Axiomatization of Interventional Probability Distributions

One of our central assumptions, Axiom 1, is that cause is transitive; see [12] for a philosophical discussion on the transitivity of the cause. Under the condition of singleton-transitivity and simple assumptions on conditional independence structure of $\mathcal{P}_{\mathrm{do}}$, we show that the causal relations are transitive; see Theorem 10. We provide a definition of causation similar to that in [24]. The concept of direct cause is defined in terms of the conditional independence properties of the interventional family (which is a departure from the widely-known definition [35, page 55]), and from this we define the intervened causal graph, and using these, we define the causal graph; see Section 4.2; this is a major relaxation of assumptions from the current paradigm where it is assumed that such a causal graph exists. The obtained causal graph allows bidirected edges and directed cycles without double edges consisting of a bidirected edge and an arrow. We call this family of graphs bowless directed mixed graphs (BDMGs). The generated graph is the “true” causal graph under the axiomatization, and we show that the definitions related to causal relationships and the graphical notions on the graph are interchangeable (Theorem 20).

One of our main theorems is that, under some additional assumptions, namely intersection and composition, intervened distributions $P_{\mathrm{do}(i)}$ in the interventional family are Markovian to the defined intervened graphs; see Theorem 26. We provide additional axioms (Axioms 2 and 3) to relate $\mathcal{P}_{\mathrm{do}}$ to an observed distribution $P$, and call the interventional family (strongly) observable. We show that the underlying distribution $P$ for an observable interventional family is Markovian to the defined causal graph; see Theorem 28. Therefore, the established theory of causality using SCMs, which mainly relies on the Markov property of the joint distribution of the SCM, could be followed from our theory.

We later provide additional axioms (Axioms 4 and 5) for the case of ancestral causal graphs, to define what we call quantifiable interventional families that allow for measuring causal effects. We show that the quantifiable interventional families are strongly observational; see Theorem 47.

We also compare and contrast our theory with the SCM setting. We show that, for SCMs with certain simple properties (which include transitivity of the cause and are implied by faithfulness), the family of interventions on each node constitutes a strongly observable interventional family, and (even without the transitivity assumption in the case of ancestral graphs) the causal graph generated by the theory presented in this paper is the same as the causal graph associated to the SCM; see Theorem 60.

Our theory is based only on intervening on single variables once at a time. We clearly identify cases (which can only be non-maximal and non-ancestral), where this theory may misidentify some direct causes; see Section 9.

To the best of our knowledge, most of the attempts to abstracting intervention or causality based on intervention in general are substantially different from our approach; see for example [28] for a category-theory approach, and more recent [20] for a measure-theoretic approach.

One such attempt is the seminal work of Dawid on the decision theoretic framework for causal inference; for example, see [8, 9]. Our approach share the same concerns and spirit as that of Dawid’s in focusing on the interventions rather than counterfactuals, as well as trying to justify the existence of the causal graph rather than assuming it, as we have heeded Dawid’s caution [6]. However, the mechanics of the two works are different. First of all, we have not provided any statistical model like Dawid does. In addition, the goal of Dawid’s work is mainly to enable “transfer of probabilistic information from an observational to an interventional setting,” whereas, here, our starting point is interventions. Finally, our approach also covers a much more general class of causal graphs than DAGs, considered by Dawid. We have not included influence diagrams, as proposed by Dawid, in our setting, but believe that it should be possible to derive them together with their conditional-independence constraints using our causal graphs and Markov properties.

A more similar approach to ours is that of Bareinboim, Brito, and Pearl [1]. Like us the authors start with a family of interventional distributions with interventions defined on single variables. A difference is that they only work on atomic interventions, which requires alternative definitions to conditional independence (which are phrased as invariances)—we believe this can be adapted to use stochastic intervention and regular conditional independence. Using this, they define the concept of direct cause, which is different, but of similar to how we define this concept. A major difference is that, in their paper, they provide different notions of compatibility of the interventional family and causal graphs by assuming certain conditions that include the global Markov property—in our work, we do not assume the global Markov property, and generate a graph directly from the interventional family by using direct cause, and prove the Markov property under certain axioms and conditions. Another important difference is that their work relies on the notion of observed initial state distribution to define interventions—as mentioned before, we do not need to rely on observational distribution to derive the causal graph.

The mentioned paper is purely on DAGs, but in a more recent work [2] the method was generalized to include arcs representing latent variables. The notion of Markov property has been replaced by semi-Markov property to deal with this generalization, but the difference between the two methods remains the same as described for the original paper.

