using these two worlds, a possible worlds
abc
µν
Figure 7: The Instrumental Scenario.
A Combinatorial Solution to Causal Compatibility |33
a
0
b
0
c
0
a
1
b
0
c
1
µ
0µ
1
ν
0ν
1
b
?b
?
c
?c
?
(a) Worlds wor(0µ0ν), and wor(1µ1ν)are initial-
ized by the observed events in Equation 23.
a
0
b
0
c
0
a
1
b
0
c
1
µ
0µ
1
ν
0ν
1
b
0b
0
c
0c
1
(b) Populating the events in wor(0µ1ν)and
wor(1µ0ν)leads to a contradiction with Equa-
tion 23.
Figure 8: A possible worlds diagram for G7(Figure 7). The worlds are colored: wor(0µ0ν)yellow, wor(1µ1ν)orange, wor(1µ0ν)
violet, wor(0µ1ν)green.
diagram can be initialized as in Figure 8a where wor(0µ0ν)is colored yellow and wor(1µ1ν)is colored orange.
In order to complete the possible worlds diagram of Figure 8a, one rst needs to specify how bbehaves in
two possible worlds: wor(0µ1ν)colored green and wor(1µ0ν)colored violet.
obsb(1µ0ν) = fb(1a0ν) =?b,
obsb(0µ1ν)=fb(0a1ν) =?b.(25)
By appealing to P(23)
abc , it must be that obsb(1µ0ν)=obsb(0µ1ν) = 0bas no other valuations for bare in
the support of P(23)
abc . Finally, the remaining ‘unknown’ observations for cin the violet world obsc(1µ0ν) =
fc(0b0ν), and green world obsc(0µ1ν) = fc(0b1ν)are determined respectively by the behavior of cin the
orange wor(1µ1ν)and yellow wor(0µ0ν)worlds as depicted in Figure 8b. Explicitly,
obsc(1µ0ν) = fc(0b0ν) = obsc(0µ0ν) = 0c,
obsc(0µ1ν) = fc(0b1ν) = obsc(1µ1ν) = 1c.(26)
Therefore the observed events in the green and violet worlds are xed to be,
obsab c(1µ0ν) = 1a0b0c,obsabc(0µ1ν) = 0a0b1c.(27)
Unfortunately, neither of theses events are in the support of P(23)
abc , which is a contradiction; therefore P(23)
abc is
possibilistically incompatible with G7.
Notice that unlike the proof from Section 4.1, here we needed to appeal to the cross-world consistency con-
straints (Equation 26) demanded by the possible worlds framework.
4.3 The Bell Structure
Consider the causal structure G9depicted in Figure 9 known as the Bell structure [7]. From the perspective of
causal inference, Bell’s theorem [7] states that any distribution compatible with G9must satisfy an inequality
constraint known as a Bell inequality. For example, the inequality due to Clauser, Horne, Shimony and Holt,
34 |T. C. Fraser
Figure 9: The Bell causal structure has variables a,b‘measuring’ hidden variable ρwith ‘measurement settings’ x,ydeter-
mined independently of ρ.
referred to as the CHSH inequality, constrains correlations held between aand bas x,yvary [15]¹⁹,
∀Pxab y ∈M(G9),S=⟨ab|0x0y⟩+⟨ab|0x1y⟩+⟨ab|1x0y⟩−⟨ab|1x1y⟩,|S|≤2(28)
Correlations measured by quantum theory are capable of violating this inequality up to S= 2√2[14]. This
violation is not maximum; it is possible to achieve a violation of S= 4 using Popescu-Rohrlich box correla-
tions [49]. The following distribution is an example of a Popescu-Rohrlich box correlation,
P(29)
xab y =1
8([0x0a0b0y] + [0x1a1b0y] + [0x0a0b1y] + [0x1a1b1y]+
+[1x0a0b0y] + [1x1a1b0y] + [1x0a1b1y] + [1x1a0b1y]).
