A Combinatorial Solution to Causal Compatibility

1 Introduction

A theory of causation specifies the effects of actions with absolute necessity. On the other hand, a probabilistic 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 probabilistic 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 effort has been put into developing a formal theory for probabilistic causation [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 parameters 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) compatible with an observed distribution. Perhaps unsurprisingly, causal inference has applications in a variety of academic disciplines including economics, risk analysis, epidemiology, bioinformatics, and machine learning [29, 42, 43, 48, 62].

For physicists, a consideration of causal influence is commonplace; the theory of special/general relativity strictly prohibits causal influences 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 satisfied by all hidden variable 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 foundations maintains two closely related causal inference research programs. The first is to develop a theory of quantum causal models in order to facilitate a causal description of quantum theory and to better understand the limitations of quantum resources [3, 6, 13, 17, 25, 30, 36, 38, 44, 47, 60]. The second is the continued study of classical causal inference with the purpose of distinguishing genuinely quantum behaviors from those which admit classical explanations [1, 2, 11, 23, 24, 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 finding new examples of such causal structures will require the development of new algorithmic strategies. As a consequence, quantum foundations research has relied upon, and contributed to, the techniques and tools used within the field 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 statistically 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 difficult.

In order to overcome these difficulties, numerous simplifications have be invoked by various authors in order to make partial progress. A particularly popular simplification 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 (efficient 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].^[1] The 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), making assumptions about the nature of the latent variables and the parameters which govern them can simplify the problem [28, 54, 56]. For instance, when the latent variables are assumed to have a known and finite cardinality^[2], it becomes possible to articulate the causal compatibility problem as a finite system of polynomial equality and inequality equations with a finite list of unknowns for which non-linear quantifier 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 techniques 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 satisfied by any compatible distribution. The majority of these methods are unsatisfactory on the basis that the derived constraints are necessary, but not sufficient. A notable exception is the Inflation Technique [60], which produces a hierarchy of linear programs (solvable using efficient algorithms [9, 18, 32, 33, 51]) which are necessary and sufficient [37] for determining compatibility.

In contrast with the aforementioned algebraic techniques, the purpose of this paper is to present the possible worlds framework, which offers 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 finite.^[3] This framework is inspired by the twin networks of Pearl [43], parallel worlds of Shpitser [52], and by some original drafts of the Inflation 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 unified 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 Inflation 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 hierarchy 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 Inflation 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 definition 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 defines its central object of study: a possible worlds diagram. Section 4 applies the possible worlds framework to prove possibilistic incompatibility between several distributions and corresponding causal structures, 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 problem. Moreover, Section 5.1 articulates how to utilize internal symmetries in order to alleviate the aforementioned 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.

2 A Review of Causal Modeling

This review section is segmented into three portions. First, Section 2.1 defines directed graphs and their properties. 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 defines the notion of a causal model and formally introduces the causal compatibility problem.

2.1 Directed Graphs

Definition 1

A directed graph 𝓖 is an ordered pair 𝓖 = (𝓠, 𝓔) where 𝓠 is a finite set of vertices and 𝓔 is a set edges, i.e. ordered pairs of vertices 𝓔 ⊆ 𝓠 × 𝓠. If (q, u) ∈ 𝓔 is an edge, denoted as q → u, then u is a child of q and q is a parent of u. A directed path of length k is a sequence of vertices q₍₁₎ → q₍₂₎ → ⋯ → q_(k) connected by directed edges. For a given vertex q, pa_𝓖(q) denotes its parents and ch_𝓖(q) its children. If there is a directed path from q to u then q is an ancestor of u and u is a descendant of q; the set of all ancestors of q is denoted an_𝓖(q) and the set of all descendants is denoted des_𝓖(q). The definition for parents, children, ancestors and descendants of a single vertex q are applied disjunctively to sets of vertices Q ⊆ 𝓠:

chG(Q)=⋃q∈QchG(q),paG(Q)=⋃q∈QpaG(q),(1)

anG(Q)=⋃q∈QanG(q),desG(Q)=⋃q∈QdesG(q).(2)

A directed graph is acyclic if there is no directed path of length k > 1 from q back to q for any q ∈ 𝓠 and cyclic otherwise. For example, Figure 1 depicts the difference between cyclic and acyclic directed graphs.

Figure 1

The difference between a directed cyclic graph and a directed acyclic graph.

Definition 2

The subgraph of 𝓖 = (𝓠, 𝓔) induced by 𝓦 ⊂ 𝓠, denoted sub_𝓖(𝓦), is given by,

subG(W)=W,E∩W×W,(3)

i.e. the graph obtained by taking all edges from 𝓔 which connect members of 𝓦.