In the next section, we provide some preliminary material including novel results on the equivalence of the pairwise and global Markov property in our newly defined class of BDMGs, which are used in the subsequent sections. In Section 3, we define foundational concepts related to interventional families, and provide conditions for the axiom of transitivity of the cause. In Section 4, we define the concepts of direct cause and intervened causal as well as causal graphs, and prove the Markov property of the intervened distribution to the intervened causal graph. In Section 5, we prove the Markov property of the underlying distribution to the causal graph under the observable interventional axioms. In Section 6, we specialize the definitions and results for directed ancestral graphs. In Section 7, we provide additional axioms of quantifiable interventional families for the purpose of measuring causal effects. In Section 8, we show how intervention on SCMs fits within the framework of this paper. In Section 9, we provide cases where only single interventions may misidentify certain direct causes. We conclude the paper in Section 10. In the Appendix, we provide certain proofs of results presented in Section 2.4, including the proof of equivalence of the pairwise and global Markov property for BDMGs.

We will work in the following setting. Let $V$ be a finite set of size $N$. Let $P$ be a probability measure on the product measurable space $\mathcal{X}=\prod_{i\in V}\mathcal{X}_{i}$. For $A\subseteq V$, we let $\mathcal{X}_{A}=\prod_{i\in A}\mathcal{X}_{i}$ and $P^{A}$ be the marginal measure of $P$ on $\mathcal{X}_{A}$ given by

for all measurable $W\subseteq\mathcal{X}_{A}$. We will use the notation $i\mbox{$\>\perp\perp$ }_{P}j$ to mean that the marginal $P^{\left\{{i,j}\right\}}=P^{i,j}$ is the product measure $P^{i}\otimes P^{j}$ on $\mathcal{X}_{i}\times\mathcal{X}_{j}$, so that if $X=(X_{1},\ldots,X_{N})$ is a random vector (defined on some probability space $(\Omega,\mathcal{F},\mathbb{P})$) taking values on $\mathcal{X}$ with law $P$, then $X_{i}$ is independent of $X_{j}$ [7, 17].

For $x_{A}\in\mathcal{X}_{A}$, we let $P(\cdot|x_{A})=\mathbb{P}(X\in\cdot|X_{A}=x_{A})$ denote a regular conditional probability [4] so that in particular, we have the disintegration

for $F\subseteq\mathcal{X}$ measurable. More generally, consider disjoint subsets $A$, $B$, and $C$ of $V$. We will often consider the marginal of a conditional measure, and have a slight abuse of notation that:

We write $A\mbox{$\>\perp\perp$ }_{P}B\,|\,C$ to denote that the measure $P^{A,B}(\cdot|x_{C})=P^{A\cup B}(\cdot|x_{C})$ is a product measure on $\mathcal{X}_{A}\times\mathcal{X}_{B}$ for $P^{C}$-almost all $x_{C}\in\mathcal{X}_{C}$, so that if $X$ has law $P$, then $X_{A}$ is conditionally independent of $X_{B}$ given $X_{C}$. Sometimes, we will simply say that $A$ is conditionally independent of $B$ given $C$ in $P$. In addition, when independence fails, we write $A\nolinebreak{\not\mbox{$\>\perp\perp$ }}_{P}B\,|\,C$ and say that $A$ and $B$ are conditionally dependent given $C$ in $P$.

A probability distribution $P$ is always a semi-graphoid [21], i.e., it satisfies the four following properties for disjoint subsets $A$, $B$, $C$, and $D$ of $V$:

1. 1.

   $A\mbox{$\>\perp\perp$ }_{P}B\,|\,C$ if and only if $B\mbox{$\>\perp\perp$ }_{P}A\,|\,C$ (symmetry);

2. 2.

   if $A\mbox{$\>\perp\perp$ }_{P}B\cup D\,|\,C$, then $A\mbox{$\>\perp\perp$ }_{P}B\,|\,C$ and $A\mbox{$\>\perp\perp$ }_{P}D\,|\,C$ (decomposition);

3. 3.

   if $A\mbox{$\>\perp\perp$ }_{P}B\cup D\,|\,C$, then $A\mbox{$\>\perp\perp$ }_{P}B\,|\,C\cup D$ and $A\mbox{$\>\perp\perp$ }_{P}D\,|\,C\cup B$ (weak union);

4. 4.

   if $A\mbox{$\>\perp\perp$ }_{P}B\,|\,C\cup D$ and $A\mbox{$\>\perp\perp$ }_{P}D\,|\,C$, then $A\mbox{$\>\perp\perp$ }_{P}B\cup D\,|\,C$ (contraction).