(29)
Unlike G7, there are conditional independence constraints placed on correlations compatible with G9, namely
the no-signaling constraints Pa|xy =Pa|xand Pb|xy =Pb|y. Because P(29)
xab y satises the no-signaling constraints,
the incompatibility of P(29)
xab y with G9is traditionally proven using Equation 28. We now proceed to prove its
incompatibility using the possible worlds framework.
Proof. Proof by contradiction; assume that a functional causal model FV={fa,fb,fx,fy}for G9exists which
supports P(29)
xab y and use the possible worlds framework. Unlike the previous proofs, we only need to consider
a subset of the events in P(29)
xab y to initialize a possible worlds diagram. Consider the following pair of events
and associated latent valuations which support them²⁰,
obsxab y(0µ0ρ0ν) = 0a0b0x0y,obsxaby (1µ1ρ1ν) = 1a0b1x1y.(30)
Using Equation 30, initialize the possible worlds diagram in Figure 10 with worlds wor(0µ0ρ0ν)colored green
and wor(1µ1ρ1ν)colored violet. An unavoidable contradiction arises when attempting to populate the val-
ues for fa(0x1ρ)in the yellow world wor(0µ1ρ1ν)and fb(0y1ρ)in the magenta world wor(1µ1ρ0ν). First,
the observed event obsxaby (0µ1ρ1ν) = 0x?a1b1yin the yellow world wor(0µ1ρ1ν)must belong to the list
of possible events prescribed by P(29)
xab y; a quick inspection leads one to recognize that the only possibility
is obsa(0µ1ρ1ν) = fa(0x1ρ)=1a. An analogous argument in the magenta world wor(1µ1ρ0ν)proves that
obsb(1µ1ρ0ν) = fb(0y1ρ) = 0b. Therefore, the observed event in the orange world wor(0µ1ρ0ν)must be,
obsab cd (0µ1ρ0ν) = 0x1a0b0y,(31)
and therefore Pxaby (0x1a0b0y)>0which contradicts P(29)
xab y. Therefore, P(29)
xab y is possibilistically²¹ incompatible
with G9.
19 The two variable correlation is dened as ⟨ab|xxxy⟩=2
i,j=1(−1)i+jPa b|xy (iajb|xxxy).
20 Clearly, the values of λµand λνthat support these worlds must be unique. Less obvious is the possibility for these worlds to
share a λρvalue. Albeit if they do, the event 0x0a1b1ybecomes possible, contradicting P(29)
xab y as well.
21 The proof holds if the probabilities of the events in P(29)
xab y are any positive value.
A Combinatorial Solution to Causal Compatibility |35
a
0b
0
x
0y
0
µ
0ν
0
ρ
0
a
0b
1
x
1y
1
µ
1ν
1
ρ
1
a
?b
?
Figure 10: An incomplete possible worlds diagram for the Bell structure G9(Figure 9) initialized by the observed events
obsxab y(0µ0ρ0ν) = 0x0a0b0yand obsxab y(1µ1ρ1ν) = 1x0a1b1y. The worlds are colored: wor(0µ0ρ0ν)green, wor(1µ1ρ1ν)
violet, wor(1µ1ρ0ν)magenta, wor(0µ1ρ1ν)yellow, and wor(0µ1ρ0ν)orange.