2.3 Causal Models and Causal Compatibility

A causal model represents a complete description of the causal mechanisms underlying a probabilistic process. Formally, a causal model is a pair of objects (𝓖, 𝓟), which will be defined in turn. First, 𝓖 is a directed acyclic graph (𝓠, 𝓔), whose vertices q ∈ 𝓠 represent random variables X_𝓠 = {X_q | q ∈ 𝓠}. The purpose of a causal structure is to graphically encode the causal relationships between the variables. Explicitly, if q → u ∈ 𝓔 is an edge of the causal structure, X_q is said to have causal influence on X_u^[5]. Consequently, the causal structure predicts that given complete knowledge of a valuation of the parental variables X_pa𝓖(u) = {X_q | q ∈ pa_𝓖(u)}, the random variable X_u should become independent of its non-descendants^[6] [43]. With this observation as motivation, the causal parameters 𝓟 of a causal model are a family of conditional probability distributions P_q|pa𝓖(q) for each q ∈ 𝓠. In the case that q has no parents in 𝓖, the distribution is simply unconditioned. The purpose of the causal parameters are to predict a joint distribution P_𝓠 over the configurations Ω_𝓠 of a causal structure,

∀xQ∈ΩQ,PQ(xQ)=∏q∈QPq|paG(q)(xq|xpaG(q)).(6)

If the hypotheses encoded within a causal structure 𝓖 are correct, then the observed distribution over Ω_𝓠 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 customary 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 ∪ 𝓛 as vpa_𝓖(q) = 𝓥 ∩ pa_𝓖(q) and analogously for the latent parents lpa_𝓖(q) = 𝓛 ∩ pa_𝓖(q).

Figure 2

The causal structure 𝓖₂ in this figure 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. v₂ experiences a direct causal influence from each of its parents, both visible vpa_𝓖2(v₂) = {v₁, v₄} and latent lpa_𝓖2(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 P_𝓥∪𝓛(x_𝓥, λ_𝓛)^[7] over the visible and latent variables, albeit an experimenter attempting to verify or discredit a causal hypothesis only has access to the marginal distribution P_𝓥(x_𝓥). If Ω_𝓛 is continuous,

∀xV∈ΩV,PV(xV)=∫λL∈ΩLdPV∪L(xV,λL)(7)

If Ω_𝓛 is discrete,

∀xV∈ΩV,PV(xV)=∑λL∈ΩLPV∪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 structure with latent variables. Nonetheless, [22] demonstrates that every causal structure 𝓖 can be converted into a standard form that is observationally equivalent to 𝓖 where the latent variables are exogenous (have no parents) and whose children sets are isomorphic to the facets of a simplicial complex over 𝓥^[8]. Appendix A summarizes the relevant results from [22] necessary for making this claim. Additionally, Appendix B demonstrates that any finite distribution P_𝓥 which satisfies 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 finite^[9]. Altogether, Appendices A and B suggest that without loss of generality, we can simplify the causal compatibility problem as follows:

Definition 5

(Functional Causal Model). A (finite) functional causal model for a causal structure 𝓖 = (𝓥 ∪ 𝓛, 𝓔) is a triple (𝓖, 𝓕_𝓥, 𝓟_𝓛) where

FV={fv:ΩpaG(v)→Ωv∣v∈V}(9)

are deterministic functions for the visible variables 𝓥 in 𝓖, and

PL=Pℓ:Ωℓ→0,1∣ℓ∈L(10)

are finite probability distributions for the latent variables 𝓛 in 𝓖. A functional causal model defines a probability distribution P_𝓥 : Ω_𝓥 → [0, 1],

∀xV∈ΩV,PV(xV)=∏ℓ∈L∑λℓ∈ΩℓPℓ(λℓ)∏v∈Lδ(xv,fv(xvpaG(v),λlpaG(v))).(11)

Definition 6

(The Causal Compatibility Problem). Given a causal structure 𝓖 = (𝓥 ∪ 𝓛, 𝓔) and a distribution P_𝓥 over the visible variables 𝓥, the causal compatibility problem is to determine if there exists a functional causal model (𝓖, 𝓕_𝓥, 𝓟_𝓛) (defined in Definition 5) such that Equation 11 reproduces P_𝓥. If such a functional causal model exists, then P_𝓥 is said to be compatible with 𝓖; otherwise P_𝓥 is incompatible with 𝓖. The set of all compatible distributions on 𝓥 for a causal structure 𝓖 is denoted 𝓜_𝓥(𝓖).