Notice that the reverse implication of contraction clearly holds by decomposition and weak union. We so use three different properties of conditional independence that are not always satisfied by probability distributions:

1. 5.

   if $A\mbox{$\>\perp\perp$ }_{P}B\,|\,C\cup D$ and $A\mbox{$\>\perp\perp$ }_{P}D\,|\,C\cup B$, then $A\mbox{$\>\perp\perp$ }_{P}B\cup D\,|\,C$ (intersection);

2. 6.

   if $A\mbox{$\>\perp\perp$ }_{P}B\,|\,C$ and $A\mbox{$\>\perp\perp$ }_{P}D\,|\,C$, then $A\mbox{$\>\perp\perp$ }_{P}B\cup D\,|\,C$ (composition);

3. 7.

   if $i\mbox{$\>\perp\perp$ }_{P}j\,|\,C$ and $i\mbox{$\>\perp\perp$ }_{P}j\,|\,C\cup\{k\}$, then $i\mbox{$\>\perp\perp$ }_{P}k\,|\,C$ or $j\mbox{$\>\perp\perp$ }_{P}k\,|\,C$ (singleton-transitivity),

where $i$, $j$, and $k$ are single elements. A semi-graphoid distribution that satisfies intersection is called graphoid. If the distribution $P$ is a regular multivariate Gaussian distribution, then $P$ is a singleton-transitive compositional graphoid; for example see [33] and [21]. If $P$ has strictly positive density, it is always a graphoid; see, for example, Proposition 3.1 in [17].

Finally, we define the concept of ordered stabilities [29]. We say that $P$ satisfies ordered upward- and downward-stability w.r.t. an order $\leq$ of $V$ if the following hold:

1. $\bullet$  

   if $i\mbox{$\>\perp\perp$ }_{P}j\,|\,C$, then $i\mbox{$\>\perp\perp$ }_{P}j\,|\,C\cup\{k\}$ for every $k\in V\setminus\{i,j\}$ such that $i<k$ or $j<k$ (ordered upward-stability);

2. $\bullet$  

   if $i\mbox{$\>\perp\perp$ }_{P}j\,|\,C$, then $i\mbox{$\>\perp\perp$ }_{P}j\,|\,C\setminus\{k\}$ for every $k\in V\setminus\{i,j\}$ such that $i\nless k$, $j\nless k$, and $\ell\nless k$ for every $\ell\in C\setminus\{k\}$ (ordered downward-stability).

We usually refer to a graph as an ordered pair $G=(V,E)$, where $V$ is the node set and $E$ is the edge set. When nodes $i$ and $j$ are the endpoints of an edge, we call them adjacent, and write $i\sim j$, and otherwise $i\nsim j$.

We consider two types of edges: arrows ($i\mbox{$\hskip 0.50003pt\frac{\hskip 4.89998pt\hskip 4.89998pt}{\hskip 4.89998pt}\!\!\!\!\!\succ\!\hskip 1.07639pt$}j$) and bidirected edges or arcs ($i\mbox{$\hskip 0.59998pt\prec\!\!\!\!\!\frac{\hskip 4.89998pt\hskip 4.89998pt}{\hskip 4.89998pt}\!\!\!\!\!\succ\!\hskip 1.07639pt$}j$). We do not consider graphs that have simultaneous third type of edge: undirected edges or lines ($i\mbox{$\,\frac{\hskip 4.89998pt\hskip 4.89998pt\;}{\hskip 4.89998pt\hskip 4.89998pt}$}j$). We only allow for the possibility of multiple edges between nodes when they are arrows in two different directions between $i$ and $j$, i.e., $i\mbox{$\hskip 0.50003pt\frac{\hskip 4.89998pt\hskip 4.89998pt}{\hskip 4.89998pt}\!\!\!\!\!\succ\!\hskip 1.07639pt$}j$ and $i\mbox{$\hskip 0.50003pt\prec\!\!\!\!\!\frac{\hskip 4.89998pt\hskip 4.89998pt}{\hskip 4.89998pt}$}j$, which we call parallel arrows. This means that we do not allow bows, i.e., a multiple edge of arrow and arc, to appear in the graph.