4.4 The Triangle Structure
Consider the causal structure G11 depicted in Figure 11 known as the Triangle structure. The Triangle has been
studied extensively in recent decades [10, 12, 23, 24, 30, 37, 55, 58, 60]. The following family of distributions
are possibilistically incompatible with G11²²,
P(32)
abc =p1[1a0b0c] + p2[0a1b0c] + p3[0a0b1c],
3
i=1
pi= 1,pi>0.(32)
Proof. Proof by contradiction: assume that a functional causal model FV={fa,fb,fc}for G11 exists support-
ing P(32)
abc and use the possible worlds framework. For each distinct event in P(32)
abc , consider a world in which it
happens denitely. Explicitly dene,
obsab c(0µ0ρ0ν)= 1a0b0c,(33)
obsab c(1µ1ρ1ν) = 0a0b1c,(34)
obsab c(2µ2ρ2ν) = 0a1b0c,(35)
corresponding to the exterior worlds in Figure 12. Consider magenta world wor(0µ1ρ1ν)with partially spec-
ied observation obsabc(0µ1ρ1ν) =?a?b1c. Recalling P(32)
abc , whenever ctakes value 1c,both aand btake
the value 0; i.e. 0a0b. Therefore, it must be that the observed event in the magenta world wor(0µ1ρ1ν)is
obsab c(0µ1ρ1ν) = 0a0b1c. An analogous argument holds for other worlds,
obsab c(0µ1ρ1ν) =?a?b1c⇒obsabc(0µ1ρ1ν) = 0a0b1c,
obsab c(2µ2ρ1ν) =?a1b?c⇒obsabc(2µ2ρ1ν) = 0a1b0c,
obsab c(0µ2ρ0ν)= 1a?b?c⇒obsabc(0µ2ρ0ν) = 1a0b0c.
(36)
ab
c
µ
ν
ρ
Figure 11: The Triangle structure G11 involving three visible variables V={a,b,c}each sharing a pair of latent variables from
L={µ,ν,ρ}.
22 The Ination Technique rst proved the incompatibility between P(32)
abc and G11 .
36 |T. C. Fraser
a
1
b
0c
0
µ
0
ν
0
ρ
0
a
0
b
0c
1
µ
1
ν
1
ρ
1
a
0
b
1c
0
µ
2
ν
2
ρ
2
a
?b
?
c
?
a
?
b
?
c
?
Figure 12: An incomplete possible worlds diagram for the Triangle structure G11 (Figure 11) initialized by the triplet of observed
events in Equation 35. The worlds are colored: wor(0µ0ν0ρ)brown, wor(1µ1ν1ρ)yellow, wor(2µ2ν2ρ)orange, wor(0µ1ν1ρ)
magenta, wor(2µ2ν1ρ)blue, wor(0µ2ν0ρ)violet, and wor(0µ2ν1ρ)green.
However, the conclusions drawn by Equation 36 predict the observed event in the central, green world
wor(0µ2ρ1ν)must be,
obsab c(0µ2ρ1ν) = 0a0b0c,(37)
and therefore Pabc (0a0b0c)>0which contradicts P(32)
abc . Therefore, P(32)
abc is possibilistically incompatible with
G11.
4.5 An Evans Causal Structure
Consider the causal structure in Figure 13, denoted G13. This causal structure, along with two others, was rst
mentioned by Evans [22] as one for which no existing techniques were able to prove whether or not it was sat-
urated; that is, whether or not all distributions were compatible with it. Here it is shown that there are indeed
distributions which are possibilistically incompatible with G13 using the possible worlds framework. As such,
this framework currently stands as the most powerful method for deciding possibilistic compatibility.
abcd
µν
ρ
Figure 13: The Evans Causal Structure G13.
A Combinatorial Solution to Causal Compatibility |37
Consider the family of distributions with three possible events:
P(38)
abc d =p1[0a0b0cyd] + p2[1a0b1c0d] + p3[0a1b1c1d],
3
i=1
pi= 1,pi>0.(38)
Regardless of the values for p1,p2,p3(and yd∈Ωdarbitrary), P(38)
abc d is incompatible with G13.
Proof. Proof by contradiction. First assume that a deterministic model FV={fa,fb,fc,fd}for P(38)
abc d exists
and adopt the possible worlds framework. Let wor(iµiνiρ)for i∈ {1,2,3}index the possible worlds which
support the events observed in Pabcd ,
obsab cd (0µ0ν0ρ) = 0a0b0cyd,
obsab cd (1µ1ν1ρ) = 1a0b1c0d,
obsab cd (2µ2ν2ρ) = 0a1b1c1d.