3 The Possible Worlds Framework

Consider the causal structure in Figure 3a denoted 𝓖_3a. For the sake of concreteness, suppose one is promised the latent variables are sampled from a binary sample space, i.e. k_μ = k_ν = 2. Let z_μ = P_μ(0_μ) and z_ν = P_ν(0_ν). The causal hypothesis 𝓖_3a predicts (via Equation 11) that observable events (x_a, x_b, x_c) ∈ Ω_a × Ω_b × Ω_c will be distributed according to,

Figure 3

A causal structure 𝓖_3a and the creation of the possible worlds diagram when k_μ = k_ν = 2.

where obs_abc(λ_μλ_ν) ∈ Ω_a × Ω_b × Ω_c is shorthand for the observed event generated by the autonomous functions f_a, f_b, f_c for each (λ_μ, λ_ν) ∈ Ω_μ × Ω_ν. In the case of 𝓖_3a,

For each distinct realization (λ_μ, λ_ν) ∈ Ω_μ × Ω_ν of the latent variables, one can consider a possible world wherein the values λ_μ, λ_ν are not sampled according to the respective distributions P_μ, P_ν, but instead take on definite values. From the perspective of counterfactual reasoning, each world is modelling a distinct counterfactual assignment of the latent variables, but not the visible variables.^[10] In this particular example, there are k_μ × k_ν = 2 × 2 = 4 distinct, possible worlds. Figure 3b represents, and uniquely colors, these possible worlds. Note that the definite valuations of the latent variables in Figure 3b are depicted using squares^[11]. Critically, regardless of the deterministic functional relationships f_a, f_b, f_c, there are identifiable consistency constraints that must hold between these worlds. For example, a is determined by a function f_a : Ω_μ → Ω_a and thus the observed value for a in the yellow (0_μ0_ν)-world must be exactly the same as the observed value for a in the green (0_μ1_ν)-world. This cross-world consistency constraint is illustrated in Figure 3c by embedding each possible world into a larger diagram with overlapping λ_μ → a subgraphs. It is important to remark that not all cross-world consistency constraints are captured by this diagram; the value of b in the yellow (0_μ0_ν)-world must match the value of b in the orange (1_μ0_ν)-world if the value of a in both possible worlds is the same.

Figure 4

A vertex of a possible worlds diagram dissected.

For comparison, in the original causal structure 𝓖_3a, 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 f_a, f_b, f_c. For example, Figure 3d populates Figure 3c with the observable events generated by the following functional dependences,

The utility of Figure 3d is in its simultaneous accounts of Equation 14, the causal structure 𝓖_3a and the cross-world consistency constraints that 𝓖_3a induces. Nonetheless, Figure 3d fails to specify the probabilities z_μ, z_ν associated 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 define the possible worlds framework.

It is important to remark that although a possible worlds diagram 𝓓 can be constructed from the pair (𝓖, 𝓕_𝓥), the two mathematical objects are not equivalent; the functional parameters 𝓕_𝓥 can contain superfluous information 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.

The remainder of this paper explores the consequences of adopting the possible worlds framework as a method for tackling the causal compatibility problem.

4 A 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.

An observed distribution P_𝓥 is said to be possibilistically compatible with 𝓖 if there exists a functional causal model (𝓖, 𝓕_𝓥, 𝓟_𝓛) for which Equation 11 produces a distribution with the same support as P_𝓥. The possibilistic variant of the causal compatibility problem is naturally related to the probabilistic causal compatibility problem defined in Definition 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 𝓖₅ depicted in Figure 5. For 𝓖₅, the causal compatibility criteria (Equation 11) takes the form,

Pabc(xaxbxc)=∑λμ∈Ωμ∑λν∈ΩνPμ(λμ)Pν(λν)δ(xa,fa(λμ))δ(xb,fb(λμ,λν))δ(xc,fc(λν)).(19)

Figure 5

A causal structure 𝓖₅ with three visible vertices 𝓥 = {a, b, c} and two latent vertices 𝓛 = {μ, ν}.

The following family of distributions for arbitrary x_b, y_b ∈ Ω_b,

Pabc(20)=z[0axb1c]+(1−z)[1ayb0c]),0<z<1,(20)

are incompatible with 𝓖₅. Traditionally, distributions like Pabc(20) are proven incompatible on the basis that they violate an independence constraint that is implied by 𝓖₅ [43], namely,

∀Pabc∈M(G5),Pac(xaxc)=Pa(xa)Pc(xc).(21)

Intuitively, 𝓖₅ provides no latent mechanism by which a and c can attempt to correlate (or anti-correlate). We now prove the possibilistic incompatibility of the support σ(Pabc(20)) with 𝓖₅ using the possible worlds framework.

Proof

Proof by contradiction; assume that a functional causal model 𝓕_𝓥 = {f_a, f_b, f_c} for 𝓖₅ exists such that Equation 19 produces Pabc(20). Since there are two distinct valuations of the joint variables abc in Pabc(20), namely 0_ax_b1_c and 1_ay_b0_c, consider each as being sampled from two possible worlds. Without loss of generality^[17], let 0_μ0_ν ∈ Ω_μ × Ω_ν denote any valuation of the latent variables such that obs_abc(0_μ0_ν) = 0_ax_b1_c. Similarly, let 1_μ1_ν ∈ Ω_μ × Ω_ν denote any valuation of the latent variables such that obs_abc(1_μ1_ν) = 1_ay_b0_c. Using these observations, initialize a possible worlds diagram using wor(0_μ0_ν), colored green, and wor(1_μ1_ν), 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 “off-diagonal” worlds, namely wor(0_μ1_ν), colored orange, and wor(1_μ0_ν), colored yellow (see Figure 6b). Regardless of this choice, the observed event obs_ac(0_μ1_ν) = 0_a0_c in the orange world wor(0_μ1_ν) predicts P_ac(0_a0_c) > 0^[18] which contradicts Pabc(20). Therefore, because the proof technique did not rely on the value of 0 < z < 1, Pabc(20) is possibilistically incompatible with 𝓖₅.□

Figure 6

The possible worlds diagram for 𝓖₅ (Figure 5) is incompatible with Pabc(20) (Equation 20).