A subgraph of a graph $G_{1}$ is graph $G_{2}$ such that $V(G_{2})\subseteq V(G_{1})$ and $E(G_{2})\subseteq E(G_{1})$ and the assignment of endpoints to edges in $G_{2}$ is the same as in $G_{1}$. An induced subgraph by nodes $A\subseteq V$ is the subgraph that contains all and only nodes in $A$ and all edges between two nodes in $A$.

A walk is a list $\langle v_{0},e_{1},v_{1},\dots,e_{k},v_{k}\rangle$ of nodes and edges such that for $1\leq i\leq k$, the edge $e_{i}$ has endpoints $v_{i-1}$ and $v_{i}$. A path is a walk with no repeated node or edge. When we define a path, we only write the nodes (and not the edges). A maximal set of nodes in a graph whose members are connected by some paths constitutes a connected component of the graph. A cycle is a walk with no repeated nodes or edges except for $v_{0}=v_{k}$.

We call the first and the last nodes endpoints of the path and all other nodes inner nodes. A path can also be seen as a certain type of connected subgraph of $G$; a subpath of a path $\pi$ is an induced connected subgraph of $\pi$. For an arrow $j\mbox{$\hskip 0.50003pt\frac{\hskip 4.89998pt\hskip 4.89998pt}{\hskip 4.89998pt}\!\!\!\!\!\succ\!\hskip 1.07639pt$}i$, we say that the arrow is from $j$ to $i$. We also call $j$ a parent of $i$, $i$ a child of $j$ and we use the notation $\mathrm{pa}(i)$ for the set of all parents of $i$ in the graph. In the cases of $i\mbox{$\hskip 0.50003pt\frac{\hskip 4.89998pt\hskip 4.89998pt}{\hskip 4.89998pt}\!\!\!\!\!\succ\!\hskip 1.07639pt$}j$ or $i\mbox{$\hskip 0.59998pt\prec\!\!\!\!\!\frac{\hskip 4.89998pt\hskip 4.89998pt}{\hskip 4.89998pt}\!\!\!\!\!\succ\!\hskip 1.07639pt$}j$ we say that there is an arrowhead at $j$ or pointing to $j$. A path $\langle i=i_{0},i_{1},\dots,i_{n}=j\rangle$ (or a cycle where $i=j$) is directed from $i$ to $j$ if all $i_{k}i_{k+1}$ edges are arrows pointing from $i_{k}$ to $i_{k+1}$. If there is a directed path from $i$ to $j$, then node $i$ is an ancestor of $j$ and $j$ is a descendant of $i$. We denote the set of ancestors of $j$ by $\mathrm{an}(j)$; unlike some authors, we do not allow $j\in\mathrm{an}(j)$. Similarly, we define an ancestor of a set of nodes $A\subset V$ given by $\mathrm{an}(A):=[\bigcup_{j\in A}\mathrm{an}(j)]\setminus A$. If necessary, we might write $\mathrm{an}_{G}$ to specify that this is the set of ancestors in $G$.

A strongly connected component of a graph is the set of nodes that are mutually ancestors of each other, or it is a single node if that node does not belong to any directed cycle. It can be observed that nodes of the graph are partitioned into strongly connected components. We denote the members of the strongly connected component containing node $i$ by $\mathrm{sc}(i)$.

A tripath is a path with three nodes. The inner node $t$ in each of the three tripaths

is a collider (or a collider node) and the inner node of any other tripath is a non-collider (or a non-collider node) on the tripath or, more generally, on any path of which the tripath is a subpath; i.e. a node is a collider if two arrowheads meet. A path is called a collider path if all its inner nodes are colliders.

The most general class of graphs that naturally arises from the theory presented in this paper is what we call the bowless directed mixed graph (BDMG), which consists of arrows and arcs, and the only multiple edges are parallel arrows (i.e., they are bowless).

Ancestral graphs [27] are graphs with arcs and arrows with no directed cycles and no arcs $ij$ such that $i\in\mathrm{an}(j)$. Acyclic directed mixed graphs (ADMGs) [26] are graphs with arcs and arrows with no directed cycles. In other words, BDMGs unify ADMGs without bows and directed cycles. BDMGs also trivially contain the class of directed ancestral graphs, i.e., ancestral graphs [27] without lines. These all also contain directed acyclic graphs (DAGs) [14], which are graphs with only arrows and no directed cycles.

The class of BDMGs is a subclass of directed mixed graphs, introduced in [3], which is a very general class of graphs with arrows and arcs that allow for directed cycles. Later on, we will use some definitions and results originally defined for directed mixed graphs.