(39)
Only two additional possible worlds are necessary for achieving a contradiction. Consulting Figure 14 for
details, these possible worlds are wor(1µ0ν2ρ)colored violet and wor(1µ2ν2ρ)colored green. Notice that the
determined value for amust be the same in both worlds as it is independent of λν:
xa=fa(1µ2ρ) = obsa(1µ0ν2ρ) = obsa(1µ2ν2ρ).(40)
There are only two possible values for xain any world, namely xa= 0aor xa= 1aas given by P(38)
abc d. First sup-
pose that xa= 0a. Then in the violet world wor(1µ0ν2ρ), the value of b, to be obsb(1µ0ν2ρ) = fb(0a0ν) = 0b
is completely constrained by consistency with the magenta world wor(0µ0ν0ρ). Therefore, obsab (1µ0ν2ρ) =
0a0b. By analogous logic, in the violet world the value of cis constrained to be obsc(1µ0ν2ρ) = fc(0b1µ) = 0c
by the orange world wor(1µ1ν1ρ). Therefore, obsabc (1µ0ν2ρ) = 0a0b0c, which is a contradiction because
0a0b0cis an impossible event in P(38)
abc d. Therefore, it must be that xa= 1a. An unavoidable contradiction
a
1b
0c
1d
0
µ
1ν
1
ρ
1
a
0b
0c
0d
y
µ
0ν
0
ρ
0
a
0b
1c
1d
1
µ
2ν
2
ρ
2
a
?
b
?c
?d
?
b
?c
?d
?
Figure 14: A possible worlds diagram for G13 initialized by the distribution in Equation 38. The worlds are colored: wor(0µ0ν0ρ)
magenta, wor(1µ1ν1ρ)orange, wor(2µ2ν2ρ)yellow, wor(1µ0ν2ρ)violet, and wor(1µ02ν2ρ)green.
38 |T. C. Fraser
follows from attempting to populate the green world wor(1µ2ν2ρ)in Figure 14 with the established knowl-
edge that obsa(1µ2ν2ρ) = 1a. The value of obsb(1µ2ν2ρ) = fb(1a1ν)has yet to be specied by any pos-
sible worlds, but choosing fb(1a1ν)=1bwould yield an impossible event obsa(1µ2ν2ρ) = 1a1b. There-
fore, it must be that fb(1a1ν)=0band obsa(1µ2ν2ρ) = 1a0b. Similarly, the orange world wor(1µ1ν1ρ)xes
fc(0b1µ)=1cand therefore obsabc (1µ2ν2ρ) = 1a0b1c. Finally, the yellow world wor(2µ2ν2ρ)already deter-
mines obsd(1µ2ν2ρ) = fd(0c2ν2ρ) = 1dand therefore one concludes that,
obsab cd (1µ2ν2ρ) = 1a0b1c1d,(41)
which is an impossible event in P(38)
abc d. This contradiction implies that no functional model FV={fa,fb,fc,fd}
exists and therefore P(38)
abc d is possibilistically incompatible with G13.
To reiterate, there are currently no other methods known [22] which are capable of proving the incompatibility
of any distribution with G13²³. Therefore, the possible worlds framework can be seen as the state-of-the-art
technique for determining possibilistic causation.
4.6 Necessity and Suciency
Throughout this section, we explored a number of proofs of possibilistic incompatibility using the possible
worlds framework. Moreover, the above examples communicate a systematic algorithm for deciding possi-
bilistic compatibility. Given a distribution PVwith support σ(PV)⊂ΩV, and a causal structure G=(V∪L,E),
the following algorithm sketch determines if PVis possibilistically compatible with G.
1. Let W=σ(PV)<|ΩV|denote the number of possible events provided by PV.
2. For each 1≤i≤W, create a possible world wor(λ(i)
L)where λ(i)
L={iℓ|ℓ∈L}, thus dening the latent
sample space ΩL.
3. Attempt to complete the possible worlds diagram Dinitialized by the worlds wor(λ(i)
L)W
i=1.
4. If an impossible event xV∈σ(PV)is produced by any “o-diagonal” world wor(. . . iℓ. . . jℓ′. . .)where
i=j, or if a cross-world consistency constraint is broken, back-track.