4.2 The Instrumental Structure

The causal structure 𝓖₇ depicted in Figure 7 is known as the Instrumental Scenario [8, 40, 41]. For 𝓖₇, Equation 11 takes the form,

Pabcxaxbxc=∑λμ∈Ωμ∑λν∈ΩνPμ(λμ)Pν(λν)δ(xa,fa(λμ))δ(xb,fb(a,λν))δ(xc,fc(b,λν)).(22)

Figure 7

The Instrumental Scenario.

The following family of distributions,

Pabc(23)=z0a0b0c+(1−z)1a0b1c,0<z<1,(23)

are possibilistically incompatible with 𝓖₇. The Instrumental scenario 𝓖₇ is different from 𝓖₅ in that there are no observable conditional independence constraints which can prove the possibilistic incompatibility of Pabc(23). Instead, the possibilistic incompatibility of Pabc(23) is traditionally witnessed by an Instrumental inequality originally derived in [41],

∀Pabc∈M(G7),Pbc|a(0b0c|0a)+Pbc|a(0b1c|1a)≤1.(24)

Independently of Equation 24, we now prove possibilistic incompatibility of Pabc(23) with 𝓖₇ using the possible worlds framework.

Proof

Proof by contradiction; assume that a functional model 𝓕_𝓥 = {f_a, f_b, f_c} for 𝓖₇ exists such that Equation 22 produces Pabc(23) (Equation 23). Analogously to the proof in Section 4.1, there are only two distinct valuations of the joint variables abc, namely 0_a0_b0_c and 1_a0_b1_c. Therefore, define two worlds one where obs_abc(0_μ0_ν) = 0_a0_b0_c and another where obs_abc(1_μ1_ν) = 1_a0_b1_c. Using these two worlds, a possible worlds diagram can be initialized as in Figure 8a where wor(0_μ0_ν) is colored yellow and wor(1_μ1_ν) is colored orange. In order to complete the possible worlds diagram of Figure 8a, one first needs to specify how b behaves in two possible worlds: wor(0_μ1_ν) colored green and wor(1_μ0_ν) colored violet.

obsb(1μ0ν)_=fb(1a0ν)=?b,obsb(0μ1ν)_=fb(0a1ν)=?b.(25)

Figure 8

A possible worlds diagram for 𝓖₇ (Figure 7). The worlds are colored: wor(0_μ0_ν) yellow, wor(1_μ1_ν) orange, wor(1_μ0_ν) violet, wor(0_μ1_ν) green.

By appealing to Pabc(23), it must be that obs_b(1_μ0_ν) = obs_b(0_μ1_ν) = 0_b as no other valuations for b are in the support of Pabc(23). Finally, the remaining ‘unknown’ observations for c in the violet world obs_c(1_μ0_ν) = f_c(0_b0_ν), and green world obs_c(0_μ1_ν) = f_c(0_b1_ν) are determined respectively by the behavior of c in the orange wor(1_μ1_ν) and yellow wor(0_μ0_ν) 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 fixed to be,

obsabc(1μ0ν)_=1a0b0c,obsabc(0μ1ν)_=0a0b1c.(27)

Unfortunately, neither of theses events are in the support of Pabc(23), which is a contradiction; therefore Pabc(23) is possibilistically incompatible with 𝓖₇.□

Notice that unlike the proof from Section 4.1, here we needed to appeal to the cross-world consistency constraints (Equation 26) demanded by the possible worlds framework.

4.3 The Bell Structure

Consider the causal structure 𝓖₉ depicted 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 𝓖₉ must satisfy an inequality constraint known as a Bell inequality. For example, the inequality due to Clauser, Horne, Shimony and Holt, referred to as the CHSH inequality, constrains correlations held between a and b as x, y vary [15]^[19],

∀Pxaby∈M(G9),S=ab|0x0y+ab|0x1y+ab|1x0y−ab|1x1y,|S|≤2(28)

Figure 9

The Bell causal structure has variables a, b ‘measuring’ hidden variable ρ with ‘measurement settings’ x, y determined independently of ρ.