In this paper, we will need global and pairwise Markov properties for BDMGs. In order to introduce the global Markov property, we need to define the concept of $\sigma$-separation for directed mixed graphs (in fact, originally defined for the larger class of directed graphs with hyperedges in [11]).

A path $\pi=\langle i=i_{0},i_{1},\cdots,i_{n}=j\rangle$ is said to be $\sigma$-connecting given $C$, which is disjoint from $i,j$, if all its collider nodes are in $C\cup\mathrm{an}(C)$; and all its non-collider nodes $i_{r}$ are either outside $C$, or if there is an arrowhead at $i_{r-1}$, then $i_{r-1}\in\mathrm{sc}(i_{r})$ and if there is an arrowhead at $i_{r+1}$ on $\pi$ then $i_{r+1}\in\mathrm{sc}(i_{r})$. For disjoint subsets $A,B,C$ of $V$, we say that $A$ and $B$ are $\sigma$-separated given $C$, and write $A\,\mbox{$\perp$}\,_{\sigma}B\,|\,C$, if there are no $\sigma$-connecting paths between $A$ and $B$ given $C$.

In the case where there are no directed cycles in the graph, $\sigma$-separation reduces to the $m$-separation of [27]; recall that $\pi$ is $m$-connecting given $C$ if all its collider nodes are in $C\cup\mathrm{an}(C)$; and all its non-collider nodes are outside $C$. In addition, if there are no arcs in the graph, i.e., the graph is a DAG, it reduces to the well-known $d$-separation [21].

We call two graphs Markov equivalent if they induce the same set of conditional separations.

A probability distribution $P$ defined over $V$ satisfies the global Markov property w.r.t. a bowless directed mixed graph $G$, or is simply Markovian to $G$, if for disjoint subsets $A$, $B$, and $C$ of $V$, we have

If $G$ is an ancestral graph (or a DAG), then $\,\mbox{$\perp$}\,_{\sigma}$ will be replaced by $\,\mbox{$\perp$}\,_{m}$ (or $\,\mbox{$\perp$}\,_{d}$) in the definition of the global Markov property.

If, in addition to the global Markov property, the other direction of the implication holds, i.e., $A\,\mbox{$\perp$}\,_{\sigma}B\,|\,C\iff A\mbox{$\>\perp\perp$ }B\,|\,C$, then we say that $P$ and $G$ are faithful. A weaker condition of adjacency-faithfulness [25, 37] states that for every edge between $k$ and $j$ in $G$, there are no independence statements $k\mbox{$\>\perp\perp$ }_{P}j\,|\,C$ for any $C$.

We now define that a distribution $P$ satisfies the pairwise Markov property (PMP) w.r.t. a bowless directed mixed graph $G$, if for every pair of non-adjacent nodes $i,j$ in $G$, we have

This is the same wording as that of the pairwise Markov property for the subclass of ancestral graphs; see [19].

We prove the equivalence of the pairwise and global Markov properties, which shall be used later for causal graphs and Theorem 28. The proofs are presented in the Appendix as they are not the main focus of this manuscript.

We also define the converse of the pairwise Markov property. We say that $P$ satisfies the converse pairwise Markov property w.r.t. $G$ if an edge between $i$ and $j$ in $G$ implies that

Notice that faithfulness and adjacency-faithfulness of $P_{\mathcal{C}}$ and $G_{\mathcal{C}}$ imply the converse pairwise Markov property; see [30].

A graph is called maximal if the absence of an edge between $i$ and $j$ corresponds to a conditional separation statement for $i$ and $j$, i.e. there exists for some $C$ a statement of form $i\,\mbox{$\perp$}\,j\,|\,C$. Notice from the definition of $\sigma$-separation that graphs with chordless directed cycles (i.e., having two non-adjacent nodes in a cycle) are not maximal.

We have the following corresponding converse to Theorem 2.

We call a non-adjacent pair of nodes which cannot be $\sigma$-separated in a non-maximal graph, regardless of what to condition on, an inseparable pair, and a non-adjacent pair of nodes which can be $\sigma$-separated in a maximal or non-maximal graph a separable pair.