Upon completing the search, there are two possibilities. The rst possibility is that the algorithm returns
a completed, consistent, possible worlds diagram D. Then by Lemma 1, PVis possibilistically compatible
with G. The second possibility is that an unavoidable contradiction arises, and PVis not possibilistically
compatible with G.²⁴
5A Complete Probabilistic Solution
In Section 4, we demonstrated that the possible worlds framework was capable of providing a complete pos-
sibilistic solution to the causal compatibility problem. If however, a given distribution PVhappens to sat-
isfy a causal hypothesis on a possibilistic level, can the possible worlds framework be used to determine if
PVsatises the causal hypothesis on a probabilistic level as well? In this section, we answer this question
armatively. In particular, we provide a hierarchy of feasibility tests for probabilistic compatibility which
converges exactly. In addition, we illustrate that a possible worlds diagram is the natural data structure for
algorithmically implementing this converging hierarchy.
23 It is worth noting we have also proven the non-saturation of the other two causal structures mention in [22] using analogous
proofs.
24 A simple C implementation of the above pseudo-algorithm for boolean visible variables (|Ωv|= 2,∀v∈V) can be found
at github.com/tcfraser/possibilistic_causality. In particular, the provided software can output a DIMACS formatted CNF le for
usage in most popular boolean satisability solvers.
A Combinatorial Solution to Causal Compatibility |39
5.1 Symmetry and Superfluity
This aforementioned hierarchy of tests, to be explained in Section 5.3, relies on the enumeration of all prob-
ability distributions PVwhich admit uniform functional causal models (G,FV,PL)for xed cardinalities
kV∪L={kq=|Ωq||q∈V∪L}. A functional causal model is uniform if the probability distributions Pℓ∈PL
over the latent variables are uniform distributions; Pℓ:Ωℓ→k−1
ℓ. Section 5.2 discusses why uniform func-
tional causal models are worth considering, whereas in this section, we discuss how to eciently enumerate
all probability distributions PVthat are uniformly generated from xed cardinalities kV∪L.
One method for generating all such distributions is to perform a brute force enumeration of all determin-
istic strategies FVfor xed cardinalities kV∪L. Depending on the details of the causal structure, the number
of deterministic functions of this form is poly-exponential in the cardinalities kV∪L. This method is inecient
because is fails to consider that many distinct deterministic strategies produce the exact same distribution
PV. There are two optimizations that can be made to avoid regenerations of the same distribution PVwhile
enumerating all deterministic strategies FV. These optimizations are best motivated by an example using the
possible worlds framework.
Consider the causal structure G15ain Figure 15a with visible variables V={a,b,c}and latent variables
L={µ,ν}. Furthermore, for concreteness, suppose that kµ=kν=ka=ka= 2 and kc= 4. Finally let
FV={fa,fb,fc}be such that,
fa(0µ) = 0a,fa(1µ) = 1a,fb(0µ) = 0b,fb(1µ) = 1b,
fc(0a0b0ν) = 2c,fc(0a0b1ν) = 0c,fc(1a1b0ν) = 3c,fc(1a1b1ν) = 1c
fc(0a1b0ν) = 0c,fc(0a1b1ν) = 1c,fc(1a0b0ν) = 2c,fc(1a0b1ν) = 3c.
(42)
The possible worlds diagram Dfor G15agenerated by Equation 42 is depicted in Figure 15b. If the latent val-
uations are distributed uniformly, the probability distribution associated with Figure 15b (as given by Equa-
tion 17) is equal to,
Pabc =1
4([wor(0µ0ν)] + [wor(0µ1ν)] + [wor(1µ0ν)] + [wor(1µ1ν)])
=1
4([0a0b2c] + [0a0b0c] + [1a1b3c] + [1a1b1c]).
(43)
The rst optimization comes from noticing that Equation 42 species how cwould respond if provided with
the valuation 1a0b1νof its parents, namely fc(1a0b1ν) =