Correlations measured by quantum theory are capable of violating this inequality up to S = 22 [14]. This violation is not maximum; it is possible to achieve a violation of S = 4 using Popescu-Rohrlich box correlations [49]. The following distribution is an example of a Popescu-Rohrlich box correlation,

Pxaby(29)=18([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 P_a|xy = P_a|x and P_b|xy = P_b|y. Because Pxaby(29) satisfies the no-signaling constraints, the incompatibility of Pxaby(29) with 𝓖₉ is 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 𝓕_𝓥 = {f_a, f_b, f_x, f_y} for 𝓖₉ exists which supports Pxaby(29) and use the possible worlds framework. Unlike the previous proofs, we only need to consider a subset of the events in Pxaby(29) to initialize a possible worlds diagram. Consider the following pair of events and associated latent valuations which support them^[20],

obsxaby(0μ0ρ0ν)_=0a0b0x0y,obsxaby(1μ1ρ1ν)_=1a0b1x1y.(30)

Using Equation 30, initialize the possible worlds diagram in Figure 10 with worlds wor(0_μ0_ρ0_ν) colored green and wor(1_μ1_ρ1_ν) colored violet. An unavoidable contradiction arises when attempting to populate the values for f_a(0_x1_ρ) in the yellow world wor(0_μ1_ρ1_ν) and f_b(0_y1_ρ) in the magenta world wor(1_μ1_ρ0_ν). First, the observed event obs_xaby(0_μ1_ρ1_ν) = 0_x?_a1_b1_y in the yellow world wor(0_μ1_ρ1_ν) must belong to the list of possible events prescribed by Pxaby(29); a quick inspection leads one to recognize that the only possibility is obs_a(0_μ1_ρ1_ν) = f_a(0_x1_ρ) = 1_a. An analogous argument in the enta world wor(1_μ1_ρ0_ν) proves that obs_b(1_μ1_ρ0_ν) = f_b(0_y1_ρ) = 0_b. Therefore, the observed event in the orange world wor(0_μ1_ρ0_ν) must be,

obsabcd(0μ1ρ0ν)_=0x1a0b0y,(31)

Figure 10

An incomplete possible worlds diagram for the Bell structure 𝓖₉ (Figure 9) initialized by the observed events obs_xaby(0_μ0_ρ0_ν) = 0_x0_a0_b0_y and obs_xaby(1_μ1_ρ1_ν) = 1_x0_a1_b1_y. The worlds are colored: wor(0_μ0_ρ0_ν) green, wor(1_μ1_ρ1_ν) violet, wor(1_μ1_ρ0_ν) magenta, wor(0_μ1_ρ1_ν) yellow, and wor(0_μ1_ρ0_ν) orange.

and therefore P_xaby(0_x1_a0_b0_y) > 0 which contradicts Pxaby(29). Therefore, Pxaby(29) is possibilistically^[21] incompatible with 𝓖₉.

□

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 𝓖₁₁^[22],

Pabc(32)=p1[1a0b0c]+p2[0a1b0c]+p3[0a0b1c],∑i=13pi=1,pi>0.(32)

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}.

Proof

Proof by contradiction: assume that a functional causal model 𝓕_𝓥 = {f_a, f_b, f_c} for 𝓖₁₁ exists supporting Pabc(32) and use the possible worlds framework. For each distinct event in Pabc(32), consider a world in which it happens definitely. Explicitly define,

obsabc(0μ0ρ0ν)_=1a0b0c,(33)

obsabc(1μ1ρ1ν)_=0a0b1c,(34)

obsabc(2μ2ρ2ν)_=0a1b0c,(35)

corresponding to the exterior worlds in Figure 12. Consider magenta world wor(0_μ1_ρ1_ν) with partially specified observation obs_abc(0_μ1_ρ1_ν) = ?_a?_b1_c. Recalling Pabc(32), whenever c takes value 1_c, botha and b take the value 0; i.e. 0_a0_b. Therefore, it must be that the observed event in the magenta world wor(0_μ1_ρ1_ν) is obs_abc(0_μ1_ρ1_ν) = 0_a0_b1_c. An analogous argument holds for other worlds,

obsabc(0μ1ρ1ν)_=?a?b1c⇒obsabc(0μ1ρ1ν)_=0a0b1c,obsabc(2μ2ρ1ν)_=?a1b?c⇒obsabc(2μ2ρ1ν)_=0a1b0c,obsabc(0μ2ρ0ν)_=1a?b?c⇒obsabc(0μ2ρ0ν)_=1a0b0c.(36)

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(0_μ0_ν0_ρ) brown, wor(1_μ1_ν1_ρ) yellow, wor(2_μ2_ν2_ρ) orange, wor(0_μ1_ν1_ρ) magenta, wor(2_μ2_ν1_ρ) blue, wor(0_μ2_ν0_ρ) violet, and wor(0_μ2_ν1_ρ) green.