For BDMGs, we define a primitive inducing path (PIP) to be a path $\langle i,q_{1},\cdots,q_{r},j\rangle$ (with at least $3$ nodes) between $i$ and $j$, where

1. (i)

   all edges $q_{m}q_{m+1}$ are either arcs or an arrow where $q_{m}\in\mathrm{sc}(q_{m+1})$ except for the first and last edges, which may be $i\mbox{$\hskip 0.50003pt\frac{\hskip 4.89998pt\hskip 4.89998pt}{\hskip 4.89998pt}\!\!\!\!\!\succ\!\hskip 1.07639pt$}q_{1}$ or $q_{r}\mbox{$\hskip 0.50003pt\prec\!\!\!\!\!\frac{\hskip 4.89998pt\hskip 4.89998pt}{\hskip 4.89998pt}$}j$ (without being in the same connected component);

2. (ii)

   for all inner nodes, we have $q_{m}\in\mathrm{an}(\{i,j\})$, i.e., they are in ancestors of $i$ or $j$.

PIPs were originally defined for the case of ancestral graphs (where they were allowed to be an edge) [27]. We show in the Appendix the result below:

In addition, notice from Proposition 3 that if $P$ is Markovian to $G$, then a pair $i,j$ being a separable pair is equivalent to the separation $i\,\mbox{$\perp$}\,_{\sigma}j\,|\,\mathrm{an}{\{i,j\}}$.

We say that a graph $G=(V,E)$ admits a valid order $\leq$ if for nodes $i$ and $j$ of $G$, $i\mbox{$\hskip 0.50003pt\frac{\hskip 4.89998pt\hskip 4.89998pt}{\hskip 4.89998pt}\!\!\!\!\!\succ\!\hskip 1.07639pt$}j$ implies that $i>j$; and $i\mbox{$\hskip 0.59998pt\prec\!\!\!\!\!\frac{\hskip 4.89998pt\hskip 4.89998pt}{\hskip 4.89998pt}\!\!\!\!\!\succ\!\hskip 1.07639pt$}j$ implies that $i$ and $j$ are incomparable. Notice that this specifies the partial order via its cover relations. In fact, this order can is used as the order w.r.t. which ordered upward- and downward-stability hold for graph separations; see [29].

Finally, for ancestral graphs, the $m$-separation holds for every set “between” parents and ancestors. We will use this fact later:

The theory in this paper does not use or assume structural causal models (SCMs) (also known as the structural equation models) [22, 32]. We define SCMs here as they are an interesting special case of our theory, for which intervention could be easily conceptualized.

Here, we define SCMs for the class of BDMGs as a simplified version of SCMs defined for directed (mixed) graphs in [3]. Consider a graph $G$ with $N$ nodes, which in the context of causal inference is often referred to as the “true causal graph.” A structural causal model $\mathfrak{C}$ associated with $G$ is defined as a collection of $N$ equations

where $\mathrm{pa}_{G}(i)$ is defined on $G$ and $\epsilon_{i}$ might be called noises; for any subsets $A$ and $B$, we require that $\epsilon_{A}\mbox{$\>\perp\perp$ }\epsilon_{B}$ if and only if, in $G$, there is no arc between any node in $A$ and any node in $B$. In this paper, we usually refer to an SCM as $\mathcal{C}$ and its joint distribution as $P_{\mathcal{C}}$.

In the more widely-used case where $G$ is a DAG, all the $\epsilon_{i}$ are jointly independent. For both mathematical and causal discussions on SCMs with DAGs, see [24]. When directed cycles are existent, some solvability conditions are required in order for the theory of SCMs to work properly; for this and for more general discussion, see [3].

Standard interventions are defined quite naturally when functional equations are specified, as in the case of SCMs: By intervening on $X_{i}$ we replace the equation associated to $X_{i}$ by $X_{i}=\tilde{X}_{i}$, where $\tilde{X}_{i}$ is independent of all other noises; it is not necessary that $\tilde{X}_{i}$ has the same distribution as $X_{i}$. We are concerned with a similar type of intervention in this paper – this is a special case of the so-called stochastic intervention [16], where some parental set of $X_{i}$ might still exist after intervention on $X_{i}$; see also [24]. A more special type of intervention is called perfect intervention (or surgical intervention), where it puts a point mass on a real value $a$ – this is the original idea of do-calculus [22], and is often denoted by $\mathrm{do}(X_{i}=a)$.

An important result for SCMs, which facilitates causal inference using them, is that the joint distribution of an SCM is Markovian to its associated graph; see [34, 22] for the case of DAGs, [30] for directed ancestral graphs, and [3] for directed mixed graphs.