However, the conclusions drawn by Equation 36 predict the observed event in the central, green world wor(0_μ2_ρ1_ν) must be,

obsabc(0μ2ρ1ν)_=0a0b0c,(37)

and therefore P_abc(0_a0_b0_c) > 0 which contradicts Pabc(32). Therefore, Pabc(32) 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 first mentioned by Evans [22] as one for which no existing techniques were able to prove whether or not it was saturated; 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.

Figure 13

The Evans Causal Structure 𝓖₁₃.

Consider the family of distributions with three possible events:

Pabcd(38)=p1[0a0b0cyd]+p2[1a0b1c0d]+p3[0a1b1c1d],∑i=13pi=1,pi>0.(38)

Regardless of the values for p₁, p₂, p₃ (and y_d ∈ Ω_d arbitrary), Pabcd(38) is incompatible with 𝓖₁₃.

Proof

Proof by contradiction. First assume that a deterministic model 𝓕_𝓥 = {f_a, f_b, f_c, f_d} for Pabcd(38) exists and adopt the possible worlds framework. Let wor(i_μi_νi_ρ) for i ∈ {1, 2, 3} index the possible worlds which support the events observed in P_abcd,

obsabcd(0μ0ν0ρ)_=0a0b0cyd,obsabcd(1μ1ν1ρ)_=1a0b1c0d,obsabcd(2μ2ν2ρ)_=0a1b1c1d.(39)

Only two additional possible worlds are necessary for achieving a contradiction. Consulting Figure 14 for details, these possible worlds are wor(1_μ0_ν2_ρ) colored violet and wor(1_μ2_ν2_ρ) colored green. Notice that the determined value for a must be the same in both worlds as it is independent of λ_ν:

xa=fa(1μ2ρ)=obsa(1μ0ν2ρ)_=obsa(1μ2ν2ρ)_.(40)

Figure 14

A possible worlds diagram for 𝓖₁₃ initialized by the distribution in Equation 38. The worlds are colored: wor(0_μ0_ν0_ρ) magenta, wor(1_μ1_ν1_ρ) orange, wor(2_μ2_ν2_ρ) yellow, wor(1_μ0_ν2_ρ) violet, and wor(1_μ02_ν2_ρ) green.

There are only two possible values for x_a in any world, namely x_a = 0_a or x_a = 1_a as given by Pabcd(38). First suppose that x_a = 0_a. Then in the violet world wor(1_μ0_ν2_ρ), the value of b, to be obs_b(1_μ0_ν2_ρ) = f_b(0_a0_ν) = 0_b is completely constrained by consistency with the magenta world wor(0_μ0_ν0_ρ). Therefore, obs_ab(1_μ0_ν2_ρ) = 0_a0_b. By analogous logic, in the violet world the value of c is constrained to be obs_c(1_μ0_ν2_ρ) = f_c(0_b1_μ) = 0_c by the orange world wor(1_μ1_ν1_ρ). Therefore, obs_abc(1_μ0_ν2_ρ) = 0_a0_b0_c, which is a contradiction because 0_a0_b0_c is an impossible event in Pabcd(38). Therefore, it must be that x_a = 1_a. An unavoidable contradiction follows from attempting to populate the green world wor(1_μ2_ν2_ρ) in Figure 14 with the established knowledge that obs_a(1_μ2_ν2_ρ) = 1_a. The value of obs_b(1_μ2_ν2_ρ) = f_b(1_a1_ν) has yet to be specified by any possible worlds, but choosing f_b(1_a1_ν) = 1_b would yield an impossible event obs_a(1_μ2_ν2_ρ) = 1_a1_b. Therefore, it must be that f_b(1_a1_ν) = 0_b and obs_a(1_μ2_ν2_ρ) = 1_a0_b. Similarly, the orange world wor(1_μ1_ν1_ρ) fixes f_c(0_b1_μ) = 1_c and therefore obs_abc(1_μ2_ν2_ρ) = 1_a0_b1_c. Finally, the yellow world wor(2_μ2_ν2_ρ) already determines obs_d(1_μ2_ν2_ρ) = f_d(0_c2_ν2_ρ) = 1_d and therefore one concludes that,

obsabcd(1μ2ν2ρ)_=1a0b1c1d,(41)

which is an impossible event in Pabcd(38). This contradiction implies that no functional model 𝓕_𝓥 = {f_a, f_b, f_c, f_d} exists and therefore Pabcd(38) 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 𝓖₁₃^[23]. Therefore, the possible worlds framework can be seen as the state-of-the-art technique for determining possibilistic causation.

5 A Complete Probabilistic Solution

In Section 4, we demonstrated that the possible worlds framework was capable of providing a complete possibilistic solution to the causal compatibility problem. If however, a given distribution P_𝓥 happens to satisfy a causal hypothesis on a possibilistic level, can the possible worlds framework be used to determine if P_𝓥 satisfies the causal hypothesis on a probabilistic level as well? In this section, we answer this question affirmatively. 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.

5.1 Symmetry and Superfluity

This aforementioned hierarchy of tests, to be explained in Section 5.3, relies on the enumeration of all probability distributions P_𝓥 which admit uniform functional causal models (𝓖, 𝓕_𝓥, 𝓟_𝓛) for fixed cardinalities k_𝓥∪𝓛 = {k_q = |Ω_q| | q ∈ 𝓥 ∪ 𝓛}. A functional causal model is uniform if the probability distributions P_ℓ ∈ 𝓟_𝓛 over the latent variables are uniform distributions; P_ℓ : Ω_ℓ → kℓ−1. Section 5.2 discusses why uniform functional causal models are worth considering, whereas in this section, we discuss how to efficiently enumerate all probability distributions P_𝓥 that are uniformly generated from fixed cardinalities k_𝓥∪𝓛.

One method for generating all such distributions is to perform a brute force enumeration of all deterministic strategies 𝓕_𝓥 for fixed cardinalities k_𝓥∪𝓛. Depending on the details of the causal structure, the number of deterministic functions of this form is poly-exponential in the cardinalities k_𝓥∪𝓛. This method is inefficient because is fails to consider that many distinct deterministic strategies produce the exact same distribution P_𝓥. There are two optimizations that can be made to avoid regenerations of the same distribution P_𝓥 while enumerating all deterministic strategies 𝓕_𝓥. These optimizations are best motivated by an example using the possible worlds framework.

Consider the causal structure 𝓖_15a in Figure 15a with visible variables 𝓥 = {a, b, c} and latent variables 𝓛 = {μ, ν}. Furthermore, for concreteness, suppose that k_μ = k_ν = k_a = k_a = 2 and k_c = 4. Finally let 𝓕_𝓥 = {f_a, f_b, f_c} 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)=1cfc(0a1b0ν)=0c,fc(0a1b1ν)=1c,fc(1a0b0ν)=2c,fc(1a0b1ν)=3c.(42)

Figure 15

Every permutation π_ℓ : Ω_ℓ → Ω_ℓ of valuations on the latent variables maps a possible worlds diagram to another possible worlds diagram with the same observed events. The worlds are colored: wor(0_μ0_ν) green, wor(0_μ1_ν) orange, wor(1_μ0_ν) yellow, and wor(1_μ1_ν) violet.

The possible worlds diagram 𝓓 for 𝓖_15a generated by Equation 42 is depicted in Figure 15b. If the latent valuations are distributed uniformly, the probability distribution associated with Figure 15b (as given by Equation 17) is equal to,

Pabc=14([wor(0μ0ν)_]+[wor(0μ1ν)_]+[wor(1μ0ν)_]+[wor(1μ1ν)_])=14([0a0b2c]+[0a0b0c]+[1a1b3c]+[1a1b1c]).(43)

The first optimization comes from noticing that Equation 42 specifies how c would respond if provided with the valuation 1_a0_b1_ν of its parents, namely f_c(1_a0_b1_ν) = 3_c. Nonetheless, this hypothetical scenario is excluded from Figure 15b (crossed out in the figure) because the functional model in Equation 42 never produces an opportunity for a to be different from b. Consequently, the functional dependences in Equation 42 contain superfluous information irrelevant to the observed probability distribution in Equation 43.

Therefore, a brute force enumeration of deterministic strategies would regenerate Equation 43 several times, once for each assignment of c’s behavior in these superfluous scenarios. It is possible to avoid these regenerations by using an unpopulated possible worlds diagram 𝓓̃ as a data structure and performing a brute force enumeration of all consistent valuations of 𝓓̃.

The second optimization comes from noticing that Equation 43 contains many symmetries. Notably, independently permuting the latent valuations, π_μ : 0_μ ↔ 1_μ or π_ν : 0_ν ↔ 1_ν, leaves the observed distribution in Equation 43 invariant, but maps the functional dependences 𝓕_𝓥 of Equation 42 to different functional dependences FVπμ and FVπν. These symmetries are reflected as permutations of the worlds as depicted in Figures 15c, and 15d.

Analogously, it is possible to avoid these regenerations by first pre-computing the induced action on 𝓓̃, and thus an induced action on 𝓕_𝓥, under the permutation group S_𝓛 = ∏_ℓ∈𝓛 perm(Ω_ℓ). Then, using the permutation group S_𝓛, one only needs to generate a representative from the equivalence classes of possible worlds diagrams 𝓓 under S_𝓛.

Importantly, the optimizations illuminated above, namely ignoring superfluous specifications and exploiting symmetries, are universal^[25]; they can be applied for any causal structure. Additionally, the possible worlds framework intuitively excludes superfluous cases and directly embodies the observational symmetries, making a possible worlds diagram the ideal data structure for performing a search over observed distributions.