Again, let $V$ be a finite set of size $N$. Consider a family of distributions $\mathcal{P}_{\mathrm{do}}=\{P_{\mathrm{do}(i)}\}_{i\in V}$, where each $P_{\mathrm{do}(i)}$ is defined over the same state space $\mathcal{X}=\prod_{i\in V}\mathcal{X}_{i}$. We refer to $\mathcal{P}_{\mathrm{do}}$ as an interventional family (of distributions). For $(\tilde{X}_{j})_{j\in V}$ a random vector with distribution $P_{\mathrm{do}(i)}$, we think of $P_{\mathrm{do}(i)}$ as the interventional distribution after intervening on some variable $X_{i}$.

For $k\in V$, we define the set of the causes of $k$ as

we rarely, simultaneously, have to consider two different interventional families at once. Thus, if $i\in\mathrm{cause}({k})$, then, for $(\tilde{X}_{j})_{j\in V}$, we have that $\tilde{X}_{k}$ is dependent on $\tilde{X}_{i}$. Notice that, by convention, $k\notin\mathrm{cause}({k})$. For a subset $A\subseteq V$, we define $\mathrm{cause}({A})=(\bigcup_{k\in A}\mathrm{cause}({k}))\setminus A$.

The definition of the cause is identical to what is know in the literature as the “existence of the total causal effect” (see, e.g., Definition 6.12 in [24]). Its combination with $\mathcal{P}_{\mathrm{do}}$ meets the intuition behind cause and intervention: after intervention on a variable $X_{i}$, it is dependent on a variable $X_{j}$ if and only if it is a cause of that variable.

The above setting is compatible with the well-known intervention for SCMs; for a comprehensive discussion on this, see Section 8. We will often illustrate our theory with simple examples of SCMs and standard intervention on a single node.

We can also define the set of effects of $i$ denoted by $\mathrm{eff}({i})$ by $i\in\mathrm{cause}({k})\iff k\in\mathrm{eff}({i})$. We take note of the following useful fact.

We call a subset $S\subseteq V$ a causal cycle if for every $i,k\in S$, we have $k\in\mathrm{cause}({i})\cap\mathrm{eff}({i})$. We write $\mathrm{cc}({i})$ to denote the causal cycle containing $i$.

Under the composition property, we have the following independence. Let $\mathrm{neff}({i}):=V\setminus\mathrm{eff}({i})$ denote the subset of $V$ that contains members that are not an effect of $i$.

We now say that $\mathcal{P}_{\mathrm{do}}$ is a transitive interventional family if the following axiom holds.

Notice also that $P_{\mathrm{do}(i)}$ in transitive interventional families are restricted: Axiom 1 places constraints between different $P_{\mathrm{do}(i)}$ since $\mathrm{cause}({k})$ depends on all $P_{\mathrm{do}(i)}$.

Under singleton-transitivity, we have sufficient conditions for Axiom 1 to hold. Notice that that these conditions are not satisfied in general, even in the case of an SCM with standard interventions.

In the next example, we show that we cannot drop the singleton transitivity assumption in Theorem 10.

Notice that only knowing the causal ordering cannot yield a graph, since, for example, for $i\mbox{$\hskip 0.50003pt\frac{\hskip 4.89998pt\hskip 4.89998pt}{\hskip 4.89998pt}\!\!\!\!\!\succ\!\hskip 1.07639pt$}j\mbox{$\hskip 0.50003pt\frac{\hskip 4.89998pt\hskip 4.89998pt}{\hskip 4.89998pt}\!\!\!\!\!\succ\!\hskip 1.07639pt$}k$, there is no way to distinguish the two graphs corresponding to whether an additional $i\mbox{$\hskip 0.50003pt\frac{\hskip 4.89998pt\hskip 4.89998pt}{\hskip 4.89998pt}\!\!\!\!\!\succ\!\hskip 1.07639pt$}k$ exists in the graph or not. For this reason, we need to define the concept of direct cause.

In order to define direct and intervened cause in the general case, we need an iterative procedure:

1. 1.

   For each $i,k\in V$, start with $\mathrm{dcause}({k}):=\mathrm{cause}({k})$, $\mathrm{\iota cause}_{i}(k):=\mathrm{cause}({k})$, and $S(\mathcal{P}_{\mathrm{do}})$ an empty graph with node set $V$;

2. 2.

   Redefine

   $$\mathrm{dcause}({k}):=\{i:\hskip 3.50006pti\in\mathrm{dcause}({k}),\hskip 3.50006pti\nolinebreak{\not\mbox{$\>\perp\perp$ }}_{P_{\mathrm{do}(i)}}k\,|\,\mathrm{\iota cause}_{i}(k)\setminus\{i\}\};$$

3. 3.