5.2 The Uniformity of Latent Distributions

The purpose of this section is motivate why it is always possible to approximate any functional causal model (𝓖, 𝓕_𝓥, 𝓟_𝓛) with another functional causal model (𝓖, 𝓕̃_𝓥, 𝓟̃_𝓛) which has latent events λ_𝓛 ∈ Ω̃_𝓛 uniformly distributed. Unsurprisingly, an accurate approximation of this form will require an increase in the cardinality |Ω̃_𝓛| > |Ω_𝓛| of the latent variables.

Definition 9

(Rational Distributions). A discrete probability distribution P over Ω is rational if every probability assigned to events in Ω by P is rational,

∀λ∈Ω,P(λ)=nλdλ,wherenλ,dλ∈Z.(44)

Definition 10

(Distance Metric for Distributions). Given two probability distributions P, P̃ over the same sample space Ω, the distance Δ(P, P̃) between P and P̃ is defined as,

Δ(P,P~)=∑x∈Ω|P(x)−P~(x)|(45)

Theorem 2

Let P_ℓ : Ω_ℓ → [0, 1] be any discrete probability distribution onΩ_ℓ, then there exists a rational approximation P̃_ℓ : Ω_ℓ → [0, 1],

∀λℓ∈Ωℓ,P~ℓ(λℓ)=1|Ωu|∑ωu∈Ωuδ(λℓ,g(ωu)),(46)

where g : Ω_u → Ω_ℓis deterministic andΔ(Pℓ,P~ℓ)≤|Ωu|−1|Ωℓ|.

Proof

The proof is illustrated in Figure 16. In the special case that |Ω_ℓ| = 1, the proof is trivial; g simply maps all values of ω_u to the singleton λ_ℓ ∈ Ω_ℓ. The proof follows from a construction of g using inverse uniform sampling. Given some ordering 1_ℓ < 2_ℓ < ⋯ of Ω_ℓ and ordering 1_u < 2_u < ⋯ of Ω_u compute the cumulative distribution function P≤ℓ(λℓ)=∑λℓ′≤λℓPℓ(λℓ′). Then the function g : Ω_u → Ω_ℓ is defined as,

g(ωu)=minλℓ∈Ωℓ∣P≤ℓ(λℓ)|Ωu|≥ωu.(47)

Figure 16

Theorem 2: Approximately sampling a non-uniform distribution using inverse sampling techniques.

Consequently, the proportion of ω_u ∈ Ω_u values which map to λ_ℓ ∈ Ω_ℓ has error ε(λ_ℓ),

ε(λℓ)=|Ωu|Pℓ(λℓ)−|g−1(λℓ)|,(48)

where |ε(λ_ℓ)| ≤ 1 for all λ_ℓ ∈ Ω_ℓ with the exception of the minimum (1_μ) and maximum (|Ω_ℓ|_ℓ) values where |ε(λ_ℓ)| ≤ 1/2. Therefore, the proof follows from a direct computation of the distance Δ(P_ℓ, P̃_ℓ),

Δ(Pℓ,P~ℓ)=∑λℓ∈Ωℓ|Pℓ(λℓ)−P~ℓ(λℓ)|,(49)

=∑λℓ∈ΩℓPℓ(λℓ)−1|Ωu|g−1(λℓ),(50)

=1|Ωu|∑λℓ∈Ωℓ|ε(λℓ)|,(51)

≤1|Ωu||Ωℓ|−2+212,(52)

=|Ωℓ|−1|Ωu|.(53)

□

In terms of the causal compatibility problem, Theorem 2 suggests that if an observed distribution P_𝓥 is compatible with 𝓖, and there exists a functional causal model (𝓖, 𝓕_𝓥, 𝓟_𝓛) which reproduces P_𝓥 (via Equation 11), then it must be close to a rational distribution P̃_𝓥 generated by a functional causal model (𝓖, 𝓕̃_𝓥, 𝓟̃_𝓛) wherein probability distributions for the latent variables 𝓟̃_𝓛 are uniform. The following theorem proves this.

Theorem 3

Let (𝓖, 𝓕_𝓥, 𝓟_𝓛) be a functional causal model with cardinalitiesc_ℓ = |Ω_ℓ| for the latent variables producing distribution P_𝓥. Then there exists a functional causal model (𝓖, 𝓕̃_𝓥, 𝓟̃_𝓛) with cardinalitiesk_ℓ = |Ω̃_ℓ| for the latent variables producing P̃_𝓥where the distributionsP~L={Uℓ:Ω~ℓ→kℓ−1∣ℓ∈L}over the latent variables are uniform. In particular, the distance between P_𝓥and P̃_𝓥is bounded by,

Δ(PV,P~V)≤ε=∑n=1L1n!L(C−1)Kn∈OLCK,(54)

whereC = max{c_ℓ | ℓ ∈ 𝓛}, K = min{k_ℓ | ℓ ∈ 𝓛}, andL = |𝓛| is the number of latent variables.

Proof

The proof relies on Theorem 2 and can be found in Appendix C.□