   Generate $S(\mathcal{P}_{\mathrm{do}})$ by setting arrows from $i$ to $k$, i.e., $i\mbox{$\hskip 0.50003pt\frac{\hskip 4.89998pt\hskip 4.89998pt}{\hskip 4.89998pt}\!\!\!\!\!\succ\!\hskip 1.07639pt$}k$ if $i\in\mathrm{dcause}({k})$;

4. 4.

   Generate the graph $S_{i}(\mathcal{P}_{\mathrm{do}})$ by removing all arrows pointing to $i$ from $S(\mathcal{P}_{\mathrm{do}})$;

5. 5.

   Redefine $\mathrm{\iota cause}_{i}(k):=\mathrm{an}_{S_{i}(\mathcal{P}_{\mathrm{do}})}(k)$;

6. 6.

   If $S(\mathcal{P}_{\mathrm{do}})$ is modified by Step 3, then go to Step 2; otherwise, output $\mathrm{dcause}({k})$, $\mathrm{\iota cause}_{i}(k)$, $S(\mathcal{P}_{\mathrm{do}})$, and $S_{i}(\mathcal{P}_{\mathrm{do}})$.

We call $\mathrm{dcause}({k})$ the set of the direct causes of $k$, and $\mathrm{\iota cause}_{i}(k)$ the set of intervened causes of $k$ after intervention on $i$. We also call $S(\mathcal{P}_{\mathrm{do}})$ the causal structure, and $S_{i}(\mathcal{P}_{\mathrm{do}})$ the $i$-intervened causal structure.

Notice that, since in the iteration $\mathrm{dcause}({k})$ is getting smaller, the procedure will stop.

Notice also that, by convention, $k\notin\mathrm{dcause}({k})\cup\mathrm{\iota cause}_{i}(k)$, for any $i\in V$. We also let $\mathrm{dcause}({A})=(\bigcup_{k\in A}\mathrm{dcause}({k}))\setminus A$, and $\mathrm{\iota cause}_{i}(A)=(\bigcup_{k\in A}\mathrm{\iota cause}_{i}(k))\setminus A$.

As seen by definition, “cause” is a universal concept: no matter how large the system of random variables is, as long as it contains the two investigated random variables, the marginal dependence of those variables stays intact. On the other hand, “direct cause” depends on the system of variables in which the two investigated variables lie.

The below example shows why we need $\mathrm{\iota cause}_{i}(k)$ as opposed to $\mathrm{cause}({k})$ in the definition of $\mathrm{dcause}({k})$; see also Section 7.2.

The next proposition can be thought of as a proxy for the pairwise Markov property, and will be used in our proofs of the Markov property for our casual graphs.

We are now ready to define a graph that demonstrates the causal relationships by capturing the direct causes as well as non-causal dependencies due to latent variables.

Given an interventional family $\mathcal{P}_{\mathrm{do}}$, and $i\in V$, we define the $i$-intervened graph, denoted by $G_{i}(\mathcal{P}_{\mathrm{do}})$ to be the $i$-intervened causal structure, where, in addition, for each pair of non-adjacent nodes $j,k\in V$ that are distinct from $i$, we place an arc between $j$ and $k$, i.e., $j\mbox{$\hskip 0.59998pt\prec\!\!\!\!\!\frac{\hskip 4.89998pt\hskip 4.89998pt}{\hskip 4.89998pt}\!\!\!\!\!\succ\!\hskip 1.07639pt$}k$, if

Thus with (3), we put an arc if one of the interventions $P_{\mathrm{do}(i)}$ suggests the presence of a latent variable.

We also define the causal graph, denoted by $G(\mathcal{P}_{\mathrm{do}})$, to be the causal structure, where, in addition, for each pair of nodes $j,k$ that are not adjacent by an arrow, we place an arc between them if the $jk$-arc exists in $G_{i}(\mathcal{P}_{\mathrm{do}})$ for every $i\in V$ that is distinct from $j$ and $k$.

For an interventional family $\mathcal{P}_{\mathrm{do}}$, our definitions now allow us to use the notions $\mathrm{pa}_{G(\mathcal{P}_{\mathrm{do}})}(k)$ and $\mathrm{dcause}({k})$ interchangeably. For a transitive $\mathcal{P}_{\mathrm{do}}$, we have that, moreover, other causal terminologies on $\mathcal{P}_{\mathrm{do}}$ and the graph terminologies on $G(\mathcal{P}_{\mathrm{do}})$ can be used interchangeably: