Causality: a decision theoretic approach

TL;DR

This paper introduces a decision theoretic framework enabling decision makers to explicitly model and identify causal relationships through their choices, extending Savage's model and connecting to Pearl's causal graphs.

Contribution

It extends Savage's decision theory to incorporate causal modeling via interventions and provides axioms to represent causal models as directed acyclic graphs.

Findings

Provides axioms for representing causal models as DAGs

Characterizes when a causal model aligns with Pearl's framework

Enables identification of causal structure from probabilistic choices

Abstract

We propose a decision theoretic framework that allows a decision maker to express its causal model of the world. We extend the model of Savage (1972) by allowing the decision maker (DM) to choose policy interventions prior to choosing acts over the nonintervened variables. We define what it means for the DM's choices to express the DM's belief that the relation between some variables is causal. We provide axioms characterizing when the DM's causal model, as expressed through the DM's choices, is represented as a directed acyclic graph. A final axiom characterizes when the DM's causal model has a representation like the one in Pearl (1995). Consequently, under this additional axiom one can apply Pearl's results to identify the DM's causal model from the DM's probabilistic model.

Equations

$$\mu (A,E,L)=\mu (A)\mu (E|A)\mu (L|A)$$

$$f{\succ }_{x_j,x_{\mathcal{K}}}g$$

$$f{\succ }_{x_j,x_{\mathcal{K}}}g\iff f{\succ }_{x_{\mathcal{K}}}g\iff f{\succ }_{x_j^{\prime },x_{\mathcal{K}}}g,$$

$$f{\succ }_{x_j,x_{\mathcal{K}}}g$$

$$\text{mathbbm}{1}_{x_{\mathcal{K}}}f{\succ }_{x_{\mathcal{H}}}\text{mathbbm}{1}_{x_{\mathcal{K}}}g\iff \text{mathbbm}{1}_{x_{\mathcal{K}}}\text{mathbbm}{1}_{x_{\mathcal{J}}}f{\succ }_{x_{\mathcal{H}}}\text{mathbbm}{1}_{x_{\mathcal{K}}}\text{mathbbm}{1}_{x_{\mathcal{J}}}g.$$

$$i{\perp }_{\mathcal{H}}\mathcal{J}|\mathcal{K}.$$

$$i{\perp }_{\mathcal{H}}j|\mathcal{K}.$$

$$f{\succ }_{p}g\iff \sum _{x_{\mathcal{N}(p)}\in X_{\mathcal{N}}(p)}{u}_p(f(x_{\mathcal{N}(p)})){\mu }_p(x_{\mathcal{N}(p)})>\sum _{x_{\mathcal{N}(p)}\in X_{\mathcal{N}}(p)}{u}_p(g(x_{\mathcal{N}(p)})){\mu }_p(x_{\mathcal{N}(p)}).$$

$${\mu }_{x_{-\{i,j\}}}(x_i|x_j)={\mu }_{x_{-\{i\}}}(x_i).$$

$$i/{\perp }_{\mathcal{H}}(Ca(i)\setminus (\mathcal{J}\cup \mathcal{H}))|\mathcal{J}$$

$$i{\perp }_{K}j|(Ca(i)\cup Ca(j)\cup \mathcal{J})\setminus \mathcal{K}$$

$$\text{mathbbm}{1}_{\{x_{Ca(i)}\}}f\succ \text{mathbbm}{1}_{\{x_{Ca(i)}\}}g\iff \text{mathbbm}{1}_{x_{Ca(i)\setminus \mathcal{J}}}f{\succ }_{x_{\mathcal{J}}}\text{mathbbm}{1}_{x_{Ca(i)\setminus \mathcal{J}}}g.$$

$$(\forall x\in {\Pi }_{i\in \mathcal{N}}{X}_i),\mu (x)={\Pi }_{i=1}^N\mu (x_i|ND(i)).$$

$$\mu (x)$$

$$(\forall ({\mathcal{T}}_i{)}_{i\in \mathcal{N}})({\mathcal{T}}_i\subset Pa(i)),\text{ if }\mu (x)$$

$$\text{mathbbm}{1}_{\{x_{Ca(i)}\}}f\succ \text{mathbbm}{1}_{\{x_{Ca(i)}\}}g\iff \text{mathbbm}{1}_{x_{Ca(i)\setminus \mathcal{J}}}f{\succ }_{x_{\mathcal{J}}}\text{mathbbm}{1}_{x_{Ca(i)\setminus \mathcal{J}}}g.$$

$${\mu }_e(l)$$

$$A=\alpha$$

$$E=2A+\epsilon$$

$$L=1000A+5\lambda +2000E$$

$$Pr(L\le l|E=e)=Pr(1000\alpha +5\lambda +2000(2\alpha +\epsilon )\le l|(2\alpha +\epsilon )=e).$$

$$A=\alpha$$

$$L=1000A+5\lambda +2000E$$

$$A=\alpha$$

$$L=1000\alpha +5\lambda +2000E$$

$${\mu }_i(x_i)=\varphi (\{\epsilon :h_i(x_{Pa(i)},{\epsilon }_i)=x_i\}),(i\in \{1,...,N\}),$$

$$i{\perp }_{\mathcal{K}}(Ca(j)\setminus \mathcal{K}\setminus Ca(i))|(Ca(i)\setminus \mathcal{K})$$

$$C{a}^0(j)$$

$$C{a}^1(j)$$

$$(...)$$

$$C{a}^k(j)$$

$$i\perp C{a}^0(j)|Ca(i).$$

$$i\perp Ca(j)|{\cup }_{n=0}^{k}C{a}^n(j).$$

$${\mu }_{x_{\mathcal{J}}}(x_i|x_{Ca(i)\setminus \mathcal{J}})=\mu (x_i|x_{Ca(i)}).$$

Full text

Keywords: causality, decision theory, subjective expected utility, axioms, representation theorem, intervention preferences, Bayesian graphs

JEL classification: D80, D81

1 Introduction

Consider an econometrician (say, Alex) who is interested in the causal relation among the intellectual ability ($A$), education level ($E$), and lifetime earnings ($L$) of a representative citizen (say, Mr. Kane). Alex maintains that both education and ability cause lifetime earnings, so that the observed dependence of $L$ on $E$ includes both the direct causal effect of $E$ and $A$ on $L$. Alex’s objective is to quantify the direct causal effect that $E$ – and $E$ alone – has on $L$. There is a plethora of models that Alex can use to isolate the direct causal effect of education on lifetime earnings; see, for example, Rosenmbaum-Rubin [16], Pearl [14], and recent books by Imbens and Rubin [9] and Hernan and Robins [7].

For Alex to decide which model best quantifies the causal effect of $E$ on $L$, Alex must first have in mind a definition of what causality means; then, Alex should choose the model of causality that best fits with her definition of causality. A natural definition of causality is that a variable (e.g., education) causes another variable (e.g., lifetime earnings) if a ceteris paribus policy on the presumed cause affects the distributions of the presumed consequence. That is, education causes earnings if a policy that exogenously changes education levels without affecting any other variable affects the distribution of earnings. As such, assessing causality requires information beyond that which is contained in observed correlation structures. Unfortunately, applied economics research is generally observational: researchers typically do not have access to a laboratory where ceteris paribus changes in causes can be implemented as a means of quantifying their effect on consequences. Instead, economists often must content themselves with observations of the realized values for education, earnings and ability, rather than direct and exogenous interventions on these variables. Economists are thus forced to conduct causal analysis based only on independence structures alone, which generates a natural disconnect with our understanding of causality as a phenomenon that transcends the information contained in correlations.333While, technically, the term correlation means a linear statistical dependency, in this paper, we use this term in its vernacular sense to denote any sort of statistical dependence.

The observational nature of applied economics research implies that there is a need for a theory that explicitly bridges the gap between quantifications of causality based on correlation structures and our natural understanding of causality, centered around ceteris paribus policy interventions. Such a theory should first provide a formal definition of causality based on the effect that the policy interventions of a set of variables have on the distribution of the nonintervened variables. Second, the theory should provide a numerical representation to quantify causal effects. Third, a theorem should prove that there is a bijection between the definition of causality and the proposed numerical quantification. Finally, for the purpose of practical applicability, the theory should provide conditions under which the quantification can be expressed purely in terms of correlation structures.

This paper proposes a decision-theoretic model in which causal effects are defined purely in terms of policy interventions affecting variables; furthermore, we provide a numerical representation of such a definition. This representation is similar to the model pioneered by Pearl [14], expressed in the language of Directed Acyclic Graphs (or DAGs). We then prove the following theorem: for any two variables (say, $X$ and $Y$), the statement “$X$ causes $Y$” (in the sense of our decision-theoretic definition) is true if, and only if, the representing DAG includes a direct arrow from $X$ to $Y$. We also provide conditions under which our decision-theoretic definition of causality can be quantified purely in terms of the correlation structure of the relevant variables. This bridges the gap between our understanding of causation as a phenomenon centered around policy interventions and the practical limitation that economics is observational in nature. As such, this exercise yields a model that can be used in empirical research together with a microeconomic foundation for why such a model should be used. Further investigation of the differences between our representation and Pearl’s model provides additional insights into Pearl’s model.

The remainder of the paper is organized as follows. Section 2 provides a heuristic overview of the paper based on the three-variable example in this introduction. Section 3 outlines the model, and Section 4 provides our formal definition of causality. Sections 5 and 6 present our axioms and representation, respectively. Section 7 contains our two representation theorems, and finally, Section 8 contains a literature review. Since this paper combines elements from different literatures, we do not yet have the tools nor the specific language to provide a detailed literature review; hence, we delay it. Readers already familiar with the Pearl model, Markov representations, and the do-probability formalism can skip to Section 8 without loss of continuity.

2 A Heuristic Analysis

This paper connects ideas from three areas of study: decision theory, Bayesian graphs, and statistical causality. This section shows how these literatures come together in our paper, using the three-variable example from the introduction (the study of the $A$bility, $E$ducation, and $L$ifetime earnings of a representative agent). We begin by illustrating how axiomatic exercises help provide foundations for statistical models – in this case, models of causality. We then provide a bird’s eye view of our model. We also briefly discuss the causal model most closely related to the one our axioms identify, Pearl [14]. In doing so, we highlight some concerns with that model and how our model addresses them.

On axiomatics.

Axiomatic exercises are useful as a guideline for selecting among different numerical models for empirical research. The purpose of such exercise is to provide a link between some numerical model and the way a rational decision maker (henceforth, DM) approaches the issue of interest (in this case, causality). In this paper, the DM is the analyst’s econometric model. Indeed, an econometric model is a tool that takes a problem of decision making under uncertainty – should a firm produce high or low output? Should we give a patient drug A or drug B? Should we implement tax reform X or maintain the status quo? – and, after a suitable treatment of the uncertainty (say, estimating parameters in some regression), produces a recommendation, the firm should produce high output, the patient should not be given the drug, tax reform X is preferred to the status quo. The role of the DM’s beliefs is played by the probability laws the researcher feeds into the numerical model (presumably, obtained via some treatment of available data), and the role of axioms is normative; they provide rules that the econometric model should follow when selecting among the different alternatives. Henceforth, all references to a “DM” refer to the researcher’s statistical model, and all references to the“DM’s beliefs” refer to the probability distributions fed into the statistical model.

The decision-theoretic framework.

In our decision-theoretic model, a DM is faced with a multidimensional state space—each dimension of which is called a variable—and solves a two-period problem. First, the DM chooses a policy intervention. The role of policy interventions is to distinguish which variables are chosen by nature and which variables are chosen by the DM. In our example, if $X=A\times E\times L$ is the state space, and a policy intervention is an element of $\mathcal{P}=(A\cup\{\emptyset\})\times(E\cup\{\emptyset\})\times(L\cup\{\emptyset\})$, where $\emptyset$ means that the variable’s value is selected by nature in the second period. For example, $p=(\emptyset,\>College,\>\emptyset)$ is a policy where both ability and lifetime earnings are chosen by nature in the second period, whereas education is chosen by the DM and takes value $p_{E}=college$; in such a case, we say that education is an intervened variable. Each choice of policy then determines a state space over the nonintervened variables; in this example, the second-period state space is $A\times L$. In the second period, the DM takes this new state space as given and behaves as in the standard Savage model; that is, the DM chooses among monetary acts defined over $A\times L$.

In this framework, we can define what causality means. Intuitively, we say that our DM—i.e., our statistical model—treats education as a cause of lifetime earnings if there is a ceteris paribus intervention on the education variable that affects the distribution of lifetime earnings. Formally, this amounts to finding two policies (say, $p$ and $p^{\prime}$) that satisfy the following four conditions: (i) Education is intervened under $p$ and $p^{\prime}$ at different levels (say, $p_{E}=$ college and $p^{\prime}_{E}=$ PhD), (ii) lifetime earnings are not intervened under either $p$ or $p^{\prime}$ (*i.e., * $p_{L}=p_{L}^{\prime}=\emptyset$), (iii) all other variables are intervened at a constant level under both $p$ and $p^{\prime}$ (i.e., $p_{A}=p^{\prime}_{A}$), and (iv) the DM’s beliefs over lifetime earnings following the choice of $p$ are different than the DM’s beliefs over lifetime earnings following $p^{\prime}$. In this case, policies $p$ and $p^{\prime}$ amount to a ceteris paribus intervention on education that impacts the distribution of lifetime earnings, thus suiting our intuitive understanding of the phrase “education causes lifetime earnings”. Section 4 formally presents this choice-theoretic definition of causality.

The DM’s choice over policies in the first period and of acts over nonintervened variables in the second period define both the DM’s causal view of the world and the DM’s beliefs over $A\times E\times L$. However, nothing so far disciplines how causality and beliefs are related. Our axioms (Axiom 1, 2, 3) discipline how our definition of causality interacts with the DM’s beliefs. Intuitively, Axiom 1 states that variables must be logically independent, while Axioms 2 and 3 jointly state that the causes of a variable are the most proximal sources of information about that variable. Section 5 formally presents the axioms.

We prove two theorems. Theorem 1 identifies a unique class of models that is consistent with both our definition of causality and axioms. Theorem 2 takes as given the model identified by Theorem 1 and shows that causal effects can be quantified in terms of correlation structures alone if, and only if, we consider an auxiliary axiom (Axiom 4). This result makes the model applicable to empirical research, since it states that what we want to calculate –causal effects– may be computed in terms of what we can calculate – independence structures.

Graphical methods and correlation structures.

Consider a distribution (say, $\mu$) over the variables in our example. Arrange the variables in a graph using the following criterion: for any two variables $X$ and $Y$, if $X$ is not adjacent to $Y$ – that is, if there is no link from $X$ to $Y$ nor from $Y$ to $X$ – then $X$ is independent of $Y$ conditional on their respective parents – that is, conditional on the variables that directly point towards $Y$ or $X$. Graphs constructed in this way encode all conditional independence properties of $\mu$ (see, for example, Dawid [1], Geiger et al. [3], Lauritzen et al. [12]). Furthermore, given any probability distribution over a set of variables, the conditional independence structure of that distribution can always be represented by some DAG as described above.

For example, suppose that $\mu$ is a distribution over $E$, $A$, and $L$ that can be factorized as follows:

[TABLE]

Equation 1 implies that under $\mu$, $E$ducation and $L$ifetime earnings are independent of each other once we condition on $A$bility. Thus, $\mu$ can be represented by either of the graphs depicted in Figure 1: in both graphs, $E$ and $L$ are non adjacent, and the set of parents of $\{E,L\}$ is $\{A\}$.

Importantly, DAGs are a depiction of conditional independence; as such, arrows have no causal connotations. Note that both graphs represent $\mu$. If arrows had causal meaning, then education would have both an indirect causal effect on earnings (via the path $E\rightarrow A\rightarrow L$ in $\hat{G}_{\mu}$) and no causal effect on earnings (evidenced by a lack of a directed path from $E$ to $L$ in $G_{\mu}$), which is a contradiction.

From correlation to causality.

While arrows in a DAG carry no causal connotations, they nevertheless form the backbone of many causal models (see, for example, Pearl [14] and follow-up work). Pearl’s model circumvents this problem by using three primitives: a distribution $\mu$ over the relevant variables, a graph $G$ that represents $\mu$, and a Markov representation of $\mu$ that is compatible with $G$ (for a formal definition of Markov representations, see Section 7.2, which can be read independently of the rest of the paper). Pearl’s model provides no definition of what causality is but assumes that, whatever causality means, it can be represented by these three objects. Under this assumption, Pearl provides tools for quantifying causal effects in terms of independence structures.

Our model takes a complementary view to Pearl’s model. First, we provide a formal definition of causality. Second, we provide normative axioms for how an econometric model should treat uncertainty. That causality can be represented via a DAG is therefore a theorem rather than an assumption. Furthermore, we provide the exact set of conditions under which causality may also be represented by a Markov representation. Again, that causality can be represented by a Markov representation is therefore a theorem, not an assumption. Furthermore, comparing both sets of axioms showcases exactly the additional assumptions required to represent causality as in Pearl’s model, relative to a treatment of causality based purely on DAGs.

3 Model and Notation

3.1 General notation

The following notation is used throughout this paper. The set $\mathcal{N}=\{1,...,N\}$ is a set of indexes. For each $\mathcal{J}\subset\mathcal{N}$, let $\{X_{j}:j\in\mathcal{J}\}$ be a family of sets indexed by $\mathcal{J}$. We denote by $X_{\mathcal{J}}=\Pi_{j\in\mathcal{J}}X_{j}$ the Cartesian products of the family and by $x_{\mathcal{J}}=(x_{j})_{j\in\mathcal{J}}$ a canonical element in $X_{\mathcal{J}}$; furthermore, we use $X$ to denote $X_{\mathcal{N}}$ to simplify notation. Moreover, all complements are taken with respect to $\mathcal{N}$: if $\mathcal{J}\subset\mathcal{N}$, then $\mathcal{J}^{\complement}\equiv\mathcal{N}\setminus\mathcal{J}$. Given any set $\mathcal{J}\subset\mathcal{N}$ and any $x\in X$, we use $x_{-\mathcal{J}}=x_{\mathcal{J}^{\complement}}$ and $X_{-\mathcal{J}}=X_{\mathcal{J}^{\complement}}$ as a way to simplify notation. Finally, if $\mathcal{J}\subset\mathcal{N}$ and $E\subset X_{\mathcal{J}}$, then $\mathbbm{1}_{E}:X_{\mathcal{J}}\rightarrow\{0,1\}$ denotes the indicator function that event $E$ has occurred; that is, $\mathbbm{1}_{E}(x_{\mathcal{J}})=1\Leftrightarrow x_{\mathcal{J}}\in E$.

The following notation refers to the graph-theoretic component of the model. A directed graph is a pair $(V,E)$ such that $V$ is a (finite) set of nodes and $E\subset V\times V$ is the set of edges. If two nodes, $i$ and $j$, satisfy that $(i,j)\in E$, we simplify the notation by writing $i\rightarrow j$. Moreover, the set of parents for a node $v\in V$ is the set $Pa(v)=\{v^{\prime}\in V:(v^{\prime},v)\in E\}$. A node $v\in V$ is a descendant of a node $v^{\prime}\in V$ whenever a directed path exists from $v^{\prime}$ to $v$. Formally, if a sequence $(v_{1},...,v_{T})\in V^{T}$ exists such that $v_{1}=v^{\prime}$, $v_{t}$ is a parent of $v_{t+1}$ for each $t\in\{1,...,T-1\}$, and $v_{T}=v$. Analogously, $v^{\prime}$ is an ancestor of $v$ whenever $v$ is a descendant of $v^{\prime}$. A directed graph is a DAG if, and only if, for all $v\in V$, $v$ is not a descendant of $v$. We denote by $D(v)$ the set of descendants of $v$ and by $ND(v)$ the set of nondescendants.

3.2 Model description

Our DM faces a variant of the standard Savage problem. The state space is $S=\Pi_{i=1}^{N}X_{i}$, where each $X_{i}$ is finite. We make this assumption for technical simplicity because causality is orthogonal to whether state spaces are finite or infinite. We let $\mathcal{N}=\{1,...,N\}$, and we call each $i\in\mathcal{N}$ a variable. Set $\mathcal{A}=\mathbb{R}^{S}$ is the set of Savage acts, and a DM has preferences $\succ$ over $\mathcal{A}$.

However, our problem differs from Savage’s since we incorporate policies that affect the states. This added language allows us to distinguish correlations from other types of relations among variables. A set of intervention policies is a set $\mathcal{P}=\Pi_{i=1}^{N}(X_{i}\cup\{\emptyset\})$. The interpretation is as follows. Let a policy $p\in\mathcal{P}$ be such that $p_{i}=\emptyset$ for some $i\in\mathcal{N}$. Then, this policy leaves variable $i$ unaffected; that is, $i$ is determined as it would have been in a standard Savage world. However, if for some $j\in\mathcal{N}$, we have $p_{j}=x_{j}\in X_{j}$, then policy $j$ forces variable $j$ to take the value $x_{j}$; that is, the value of variable $j$ is not determined as it would have been in a Savage problem but is chosen by the DM. Therefore, each policy implies a collection of interventions on the state space. Our model is one where the DM first chooses a policy from the set of all policies and then chooses a Savage act from the set of acts defined over the nonintervened variables.

We now define the primitive choice domain for our DM. Let $p\in\mathcal{P}$ be any policy, and let $\mathcal{N}(p)=\{i\in\mathcal{N}:p_{i}=\emptyset\}$. That is, $\mathcal{N}(p)$ are the variables that $p$ leaves unaffected. Furthermore, let $\mathcal{A}(p)\equiv\mathbb{R}^{X_{\mathcal{N}(p)}}$ be the set of acts defined over the variables that $p$ leaves unaffected. Then, the primitive domain of choice for the DM is the set $\{(p,a):p\in\mathcal{P},a\in\mathcal{A}(p)\}$. That is, our DM’s problem is to select an intervention policy and a Savage act over the nonintervened variables. We endow this DM with a preference relation $\bar{\succ}$ on $\{(p,a):p\in\mathcal{P},a\in\mathcal{A}(p)\}$.

Given $\bar{\succ}$, each $p$ induces an intervention preference on $\mathcal{A}(p)$: for each $p\in\mathcal{P}$ and each $f,g\in\mathcal{A}(p)$, we say that $f\succ_{p}g$ if, and only if, $(p,f)\bar{\succ}(p,g)$. Since our axioms focus on the DM’s intervention preferences, it is convenient to express intervention preferences explicitly in terms of the values at which the DM intervenes on the variables. For each policy $p\in\mathcal{P}$, if $p_{-\mathcal{N}(p)}=x_{-\mathcal{N}(p)}$, we use $\succ_{x_{-\mathcal{N}(p)}}$ to denote $\succ_{p_{-\mathcal{N}(p)}}$. The special case where $p=(\emptyset,...,\emptyset)$, so that no variables are intervened on, corresponds to the DM’s preferences in a standard Savage world. For such a $p$, we use $\succ_{(\emptyset,...,\emptyset)}=\succ$ for notational simplicity.

From intervention preferences, we obtain intervention beliefs. For each $p\in\mathcal{P}$, we say that $\succ_{p}$ has a belief representation if there is a probability distribution $\mu_{p}$ on $X^{\mathcal{N}(p)}$ such that for all $E,F\subset X^{\mathcal{N}(p)}$, $\mu_{p}(E)>\mu_{p}(F)$ if, and only if, $\mathbbm{1}_{E}\succ_{p}\mathbbm{1}_{F}$. When such a representation exists, we say that $\mu_{p}$ is an intervention belief.

Intervention preferences (resp. beliefs) resemble Savage conditional preferences (resp. beliefs) but have important differences. Savage conditional preferences capture betting behavior conditional on the DM observing that a certain event was realized, whereas intervention preferences (beliefs) capture betting behavior after a controlled intervention on the relevant variables. To illustrate the difference, consider Example 1 below. Conditional preferences (resp. beliefs) are statements about item $[1.]$, whereas intervention preferences (resp. beliefs) are statements about item $[2.]$. These are clearly different statements that do not imply one another. Therefore, we need language to distinguish these two decision problems, and intervention preferences provide such language.

Example 1**.**

Let $f$ and $g$ be acts over Mr. Kane’s lifetime earnings defined as follows. Act $f$ pays \1$if Mr. Kane’s lifetime earnings are greater than$$100K$per year and$-$1$otherwise. Act$g$is the opposite: it pays$-$1$if lifetime earnings are greater than$$100K$per year and$$1$ otherwise. Consider the following statements:

“Having observed that Mr. Kane earned a college degree (of his own free will and ability), Alex prefers $f$ to $g$.”

• 2.

“Having forced Mr. Kane to obtain a college degree (regardless of his desire or ability to do so) Alex prefers $f$ to $g$”.

Because this paper is concerned with understanding what a rational agent’s approach to causality is, the role of axioms is exclusively normative. Whether actual humans adhere to these axioms is orthogonal to this paper. While the counterfactual-based setup presented above may seem hard to test in a laboratory with actual human subjects, this is not the objective of the exercise at hand. Since the DM in our paper is an analyst’s econometric model of the world, the only question that matters is whether the analyst finds the axioms normatively appealing. Moreover, econometric models (as opposed to human subjects in a laboratory) are naturally built to handle counterfactual analysis of the sort presented above.

4 Definition of Causal Effect

In this section, we introduce the definition of causal effect, which formalizes the intuitive definition given in Section 2.

We begin by introducing the definition of intervention independence. Consider a set of variables $\mathcal{K}$ and two variables $i,j\notin\mathcal{K}$. Informally, $i$ is $\mathcal{K}$-independent of $j$ if, after eliminating the possibility that $i$ and $j$ are related through variables in $\mathcal{K}$, the choice of acts over $i$ is insensitive to interventions on $j$. Formally, we say that $i$ is $\mathcal{K}$-independent of $j$ if the following holds: $(\forall x_{\mathcal{K}}\in X_{\mathcal{K}})$, $(\forall x_{j},x_{j}^{\prime}\in X_{j})$, and $(\forall f,g\in\mathbb{R}^{X_{i}})$,

[TABLE]

The first line indicates that having intervened on $\mathcal{K}$ at value $x_{\mathcal{K}}$, intervening on $j$ at different values does not affect the DM’s choice of act in $\mathbb{R}^{X_{i}}$. The second line indicates that having intervened on $\mathcal{K}$, the ability to intervene on $j$ at all, regardless of the values at which it is intervened on, does not affect the DM’s choice of act in $\mathbb{R}^{X_{i}}$. Note that the second of these conditions implies the first. Indeed, if the second condition holds, then we have that for all $x_{j},x_{j}^{\prime}$,

[TABLE]

so the first equation also holds. This motivates the formal definition of intervention independence.

Definition 1**.**

For all $i,j\in\mathcal{N}$ and $\mathcal{K}$ such that $i,j\notin\mathcal{K}$, we say that variable $i$ is $\mathcal{K}$-independent of variable $j$ if for all $f,g\in\mathbb{R}^{X_{i}}$,

[TABLE]

To illustrate Definition 1, consider a DM who believes that $A$bility has a direct impact on $E$ducation and that education has a direct impact on $L$ifetime earnings but that ability has no direct impact on lifetime earnings. This is depicted in Figure 2 below. If $a,a^{\prime}\in A$ are two ability levels and $f$ and $g\in\mathbb{R}^{L}$ are two acts on lifetime earnings, we might have the DM behave as follows: $f\succ_{a}g$ and $g\succ_{a^{\prime}}f$. This reversal indicates that $A$ and $E$ are not $\{\emptyset\}$-independent, which is intuitive: interventions on $A$ affect beliefs about $E$, and beliefs about $E$ affect beliefs about $L$. However, this is an effect of $A$ on $L$ that is mediated through $E$. As such, we do not want to use this as a basis to claim that $A$ causes $L$. The correct way to capture the causal effect of $A$ on $L$ is to regard intervention preferences $\succ_{(a,e)}$ as a function of $a$ for each fixed $e\in E$. In other words, we want to ask if $A$ and $L$ are $\{E\}$-independent. This motivates the formal definition of causal effect.

Definition 2**.**

*For all $i,j\in\mathcal{N}$, we say that variable $j$ causes variable $i$ if $i$ is not $\{i,j\}^{\complement}$-independent of $j$.

Let Ca(i)=\{j\in\mathcal{N}:\text{j$causes$i}\} denote the causal set of $i$.

Finally, we say that $j$ is an indirect cause of $i$ if there is a sequence $j_{0},...,j_{T}$ such that, for all $t\in\{0,...,T-1\}$, $j_{t}$ causes $j_{t+1}$, $j_{0}=j$ and $j_{T}=i$. We denote the set of indirect causes of a variable $i$ by $ICa(i)$.*

Finally, if a variable $i$ is such that $Ca(i)=\emptyset$, we say that $i$ is an exogenous primitive; otherwise, we say that it is an endogenous variable. Indeed, when a DM forms a causal model of the world, the set of primitives of such a model is precisely the set of variables that are not caused by any other variable in the model. Exogenous primitives are relevant in our discussion of Axiom 1.

We conclude this section by defining the causal graph associated with a preference, $\bar{\succ}$. Causal graphs are an integral part of our representation, which is introduced in Section 6. Given $\bar{\succ}$, draw a graph by letting the set of nodes be the set of variables and the set of arrows be defined by the causal sets, that is, by letting $j\rightarrow i\Leftrightarrow j\in Ca(i)$. This graph is well defined because $Ca(i)$ is well defined for each $i\in\mathcal{N}$. We denote such a graph as $G(\bar{\succ})$.

Definition 3**.**

Let $\bar{\succ}$ be a preference and $\{Ca(i):i\in\mathcal{N}\}$ be the collection of causal sets derived from $\bar{\succ}$. Define $G(\bar{\succ})=(V,E)$ by setting $V=\mathcal{N}$ and $E=\{(j,i):j\in Ca(i)\}$.

5 Axioms

Our axioms are normative statements about how the DM should treat uncertainty as a function of the DM’s causal model. Hence, our axioms tackle variations of the following question: given the DM’s causal graph as per Definition 3, what normative restrictions should we impose on the DM’s intervention beliefs?

As such, the axioms are about conditional independence properties of the various intervention preferences, $\succ_{p}$. Since the act notation for conditional independence is somewhat burdensome, we use the following simplifying notation.

Definition 4**.**

Let $i\in\mathcal{N}$ and let $\mathcal{J},\mathcal{K},\mathcal{H}\subset\mathcal{N}$ be disjoint sets such that $i\notin\mathcal{J}\cup\mathcal{K}\cup\mathcal{H}$. We say that $i$ is independent of $\mathcal{J}$ conditional on $\mathcal{K}$ after intervening on $\mathcal{H}$ if the following is true for all $x_{(\mathcal{J}\cup\mathcal{K}\cup\mathcal{H})}\in X^{(\mathcal{J}\cup\mathcal{K}\cup\mathcal{H})}$ and all $f,g\in\mathbb{R}^{X_{i}}$:

[TABLE]

When the above holds, we write

[TABLE]

In the case in which $\mathcal{J}$ is a singleton, $\mathcal{J}=\{j\}$, we simply write

[TABLE]

In terms of behavior, conditional independence says the following: $i$ and $j$ are independent if a DM would never pay for information about $j$ when the DM’s task is to predict $i$. Imagine a DM intervened variables $\mathcal{H}$ to a specific level, $x_{\mathcal{H}}$. For instance, the DM might have carried out a controlled experiment, or this could simply be a thought experiment. Imagine further that the DM observed a specific realization of variables in $\mathcal{K}$. In this context, if the DM had to choose between $f$ and $g$, he would have to compare $\mathbbm{1}_{x_{\mathcal{K}}}f$ with $\mathbbm{1}_{x_{\mathcal{K}}}g$ using preferences $\succ_{x_{\mathcal{H}}}$. To aid the DM’s decision, someone offers to reveal the DM the value of the variables in $\mathcal{J}$ for a fee $\varepsilon>0$. Is there an $\varepsilon$ small enough that the DM would purchase this information? If the DM bought this information about $\mathcal{J}$, then his problem becomes to compare $\mathbbm{1}_{x_{\mathcal{K}}}\mathbbm{1}_{x_{\mathcal{J}}}f$ with $\mathbbm{1}_{x_{\mathcal{K}}}\mathbbm{1}_{x_{\mathcal{J}}}g$ using preferences $\succ_{x_{\mathcal{H}}}$. Since the definition states that the DM’s choice under both situations is the same, the information is useless. Thus, the DM would not accept any price $\varepsilon>0$.

We start with Assumption 1, which defines all relevant aspects of the DM’s probabilistic model. To state Assumption 1, we recall the definition of a subjective expected utility preference. We say that a preference $\succ_{p}$ is a (monotone) subjective expected utility preference if there exists a unique probability distribution, $\mu_{p}\in\Delta(X^{\mathcal{N}(p)})$, and a (monotone increasing) function $u_{p}:\mathbb{R}\rightarrow\mathbb{R}$ such that for all acts, $f,g\in\mathbb{R}^{X^{\mathcal{N}(p)}}$ condition 5 below holds. There are many axiomatizations of monotone expected utility preferences that fit the framework of our model, such as Gul [4], Fishburn [2], and Theorem $3$ in Karni [10], among others. We let the readers select their preferred axiomatization.

[TABLE]

Assumption 1**.**

For each $\mathcal{J}\subset\mathcal{N}$, the following are true.

• i-

For each $p\in\mathcal{P}$, the preferences $\succ_{p}$ are monotone subjective expected utility preferences.

• ii-

The state space is complete: $(\forall i,j\in\mathcal{N})$, $(\forall x_{\mathcal{N}\setminus\{i\}}\in X_{\mathcal{N}\setminus\{i\}})$, and $(\forall f,g\in\mathbb{R}^{X_{i}})$; if $j\in Ca(i)$, then $f\succ_{x_{\mathcal{N}\setminus\{i\}}}g\Leftrightarrow\mathbbm{1}_{x_{j}}f\succ_{x_{\mathcal{N}\setminus\{i,j\}}}\mathbbm{1}_{x_{j}}g$.

• iii-

There are no null states: for all $x\in X$, $\mathbbm{1}_{x}\succ\mathbbm{1}_{X}0$.

• iv-

Policies do not affect preferences: $(\forall x,y\in\mathbb{R})$, $(\forall p,p^{\prime}\in\mathcal{P})$, $\mathbbm{1}_{X_{\mathcal{N}(p)}}x\succ_{p}\mathbbm{1}_{X_{\mathcal{N}(p)}}y\Leftrightarrow\mathbbm{1}_{X_{\mathcal{N}(p^{\prime})}}x\succ_{p^{\prime}}\mathbbm{1}_{X_{\mathcal{N}(p^{\prime})}}y$

Assumption 1 captures the basics of the DM’s beliefs. As such, it is orthogonal to issues of causation, which is why we do not refer to it as an axiom per se. Below, we examine each of the restrictions in Assumption 1.

As noted above, many axiomatizations exist that will deliver item $[i.]$, each with its own advantages and disadvantages. We let readers choose their preferred axiomatization of monotone expected utility. The importance of item $[i.]$ is that all intervention preferences are probabilistically sophisticated, so intervention beliefs are always well defined. Item $[iii.]$ states that being paid \1$if realization$x\in X$occurs is strictly preferred to receiving$$0$for sure, thus guaranteeing that all states receive positive probability. Item$[iv.]$rules out the possibility that policies have a direct impact on the Bernoulli utility indexes, thus making$u_{p}=u_{p}^{\prime}$for all$p,p^{\prime}\in\mathcal{P}$.555 This assumption is not strictly needed, but it simplifies notation in some proofs. Since causality is orthogonal to whether Bernoulli utilities are constant in$\mathcal{P}$, we feel comfortable maintaining this assumption.

Item $[ii.]$ in Assumption 1 says that the state space is complete: given two variables (say, $i$ and $j$), the state space includes all variables that could mediate effects between $i$ and $j$. Once all variables $k\neq i,j$ are intervened on, either $i$ causes $j$, $j$ causes $i$, or $i$ and $j$ are independent. If $j$ causes $i$, since all possible confounding variables have been intervened on, observing that $x_{j}\in X_{j}$ was realized or intervening on variable $j$ and shifting its value to $x_{j}$ should lead to the same preference over $\mathbb{R}^{X_{i}}$. Violations of this axiom are reasonable only if the state space is missing some potential confounding variables. In line with Savage [17], we assume that the state space is complete, so there are no missing confounding variables.

That the state space is complete has the following implication for the representation of preferences. Take two variables (say, $i$ and $j$) and assume that all other variables are intervened on at a level $x_{-\{i,j\}}$. Let $\mu_{x_{-\{i,j\}}}$ be the belief representation of $\succ_{x_{-\{i,j\}}}$. In this $2$-variable environment, correlation and causation should coincide. Therefore, if $i$ causes $j$, conditioning on $j$ or intervening on $j$ should lead to the same posteriors about $i$. Namely,

[TABLE]

Importantly, this relation is not symmetric. If $j$ does not cause $i$, then the symmetric expression $\mu_{x_{-\{i,j\}}}(x_{j}|x_{i})=\mu_{x_{-\{i,j\}},x_{i}}(x_{j})$ is false. The left-hand side is a nonconstant function of $x_{i}$, whereas the right-hand side is constant in $x_{i}$. Thus, under complete state spaces, equation 6 identifies the direction of causality. We will use this observation in Section 6 when defining when a graph represents a preference.

That the state space is complete does not imply that the DM must know what all the relevant variables are. For instance, assume that Alex from the introduction is worried that the interaction among ability, education and lifetime earnings might be affected by some other variable. Concretely, Alex thinks that some other variable might influence education: Alex does not know what this variable is but believes that it exists. For concreteness, denote this variable as an “unknown but possibly exiting variable”. Assumption 1 says that Alex’s state space should include such a variable. Therefore, the state space should not be $A\times E\times L$ but rather $A\times E\times L\times U$, where $U$ stands for “unknown but possibly existing variable”. In short, Assumption 1 does allow the econometrician to add variables that act as proxies for unknown shocks to the system. Indeed, modeling a potential unknown confounder as exogenous noise shocks is a common way to proceed in empirical studies.

Assumption 1 states that the DM is probabilistically sophisticated but is silent about the statistical properties of causal sets. Axioms 1 through 3 discipline how causation interacts with the DM’s probabilistic beliefs. In other words, they discipline how correlation and causation interact.

Axiom 1**.**

$\mathcal{I}\subset\mathcal{N}$, there exists $i\in\mathcal{I}$ such that $Ca(i)\cap\mathcal{I}=\emptyset$

Axiom 1 states that variables are logically independent of each other. If the DM is asked to explain the relation between variables in $\mathcal{I}$ and only those in $\mathcal{I}$, Axiom 1 states that the DM has an explanation that involves at least one exogenous primitive relative to $\mathcal{I}$. Models without primitives describe identities rather than relations among logically independent variables. Therefore, Axiom 1 states that the DM’s state space includes only logically independent variables.

A potential critique of this axiom is that certain systems are inherently cyclical. For instance, the relation among the speed of a car, the distance traveled by the car, and the time traveled by the car is inherently circular: any two determine the third. The problem with this system is that speed is not caused by distance and time traveled; rather, speed is defined in terms of distance and time traveled. Therefore, the model includes variables that are not logically independent of one another. The correct model to analyze this situation is one in which the only variables are time and distance traveled by the car, as these variables are the only logically independent variables. In this sense, the assumption that no causal cycles exist is sensible.

A related critique of Axiom 1 is that it precludes the DM from viewing the world as a system of recursive structural equations. As such, Axiom 1 could be seen as precluding the DM from reasoning in terms of equilibrium equations (see, for example, the critique in Heckman and Pinto [6]). This assessment stems from interpreting functional relations as causal relations. However, the equations in a model (in particular, equilibrium equations) are succinct descriptions of the specific values that the variables may obtain; they say nothing of how those values are achieved. As such, causality and equilibrium equations are orthogonal issues.

To make the above discussion concrete, consider a general equilibrium model with aggregate demand curve $D$ and aggregate supply curve $S$. The equilibrium is defined as follows: $(p^{*},q^{*})$ constitutes an equilibrium if $D(p^{*})=q^{*}$ and $S(p^{*})=q^{*}$. Note that this is a definition; as such, the equilibrium price and equilibrium quantity are not logically independent. These equations describe the values one should expect for prices and quantities but are silent regarding the mechanism that generated them. This silence motivates the equilibrium convergence literature. For example, a tatônnement convergence process is compatible with the general equilibrium equations without invoking feedback loops: a DM posits that prices in period $t$ cause quantities in period $t$ (via consumer/producer optimization) and that quantities in period $t$ cause prices in period $t+1$ (through a process that increases/decreases the price in response to excess demand/supply). That the system stabilizes at a point where $p_{t}=p_{t+1}=p^{*}$ and $q_{t}=q_{t+1}=q^{*}$ is orthogonal to the issue of causation. In short, one should not mistake functional equations, which simply describe relations between variables, for causal statements.

Axiom 2**.**

For all $i\in\mathcal{N}$, if $\mathcal{J}\subset Ca(i)$ and $\mathcal{H}\subset\mathcal{N}\setminus\{i\}$ is disjoint from $\mathcal{J}$,

[TABLE]

Axiom 2 captures the following normative property about causation: the causes of a variable (say, $i$) are the most proximal sources of information about $i$. Suppose that one were to slice the set $Ca(i)$ into three slices: $\mathcal{J}$, $\mathcal{H}$ and $Ca(i)\setminus\mathcal{J}\setminus\mathcal{H}$. If one intervened on slice $\mathcal{H}$ and observed the value of variables in $\mathcal{J}$, would one pay for information about the remaining variables $j\in Ca(i)\setminus\mathcal{H}\setminus\mathcal{J}$? If $j$ is the most proximal source of information about $i$, then there should be an $\varepsilon>0$ small enough that the DM should pay $\varepsilon$ in exchange for information on the value of these variables $j$. If $j$ were rendered useless for prediction after information on the other causes were obtained, then $j$ would not itself be a proximal source of information about $i$ and therefore should not be called a “cause” of $i$. As a final remark, note that Axiom 2 is symmetric in the following sense: the only fundamental sources of information about $i$ are causes of $i$ and those variables that are directly caused by $i$.

Axiom 3**.**

$(\forall i,j\in\mathcal{N})$, $(\forall\mathcal{K}\text{, }\mathcal{J}\subset\mathcal{N}\setminus\{i,j\})$, $(\mathcal{J}\cap\mathcal{K}=\emptyset)$, if $i\notin Ca(j)$ and $j\notin Ca(i)$, then

[TABLE]

While Axiom 2 describes the conditional independence properties of variables that are directly related to each other, Axiom 3 analyzes the independence properties of variables that are not directly related to each other. Axiom 3 states that two variables that do not cause each other are independent of each other once we condition on any set that includes $i$ and $j$’s causes.

To understand Axiom 3’s normative appeal, consider the DAG in Figure 3 below, where arrows point from cause to effect. If a DM had to predict the value of $i$ and knew the realizations $x_{b}$ and $x_{c}$ (and perhaps some other noncauses, say $k$), should the DM pay for information about the realization of $j$? Our axiom says the DM should not pay for this information, which is quite sensible: once the DM knows the values of $b$ and $c$, he knows all there is to know about the relation between $i$ and $j$. Thus, extra information on $j$ is useless for predicting the value of $i$.

A more general analysis of Axiom 3 proceeds in three steps. First, it is normatively appealing to say that $i$ and $j$ are not independent: because $i$ causes $c$ and $c$ causes $j$, it stands to reason that any information we have about $i$ will (via $c$) provide information about $j$. Similarly, $b$ provides another link between $i$ and $j$: since $b$ is a common cause of both, then any information we have about $i$ should allow us to make inferences about $b$ and, in turn, inferences about $j$. Second, because neither $i$ causes $j$ nor vice versa, any information $i$ provides about $j$ will be mediated by some variable. Third, the mediating variable will either be a cause of $i$, a cause of $j$, or both. Indeed, if $i$ provided information about $j$ that is not mediated by any cause of $j$, then $i$ is providing information about $j$ that is more proximal than the information contained by any cause of $j$. Therefore, $i$ should itself be a cause of $j$, which it is not. Combining these three observations implies that if we condition on the causes of both $i$ and $j$, then $i$ and $j$ should be conditionally independent. Finally, Axiom 3 states that the same condition would remain true if we were to make the choice of $f$ vs. $g$ contingent on some additional noncauses, $\mathcal{J}$.

While Axioms 1 through 3 are our basic axioms, Axiom 4 is a supplementary axiom that is relevant for Theorem 2. We present it here in the interest of containing all axioms and their corresponding discussions within a single section.

Axiom 4**.**

$(\forall i\in\mathcal{N})$, $(\forall\mathcal{J}\subset\mathcal{N}\setminus\{i\})$ $(\forall f,g\in\mathbb{R}^{X_{i}})$, $(\forall x_{Ca(i)\cup\mathcal{J}}\in X_{Ca(i)\cup\mathcal{J}})$,

[TABLE]

Axiom 4 states that the following two decision problems are equivalent. Given a variable $i$ and acts $f,g\in\mathbb{R}^{X_{i}}$, the first problem is to choose $f$ or $g$ when their payments are contingent on the causes of $i$ obtaining a particular value, $x_{Ca(i)}$. In the second decision problem, the DM intervenes on a subset of causes of $i$ (say, moving $\mathcal{J}\subset Ca(i)$ to the value $x_{\mathcal{J}}$), and the payments of $f$ and $g$ are now contingent on the values of the nonintervened causes, $x_{-\mathcal{J}}$, being realized. From a numerical standpoint, both these situations result in the same value for the causes of $i$ (namely, $x_{Ca(i)}$); the difference is how those values are achieved. In the first problem, it is simply by selecting a standard Savage conditional act, while in the second problem, it is by a combination of interventions and Savage conditional acts. Because Axiom 4 requires that these two problems be treated identically, Axiom 4 implies that the only aspect of interventions that matters is the value the intervention sets for the variable. In other words, the act of intervening on a variable does not, in itself, change the DM’s structural view of the world.

We use Figure 4 below to illustrate the normative appeal of Axiom 4.

First, we explain why Axiom 4 involves sets that are weakly larger than $Ca(i)$. Suppose that a DM has to choose between two acts over $i$ (say, $f,g\in\mathbb{R}^{X_{i}}$), the payments of which are contingent on $j$ taking value $x_{j}$. That is, the DM has to choose between $\mathbbm{1}_{x_{j}}f$ and $\mathbbm{1}_{x_{j}}g$. Note that $\{j\}$ is a strict subset of $Ca(i)$. Observing that $j$ takes the value $x_{j}$ gives the DM information about the value of $k$; in turn, this information about $k$ gives the DM information about $w$, which ultimately gives the DM information about $i$. Thus, observing that $j$ took the value $x_{j}$ is informative about $i$ in two ways: directly, because $j\in Ca(i)$, and indirectly, via $k$ and $w$. If the DM intervenes on $j$ at value $x_{j}$, the DM receives the same direct information about $i$ but loses the indirect information mediated via $k$ and $w$. Thus, the DM could say that $\mathbbm{1}_{x_{j}}f\succ\mathbbm{1}_{x_{j}}g$ but $g\succ_{x_{j}}f$. Clearly, observing $x_{j}$ or intervening on variable $j$ and moving it to value $x_{j}$ are different problems in terms of the DM’s updating.

Now, consider the situation above but where the payments of $f$ and $g$ involve all causes $i$, $j$ and $w$. That is, for some $x_{j}$ and $x_{w}$, the DM chooses between $\mathbbm{1}_{x_{j},x_{w}}f$ and $\mathbbm{1}_{x_{j},x_{w}}g$. For concreteness, suppose that $\mathbbm{1}_{x_{j},x_{w}}f\succ\mathbbm{1}_{x_{j},x_{w}}g$. If the DM intervened on $j$ and shifted it to the value $x_{j}$ and then had to choose between $\mathbbm{1}_{x_{w}}f$ and $\mathbbm{1}_{x_{w}}g$, would the DM lose any information? In other words, if at a cost $\varepsilon>0$, the DM could intervene and shift the value of $j$ to $x_{j}$, rather than simply conditioning his choice on value $x_{j}$ being realized, is there a $\varepsilon>0$ small enough that the DM would pay for this option? In both problems, $w$ is observed to take value $x_{w}$; therefore, any information that $j$ could indirectly provide about $i$ through $w$ is still directly captured in the observation of $x_{w}$. Thus, intervening on $j$ entails no information loss relative to simply observing that $j$ took value $x_{j}$. Thus, the DM has the same information in both problems and should thus treat the problems equivalently. This result is precisely what Axiom 4 requires.

Both of the above discussions addressed $\mathcal{J}\subset Ca(i)$, but to complete our discussion of Axiom 4, we must allow that $\mathcal{J}$ contains noncauses of $i$. Axiom 4 states that once we know the value of all the causes of $i$, intervening variables that are not causes of $i$ are uninformative about $i$. In Figure 4, if an act’s payments are contingent on $x_{w}$ and $x_{j}$, then intervening and shifting the value of $k$ to some $x_{k}$ is uninformative about $i$.

6 Representation

In this section, we define the representation we seek for $\bar{\succ}$. Since $\bar{\succ}$ will ultimately be associated with a collection of probability distributions, we proceed in two steps. First, define what it means for a DAG to represent a single probability distribution. Then, we generalize to a family of probability distributions. For a reminder of our graph-theoretic notation, see Section 3.1.

Lauritzen et al. [12] provide a definition for when a DAG represents a probability distribution, say $\mu\in\Delta(\Pi_{i\in\mathcal{N}}X_{i})$. The objective of such a definition is to graphically represent the conditional independence structure of $\mu$. Let $\mu\in\Delta(\Pi_{i\in\mathcal{N}}X_{i})$, and let $G=(\{1,...,N\},E)$ be a DAG. The chain rule implies the following

[TABLE]

Now, consider the DAG in Figure 5 below.

In a DAG such as the one above, an arrow between two nodes indicates that the two nodes are never statistically independent. In this way, arrows encode which variables provide fundamental information about other variables, in the sense that the information transmitted by the source is not contained in any other variable. For instance, the DAG in Figure 5 conveys that $w$ and $j$ contain fundamental information about $i$ and thus that $i$ is never independent of $\{w,j\}$. Similarly, $i$ is never independent of its direct descendant, $k$. Now, consider a variable that is an ancestor of $i$; for example, $a$. Clearly, $a$ and $i$ are not independent: $a$ provides fundamental information about $j$, which provides fundamental information about $i$. However, any information that $a$ has about $i$ is implicitly encoded in $j\in Pa(i)$. Indeed, if $a$ carries fundamental information about $i$, there should be an arrow $a\rightarrow i$, but such an arrow is absent. Similarly, $b$ provides information about $i$: $b$ is informative about $\{a,w\}$, both of which are informative about $i$. However, any information that $b$ has about $i$ is encoded in $\{j,w\}$. What this implies is that once we condition on the parents of $i$ (in this case, $\{w,j\}$), all nondescendants of $i$ are conditionally independent of $i$. Therefore, the terms $\mu(x_{i}|ND(i))$ in equation 7 simplify to $\mu(x_{i}|Pa(i))$. This observation motivates Definition 5 below.

Definition 5**.**

Let $\mu\in\Delta(\Pi_{i\in\mathcal{N}}X_{i})$. A DAG $(\{1,...,N\},E)$ represents $\mu$ if, and only if, the following hold: $(\forall x\in\Pi_{i\in\mathcal{N}}X_{i})$,

[TABLE]

Definition 5 makes two statements. First, a DAG represents a probability distribution if, and only if, the DAG summarizes the conditional independence properties of $\mu$ in the sense discussed previously. Second, the set of parents is the smallest set that allows for such a decomposition. Indeed, consider a set of nodes $V=\{a,b,c\}$ and a probability distribution $\mu(x_{a},x_{b},x_{c})=\mu(x_{a})\mu(x_{b})\mu(x_{c})$. Since all variables are statistically independent, both DAGs in Figure 6 represent this $\mu$. Indeed, both $\mu(x_{a},x_{b},x_{c})=\mu(x_{a})\mu(x_{b}|x_{a})\mu(x_{c}|x_{a})$ and $\mu(x_{a},x_{b},x_{c})=\mu(x_{a})\mu(x_{b})\mu(x_{c})$ are true statements. However, the first representation includes irrelevant arrows: the minimality requirement prevents this.

Using Definition 5, we can define when a graph represents a standard Savage preference. Suppose that $\succ$ were the DM’s Savage preference defined on $\mathbbm{R}^{X}$. Under Assumption 1, there is a well-defined belief representation of $\succ$, $\mu\in\Delta(X)$. We can then say that a graph $G$ represents $\succ$ if $G$ represents $\mu$, in the sense of Definition 5.

However, Definition 5 is insufficient to define when a graph represents a preference $\bar{\succ}$. Indeed, a preference $\bar{\succ}$ is associated with the collection of induced Savage preferences $\{\succ_{p}:p\in\mathcal{P}\}$. As such, $\bar{\succ}$ is associated with a family of beliefs, rather than a single belief, as in Savage’s model. Thus, to define when a DAG represents preferences $\bar{\succ}$, we first define what it means for a DAG to represent a collection of probability distributions rather than a single probability distribution.

To define when a DAG represents preferences $\bar{\succ}$, we first define the truncation of a DAG. Let $G=(V,E)$ be a DAG, and let $W\subsetneq V$. The $W$-truncated DAG, $G_{W}$, is the DAG obtained by eliminating all nodes in $W$, together with their incoming and outgoing arrows. Formally, $G_{W}=(V\setminus W,E\cap W^{\complement}\times W^{\complement})$. This DAG is a useful representation of intervention beliefs. After variables in $W$ are intervened on, they no longer form part of the DM’s statistical model; they are now deterministic objects that are statistically uninformative about the value of their parents. Thus, we exclude these variables from the corresponding DAG. In the context of our running example, if Alex observes that Mr. Kane obtained a college degree, his education is no longer random, but Alex can still make inferences about Mr. Kane’s intellectual ability. Thus, education remains a legitimate element of Alex’s statistical model. However, if Mr. Kane’s education is intervened on and shifted to “college degree”, then his education level is no longer random and, furthermore, is uninformative about his ability level. Thus, we exclude education from the DM’s post-intervention model.

We can now define when a DAG, $G$, represents a preference $\bar{\succ}$. We then say that a graph represents a preference if the appropriately truncated subgraph represents the corresponding intervention preference and the arrows are consistent with the direction of causality. This is formally presented in Definition 6 below.

Definition 6**.**

Let $G=(\mathcal{N},E)$ be a DAG and $\bar{\succ}$ be a DM’s preference. Assume that for each $\mathcal{T}\subset\mathcal{N}$ and each $x_{\mathcal{T}}\in X_{\mathcal{T}}$, $\succ_{x_{\mathcal{T}}}$ has a well-defined belief representation; let $\mu_{x_{\mathcal{T}}}$ be the corresponding belief representation. We say that $G$ represents $\bar{\succ}$ if the following are true for each $\mathcal{T}\subset\mathcal{N}$ and each $x\in X$:

• i

$G_{\mathcal{T}}$* represents $\mu_{x_{\mathcal{T}}}$,*

• ii

If $(i,j)\in E$ then $\mu_{x_{-\{i,j\}}}(x_{j}|x_{i})=\mu_{x_{-\{j\}}}(x_{j})$.

Note that nothing in this section is related to causality. Indeed, the statement that a graph represents a probability distribution is purely a statement about statistical independence. As such, the representation of a probability by a DAG is a statement about correlation, not causation. At this point in the exposition, DAGs used to represent probability distributions and DAGs used to represent causal statements are completely unrelated. It is precisely the job of Theorems 1 and 2 to show the exact conditions under which a DAG can simultaneously represent the DM’s beliefs and the DM’s causal model.

7 Results

Our first theorem is Theorem 1, stated below.

Theorem 1**.**

Let $\bar{\succ}$ satisfy Assumption 1. The following are equivalent:

• i

Axioms 1 through 3 hold, and

• ii

$(\exists G)$* such that $G$ is a DAG and represents $\bar{\succ}$ in the sense of Definition 6.*

Furthermore, if $G$ represents $\bar{\succ}$, then $G=G(\bar{\succ})$.

The literature on Bayesian graphs assumes that causal DAGs fulfill a dual role: they represent both causal assumptions and assumptions on conditional independence. This is clearly a joint assumption about how the analyst defines causality and how the analyst’s definition of causality interacts with statements of conditional independence. Theorem 1 states the exact conditions under which a DAG can fulfill this dual role.

In particular, the uniqueness result implies that Definition 2 is the only definition of causality that satisfies our axioms. Suppose that a researcher has a definition of causality in mind (say, $C$) such that statements of the form “$i$ causes $j$ according to criterion $C$” are well defined. If $C$ satisfies our axioms, then $C$ can be represented via a DAG, $G$, such that $i\rightarrow j$ holds if, and only if, “$i$ causes $j$ according to criterion $C$”. The uniqueness claim in Theorem 1 says that $G=G(\bar{\succ})$. Therefore, $i\rightarrow j$ also holds if, and only if, $i$ causes $j$ in the sense of Definition 2. Thus, $C$ must coincide with Definition 2.

Theorem 1 also provides a foundation for unifying and structuring our understanding of causation. The theorem states that any formal discussion of causality (as understood by Definition 2) begins with two items: a collection of probability laws, $\{\mu_{p}\in\Delta(X_{\mathcal{N}(p)}):p\in\mathcal{P}\}$, and a DAG, $G$, that represents those laws. Models that include these components can legitimately be called models of “causation” regardless of any other details the model might include. However, models that cannot be phrased in terms of intervention beliefs and their representing DAG are not models of causality (again, as understood by Definition 2). In short, researchers who find our axioms normatively appealing and who agree that Definition 2 is a sensible definition of causal effect are encouraged to use DAG-based models for conducting causal inference. Researchers who find our axioms normatively unappealing, or disagree with Definition 2 as a sensible definition of causal effect, are encouraged to stay away from DAG-based models. In this way, Theorem 1 provides a foundation for selecting among models with which to empirically study causal effects.

7.1 Identification of intervention beliefs

In this section, we consider the following question. Let $\mu\in\Delta(X)$ be the DM’s beliefs elicited from his Savage preference and $\mu_{p}$ be the DM’s beliefs elicited from an intervention preference $\succ_{p}$. When can we express $\mu_{p}$ as a function of $\mu$? Theorem 2 in this section answers this question, and Proposition 2 in Appendix B provides further results along the same lines.

Answering the question above is useful to make the model applicable to empirical research. When $\mu_{p}$ is expressed in terms of $\mu$ (henceforth, when $\mu_{p}$ is identified), any information that allows a DM to update his Savage beliefs, $\mu$, also allows the DM to update his intervention beliefs, $\mu_{p}$. If an analyst had access to a perfectly controlled setting, the analyst could directly estimate each $\mu_{p}$, and models of causal inference would be unnecessary. However, most empirical work in economics is observational, in the sense that direct policy interventions are unavailable to the researcher. Proposition 2 and Theorem 2 bridge the gap between intervention beliefs – what the econometrician wants to calculate – and standard conditional probabilities – what the econometrician can calculate.

When added to Axioms 1 through 3, Axiom 4 yields a model in which different intervention beliefs, $\mu_{p}$, can be expressed in terms of $\mu$. In what follows, we remind the reader of Axiom 4 and illustrate Theorem 2 by means of two simple examples. Then, we state and discuss the general form of Theorem 2.

Axiom 4.

$(\forall i\in\mathcal{N})$, $(\forall\mathcal{J}\subset\mathcal{N}\setminus\{i\})$, $(\forall f,g\in\mathbb{R}^{X_{i}})$, $(\forall x_{Ca(i)\cup\mathcal{J}}\in X_{Ca(i)\cup\mathcal{J}})$,

[TABLE]

Example 2**.**

Blake is an econometrician who believes that ability causes both education and lifetime earnings and that education causes lifetime earnings. This model is graphically depicted in Figure 8. To understand the direct effect of education on lifetime earnings, Blake has to understand how $\mu_{a,e}(\cdot)$ changes with $e\in E$ for each fixed $a\in A$. However, Blake cannot access a controlled environment, so Blake has no data on $\mu_{(a,e)}$. However, under Axiom 4, data from controlled environments are unnecessary. When $\mathcal{J}=\{A,E\}$, Axiom 4 implies $\mu_{(a,e)}(\cdot)=\mu(\cdot|a,e)$. Thus, the direct causal effect of education on lifetime earnings is calculated by computing how $\mu(\cdot|a,e)$ varies with $e$ for each value of $a$. Note that $\mu(\cdot|a,e)$ is a standard conditional probability, and data on this quantity are available with access to observational datasets. Blake can therefore use data from outside a controlled environment to form his intervention beliefs.

Example 2 above provides a simple case in which one can directly substitute interventions for conditionals. Example 3 below shows an example where slightly more work is involved in identifying intervention beliefs.

Example 3**.**

Charlie is a colleague of Blake. However, Charlie believes that people are not born with intrinsic ability. On the contrary, education causes ability, and this ability is the sole cause of lifetime earnings. Charlie’s causal DAG is depicted in Figure 9. Charlie is interested in studying the indirect effect that education policies have on lifetime earnings, which can be done by applying Axiom 4 twice. First, set $\mathcal{J}=\{E\}$, $i=A$ to obtain $\mu_{a}(e)=\mu(e|a)$ for each $(a,e)\in A\times E$. Second, set $\mathcal{J}=\{E\}$, $i=L$ to obtain $\mu_{e}(l|a)=\mu(l|a)$ for each $(e,a,l)\in E\times A\times L$. Finally, we obtain the following derivation.

[TABLE]

Thus, calculating the indirect effects of $E$ and $L$ requires computing $\mu(l|a)$ and $\mu(a|e)$, both of which can be computed with data from observational studies. Even if access to a controlled environment is unavailable, the identification of $\mu_{e}$ implies that such data are unnecessary.

The examples highlight two simple cases in which intervention beliefs are identified. First, if $j$ is a cause of $i$, then the direct causal effect that $j$ has on $i$ is identified via the formula $\mu_{x_{-\{i,j\}},x_{j}}(x_{i})=\mu(x_{i}|x_{j},x_{Ca(i)\setminus\{j\}})$. Thus, one can obtain the direct causal effect of $j$ on $i$ by conditioning on all causes of $i$ and analyzing how that conditional probability varies with $x_{j}$. Similarly, if $j$ causes $k$, $k$ causes $i$, and this is the only connection between $j$ and $i$, the indirect causal effect of $j$ on $i$ is calculated by following the chain rule: $\mu_{x_{j}}(x_{i})=\sum_{x_{k}}\mu(x_{i}|x_{k})\mu(x_{k}|x_{j})$. However, other intervention beliefs may also be identified. The rest of this section is devoted to understanding the exact conditions under which intervention beliefs are identified.

7.2 Markov representations and do-probabilities

Before we can present Theorem 2, we need to define two auxiliary concepts: Markov representations and do-probability. We do this by means of a simple numerical example first, where we argue how Markov representations are related to a specific view of causality. We then provide a formal definition of both Markov representations and do-probabilities and state Theorem 2

Consider a distribution $\mu$ and a graph $G$ as in Figure 10: Ability is a common cause of education and lifetime earnings, and education is a further cause of lifetime earnings.

In a Markov representation of $\mu$ that is compatible with $G$, each variable is assumed to be written as a deterministic function of its parents in the graph and its own idiosyncratic noise. One way to understand such functions is as production functions: in this example, ability is exogenous, education is stochastically “produced” with units of ability, and lifetime earnings are stochastically “produced” with units of education and ability. For example, the following is a Markov representation of a distribution $\mu$ compatible with the graph $G$ in Figure 10.

[TABLE]

In particular, the distribution of $L$ conditional on $E=e$ is given by the following expression:

[TABLE]

One may express the above system of equations purely in terms of the independent noise shocks $\alpha$, $\varepsilon$ and $\lambda$ and thus obtain the joint distribution $\mu\in\Delta(A\times E\times L)$.

Do-probabilities are calculated from Markov representations, and they are the tool Pearl’s model uses to quantify causal effects. Suppose that we want to calculate the direct effect of education on lifetime earnings; that is, we want to calculate what weight arrow $E\rightarrow L$ carries without including the effect of path $E\leftarrow A\rightarrow L$. Doing this requires eliminating the dependence of $E$ on $A$. To eliminate the dependence of $E$ on $A$, we first eliminate the equation that determines the value of $E$ in the Markov representation, namely, $E=2A+\varepsilon$. Second, any time that $E$ would appear in any other equation, we treat $E$ as a deterministic value. Finally, we calculate the joint distribution of $A$ and $L$ for each deterministic value of $E$. Concretely, the distribution of $L$ do-$E$ is obtained by solving the following system, where $E$ is treated as a number rather than a random variable:

[TABLE]

which, purely in terms of the random shocks, simplifies to the following:

[TABLE]

In the expression above, the distribution of $L$ do-$E=e$, denoted as $\mu(L\>|:do(E=e))$, is given by $\mu(L\leq l\>|\>do(E=e))=\mu(100\alpha+200\lambda\leq l-2000e)$. The dependence of this expression on the value $e$ indicates that education indeed has a causal effect on $L$; furthermore, this expression differs from $\mu(L:|\>E=e)$ because $\mu(L\>|\>do(E=e))$ disregards the confounding effect of the arch $E\leftarrow A\rightarrow L$.

Theorem 2 shows that under Axioms 1, 2, 3 and 4, the DM’s beliefs, $\mu$, can be represented by a Markov representation. In particular, intervention beliefs exactly coincide with do-probabilities, as illustrated above, so all the tools from Pearl [14] apply to intervention beliefs.

Markov representations, do-probabilities, and Theorem 2

.

Below, we present a formal definition of a Markov representation.

Definition 7**.**

Let $\mu\in\Delta(X)$. For each $i\in\mathcal{N}$, let $\mu_{i}$ be the marginal over $X_{i}$. For each $i\in\mathcal{N}$, let $\varepsilon_{i}$ be a random variable with range $\mathcal{E}_{i}$, let $G$ be the DAG defined by a family of sets of parents $(Pa(i))_{i\in\mathcal{N}}$, and let $h_{i}$ be a function $h_{i}:X_{Pa(i)}\times\mathcal{E}_{i}\rightarrow X_{i}$. Let $\phi$ be the joint distribution of the vector $(\varepsilon_{1},...,\varepsilon_{N})$. A Markov representation of $\mu$ compatible with $G$ is a tuple $((h_{1},...,h_{N})\>,\>(\varepsilon_{1},...,\varepsilon_{N}))$ that satisfies the following:

• •

$(\forall i,j)$, $\varepsilon_{i}$ is independent of $\varepsilon_{j}$,

• •

$\mu$* can be recovered implicitly as a solution to the following system of equations:*

[TABLE]

Markov representations are used in statistical causality to numerically represent causal effects (see Pearl [14]) via do-probabilities, defined below.

Definition 8**.**

Let $\mu\in\Delta(X)$ be a probability distribution, $G$ be a DAG that represents $\mu$, and $((h_{1},...,h_{N}),(\varepsilon_{1},...,\varepsilon_{N}))$ be a Markov representation of $\mu$ compatible with $G$. Given two disjoint sets of variables, $\mathcal{I}$ and $\mathcal{J}$, the do-probability $\mu(x_{\mathcal{I}}|do(x_{\mathcal{J}}))$ is calculated as follows:

• 1

For all $j\in\mathcal{J}$, eliminate from system (8) in Definition 7 all the formulas $\mu_{j}(x_{j})=\phi(\{\varepsilon:h_{j}(x_{Pa(j)},\varepsilon_{i})=x_{j}\}$.

• 2

For each $i\notin\mathcal{J}$ and for each $j\in Pa(i)\cap\mathcal{J}$, input value $x_{j}$ into the corresponding formula in system (8) of Definition 7.

• 3

Calculate the probability of realization $x_{\mathcal{I}}$ in the model resulting from applying steps $1$ and $2$ above.

Having defined Markov representations and do-probabilities, we can now state Theorem 2.

Theorem 2**.**

Let $\bar{\succ}$ satisfy Assumption 1, and let $(\mu_{x_{\mathcal{I}}})_{\mathcal{I}\subset\mathcal{N}}$ be the subjective beliefs elicited from $\bar{\succ}$. The following statements are equivalent:

• •

Axioms 1 through 3 and 4 hold,

• •

There exists a Markov representation of $\mu$, $(G,(h_{1},...,h_{N}),(\varepsilon_{1},...,\varepsilon_{N}))$, such that

• –

$(\forall\mathcal{J}\subset\mathcal{N})$, $(\forall x_{\mathcal{J}}\in X_{\mathcal{J}})$; $\mu_{x_{\mathcal{J}}}=\mu(\cdot|do(x_{\mathcal{J}}))\in\Delta(X_{-\mathcal{J}})$,

• –

$G$* represents $\bar{\succ}$.*

Furthermore, if $G$ represents $\bar{\succ}$, then $G=G(\bar{\succ})$.

The crucial contribution of Theorem 2 is that it clarifies the role of do-probabilities in the understanding of causal effects. Do-probabilities are often presented as the definition of a causal effect. As Pearl writes in [15]: “* the definition of a “cause” is clear and crisp; variable $X$ is a probabilistic-cause of variable $Y$ if $P(y|do(x))\neq P(y)$ for some values $x$ and $y$*.” Theorem 1 states that one can legitimately represent causal effects based on interventions via a DAG, which, nonetheless, is incompatible with any system of do-probabilities. The causal DAG will be compatible with a set of do-probabilities only when adding Axiom 4 to the list of basic axioms. This result is analogous to the exercise conducted by Machina-Schmeidler [13]: just as expected utility and probabilistic sophistication can be behaviorally separated, we show that the graph-theoretic aspects of Pearl-like models can be separated from the do-probability formalism. The substantive assumptions about causality are conveyed by the DAG, while do-probabilities represent an additional assumption about when interventions and simple observations can be used interchangeably. In short, the notion of causality represented by a do-probability is strictly stronger than the notion of causality represented by a DAG.

Theorem 2 further clarifies that Axiom 4 is the fundamental property that links do-probabilities with intervention beliefs. When defining Markov representations, the functions $h(\cdot)$ are not indexed by whether their arguments have been observed or intervened on. The functions $h(\cdot)$ only concern the numerical values of their arguments and not the method through which these numerical values are obtained. This is an implicit assumption of Pearl’s model, and it is delivered by Axiom 4. Furthermore, note that Definition 7 implicitly requires that the Markov representation that defines do-probabilities has a unique solution. While this characteristic has sometimes been pointed to as a limitation of the theory (see Halpern [5]), under Axiom 4, this result is without loss of generality.

Finally, while do-probabilities are commonly referred to as the causal effect of one variable on another, it is important to be cautious with language. Do-probabilities reflect the effect that an intervention on a set of variables has on the whole system of equations; that is, do-probabilities capture both the direct and indirect effects of interventions. For example, consider the DAG in Figure 11. This DAG states that there is no direct causal effect of $A$ on $C$; however, $Pr(x_{C}|do(x_{A}))=Pr(x_{C}|x_{A})$, which is a nontrivial function of $x_{A}$. Indeed, intervening on $A$ has an effect on $B$, which in turn affects $C$. In this example, $Pr(x_{C}|do(x_{A}))$ captures this indirect effect. In line with our definition of causal effects, the causal effect of $A$ on $C$ is given by how $Pr(x_{C}|do(x_{A},x_{B}))$ changes with $x_{A}$. In this case, $Pr(X_{c}|do(x_{A},x_{B}))$ is a constant function of $x_{A}$, which is consistent with $A$ having no direct causal impact on $x_{C}$.

8 Literature Review

In economic theory, the work most closely related to ours is a series of papers by Spiegler ([18], [19], [20]). The main difference is the focus of the papers. Spiegler’s work does not provide a definition of the term “causal effect”, except that it can be represented via a DAG that satisfies two properties. First, the DAG factorizes the correlation structure in the DM’s beliefs; second, the arrows in the DAG are interpreted as pointing from cause to effect. Given these assumptions, Spiegler asks what types of mistakes a DM with a misspecified causal model might make. In our paper, we first define what a causal relation is and then seek to understand which axioms on behavior allow us to represent causal effects in the language of DAGs that factorize the DM’s beliefs. The uniqueness claim in Theorem 1 provides the point of contact between the two papers. Under our definition of causality, a DAG can simultaneously factorize the DM’s beliefs while retaining a causal interpretation only if Axioms 1 through 3 hold. Furthermore, under the axioms, a graph $G$ both represents a DM’s correlation structure and is interpreted causally (in the sense that arrows point from cause to effect); only the definition of causal effect is as in Definition 2.

In decision theory, Karni ([10], [11]) explores models where a DM can affect the states that are realized. In those papers, the primitive objects are a set of actions and a set of consequences. States of nature are defined as mappings from actions that a DM might take to consequences that arise from those actions; that the mapping $Action\rightarrow Outcome$ is stochastic reflects that states are stochastically realized. A DM can affect the states that occur by making an appropriate choice of action. This idea is similar to our idea of a policy intervention, since a policy $p$ can be seen as an action that the DM takes that affects the realization of states. Indeed, we can map Karni’s set of actions to our set $\mathcal{P}$ and Karni’s outcomes to realizations of our state space, $X$, and a Karni state is a mapping $s:\mathcal{P}\rightarrow X$. The main difference arises in that we impose – objectively – a consistency condition: if a policy $p$ intervenes and shifts variable $j$ to value $x_{j}$, a state $s$ cannot map this policy to a realization $x^{\prime}_{j}\neq x_{j}$. Karni has a version of this condition, but it is imposed subjectively on preferences. Moreover, the focus of Karni’s paper is not on using these ideas to discuss causal effects or understand what types of models reflect normative definitions of causality. Rather, Karni focuses on obtaining subjective expected utility representations of his preferences. For this reason, while a formal connection exists, the substance of the research agenda is different.

The statistics and computer science literature includes research that uses graphical methods to represent the conditional independence structure of any given joint probability law (see Dawid [1], Geiger et al. [3], and Lauritzen et al. [12]). Specifically, Dawid [1] and Geiger et al. [3] show that, given a probability distribution over a set of variables, $p(\cdot)$, and given a graph $G$ that represents $p$, the D-separation criterion for graphs (see Definitions 11 and 9) summarizes the independence structure of $p$. Our proof relies on the one-to-one correspondence between variables that satisfy the D-separation criterion and variables that are conditionally independent. This is the main point of contact between our paper and that body of work. Lauritzen et al. [12] provide alternative graphical tests for D-separation that may be used to obtain alternative proofs of our results.

In causal statistics, the most closely related papers are those in the Bayesian networks literature (see Spirtes [21], Pearl [14], and follow-up work). Two main points of contact exist between that literature and our paper. First, the statistical causality literature offers no formal definition of the term “causal relation”, and the exact meaning of this phrase is left to the researcher’s common sense. As Pearl states, “The first step in this analysis is to construct a causal diagram such as the one given in Fig. [1] (sic. ), which represents the investigator’s understanding of the major causal influences among measurable quantities in the domain” and later “ The purpose of the paper is not to validate or repudiate such domain-specific assumptions but, rather, to test whether a given set of assumptions is sufficient for quantifying causal effects from non-experimental data, for example, estimating the total effect of fumigants on yields”. Second, the numerical value of the causal effect of one variable on another (e.g., education on lifetime earnings) is given by do-probability formalism. As Pearl writes in [15], “* the definition of a “cause” is clear and crisp; variable $X$ is a probabilistic-cause of variable $Y$ if $P(y|do(x))\neq P(y)$ for some values $x$ and $y$*.” By contrast, we show that, under Axioms 1 through 4, there exists a unique definition of causal effect that is both representable via a DAG and consistent with an interventionist perspective on causality. Thus, we show that causal models based on causal diagrams implicitly impose a specific definition of causality. Moreover, Axioms 1 through 4 neither imply nor are implied by a representation of causality in terms of do-probabilities. Contrary to Pearl’s quote, do-probabilities neither define nor are defined by the definition of causality embodied by the causal diagram. Theorem 2 shows that under Axioms 1 through 4, causal effects are represented via a DAG that is compatible with the do-probability formulas. This makes explicit the fundamental restrictions imposed by using do-probabilities to numerically quantify causal effects.

Appendix A Proofs

Proposition 1**.**

Let $\bar{\succ}=(\succ_{\mathcal{J}})_{\mathcal{J}\subset\mathcal{N}}$ be a DM’s preferences, and let $G(\bar{\succ})=(\mathcal{N},E)$ be the directed graph defined by setting $Pa(i)=Ca(i)$ for each $i\in\mathcal{I}$. If $\bar{\succ}$ satisfies Assumption 1, then the following are true:

• •

If $G=(\mathcal{N},F)$ is a directed graph that represents $\bar{\succ}$, then $(j,i)\in F\Rightarrow j\in Ca(i)$.

• •

If $G=(\mathcal{N},F)$ is a directed graph that represents $\bar{\succ}$, then $j\in Ca(i)\Rightarrow(j,i)\in F$ or $i\in Ca(j)$.

Proof.

Let $\bar{\succ}$ be as in the statement of the proposition, $G(\bar{\succ})$ be the directed graph defined by setting $Pa(i)=Ca(i)$ for each $i\in\mathcal{N}$, and $G=(\mathcal{N},F)$ be any other directed graph that represents $\bar{\succ}$. For each $\mathcal{I}\subset\mathcal{N}$ and each realization $x_{\mathcal{I}}\in X_{\mathcal{I}}$, let $\mu_{x_{\mathcal{I}}}\in\Delta(X_{-\mathcal{I}})$ represent beliefs obtained from $\succ_{x_{\mathcal{I}}}$.

We first show that $j\in Ca(i)\Rightarrow(j,i)\in F$ or $i\in Ca(j)$. If $j\in Ca(i)$, then the function $T:X_{j}\rightarrow\mathbb{R}$ defined as $T(x_{j})=\mu_{x_{-\{i,j\}},x_{j}}(x_{i})$ is not constant in $x_{j}$. Additionally, by Assumption 1‘, $\mu_{x_{-\{i,j\}}}(x_{i}|x_{j})=T(x_{j})$. Thus, $i$ and $j$ are not independent after intervening on $\{i,j\}^{\complement}$. Because $G$ represents $\bar{\succ}$, then $G_{-\{i,j\}}$ represents $\succ_{-\{i,j\}}$. Thus, either $(i,j)\in F$ or $(j,i)\in F$ (if not, $G_{-\{i,j\}}$ would treat $i$ and $j$ as independent, which is a contradiction). If $(j,i)\in F$, the proof concludes. Therefore, let $(j,i)\notin F$ so that $(i,j)\in F$. Because $G$ represents $\bar{\succ}$, this means that $\mu_{x_{-\{i,j\}}}(x_{j}|x_{i})=\mu_{x_{-\{j\}}}(x_{j})$. By definition, the above equation says $i\in Ca(j)$, as desired.

We now show $(j,i)\in F\Rightarrow j\in Ca(i)$. First, note that for all $x\in X$, $\mu_{x_{-\{i,j\}}}(x_{i},x_{j})=\mu_{x_{-\{i,j\}}}(x_{j})\mu_{x_{-\{i,j\}}}(x_{i}|x_{j})$. Because $G$ represents $\bar{\succ}$, $(j,i)\in F$ and the minimality condition in Definition 5, jointly imply that $i$ and $j$ are not independent after intervening on $\{i,j\}^{\complement}$. That is, $\mu_{x_{-\{i,j\}}}(x_{i}|x_{j})$ is not constant in $x_{j}$. Moreover, because $G$ represents $\bar{\succ}$ and $(j,i)\in F$, we obtain that $\mu_{x_{-\{i,j\}}}(x_{i}|x_{j})=\mu_{x_{-\{i\}}}(x_{i})$. Therefore, there is a value of $x_{-\{j\}}$ for which $T(x_{j})=\mu_{x_{-\{i\}}}(x_{i})$ is not constant in $x_{j}$. Therefore, $j\in Ca(i)$. ∎

Remark 1**.**

Without axiom $2$, any representing graph must include the causal links in the sense of Definition 2 (i.e., $(j,i)\in F\Rightarrow j\in Ca(i)$), but $F$ could omit some arrows. However, only arrows involved in 2 cycles are omitted.

Before proving Theorem 1, we need two Lemmas. Let $i$ be a variable and $\mathcal{I},\mathcal{J}$ be two disjoint sets of variables that do not contain $i$. It is known from Dawid ([1]) and Pearl ([14]) that $i$ is independent of $\mathcal{I}$ conditional on $\mathcal{J}$ if, and only if, $\mathcal{J}$ D-separates $\{i\}$ from $\mathcal{I}$ (see below for a definition of D-separation). The next two lemmas prove that, for each variable $i$, $Ca(i)$ D-separates $\{i\}$ from all sets $\mathcal{J}$ that satisfy $\mathcal{J}\subset ND(i)$, where $ND(i)$ is the set of nondescendants of $i$. Furthermore, $Ca(i)$ is the smallest set that has this property.

Definition 9**.**

Let $\mathcal{I},\mathcal{J},\mathcal{K}\subset\mathcal{N}$ be three disjoint sets of variables. We say that $\mathcal{K}$ D- separates $\mathcal{I}$ from $\mathcal{J}$ if for each undirected path between a variable in $\mathcal{I}$ and a variable in $\mathcal{J}$, one of the following properties holds:

• •

There is a node $w$ along the path such that $w$ is a collider (that is, there are nodes $w_{0},w_{1}$ in the path such that $w_{0}\rightarrow w\leftarrow w_{1}$), such that $w\notin\mathcal{K}$ and $\mathcal{K}\subset ND(w)$.

• •

There is a node $w$ along the path such that $w$ is not a collider and such that $w\in\mathcal{K}$.

Lemma 1**.**

Fix $\mathcal{K}\subset\mathcal{N}$ and $x_{\mathcal{K}}\in X_{\mathcal{K}}$. Let $G_{\mathcal{K}}$ represent $\succ_{x_{\mathcal{K}}}$. For each $i\in\mathcal{N}$, $Ca(i)\setminus\mathcal{K}$ D-separates $\{i\}$ from ND(i)\setminus\mathcal{K}\equiv\{\hat{j}\in\mathcal{K}^{\complement}:\text{ i$is not an indirect cause of$\hat{j}. }\}.

Proof.

Choose j\in\{\hat{j}\in\mathcal{K}^{\complement}:\text{ i$is not an indirect cause of$\hat{j}. }\} Choose an undirected trail $t$ from $j$ to $i$. That is, $t=(i_{0},...,i_{N})$ where $i_{0}=j$, $i_{N}$=i, and, for each $n\in\{1,...,N\}$, either $(i_{n-1},i_{n})\in E$ or $(i_{n},i_{n-1})\in E$. First, since $i$ is not an indirect cause of $j$, then $t$ cannot be a directed path from $i$ to $j$. That is, $t$ cannot be such that $(i_{n},i_{n-1})\in E$ for each $n$. Second, if $t$ is a directed path from $j$ to $i$ (that is, $(i_{n-1},i_{n})\in E$ for each $n$), then $t$ is blocked by $i_{N-1}\in Ca(i)\setminus\mathcal{K}$. Third, assume that $t$ is not directed in any direction. Then, $t$ has colliders and/or tail-to-tail nodes. Let $i_{n}$ be the last node that is either a collider or a tail-to-tail node. Let $q=(i_{n},...,i_{N})$ be the trail starting at $i_{n}$. By the definition of $i_{n}$, $q$ must be directed. Assume that $q$ is directed from $i_{n}$ to $i$. Then, $i_{n}$ is tail to tail. Then, $t$ is blocked by $i_{N-1}$. Finally, assume that $q$ is directed from $i$ to $i_{n}$. Then, $i_{n}$ is a collider. If $i_{n}\in Ca(i)\setminus\mathcal{K}$, then $(i_{n},i,q)$ is a cycle. Thus, $i_{n}\notin Ca(i)\setminus\mathcal{K}$. By a similar argument, no descendants of $i_{n}$ can be in $Ca(i)\setminus\mathcal{K}$. Therefore, $i_{n}$ blocks $t$. Since each trail joining $j$ to $i$ is blocked, this concludes the proof. ∎

Lemma 2**.**

Fix $\mathcal{K}\subset\mathcal{N}$, $x_{\mathcal{K}}\in X_{\mathcal{K}}$, and $i\in\mathcal{K}^{\complement}$. Let $G_{\mathcal{K}}$ represent $\succ_{x_{\mathcal{K}}}$. If $\mathcal{T}\subset\mathcal{K}^{\complement}$ satisfies that $\mathcal{T}$ D-separates $\{i\}$ from $ND(i)$, then $Ca(i)\setminus\mathcal{K}\subset\mathcal{T}$

Proof.

Let $\mathcal{K}$, $i$, and $\mathcal{T}$ be as in the statement of the Lemma. Assume that $w\in Ca(i)\setminus\mathcal{K}$. Then, $w\in ND(i)$ because otherwise, $G_{\mathcal{K}}$ would not be acyclic. Consider the path $w\rightarrow i$. Then, $\mathcal{T}$ can D-separate this path only if $w\in\mathcal{T}$. Thus, $Ca(i)\setminus\mathcal{K}\subset\mathcal{T}$. ∎

Lemma 3**.**

Assume that Axiom 3 holds. Then, $(\forall i\in\mathcal{N})$, $(\forall j\text{ such that }i\notin ICa(j))$ and $(\forall\mathcal{K}\subset\mathcal{N}\setminus\{i,j\})$,

[TABLE]

Proof.

Let $i\in\mathcal{N}$ be arbitrarily chosen and $j$, $\mathcal{K}$ be as in the statement of the theorem. Define the following sets:

[TABLE]

Axiom 3 implies the following:

[TABLE]

The above follows because $i\notin Ca^{0}(j)$ (since $i\notin ICa(j)$) and because $Ca(Ca^{0}(j))=\emptyset$. By induction, Axiom 3 implies the following for all $k$:

[TABLE]

We have already shown that the above is true for $k=0$. Assume that it is true for some value $k$. We need to show that this is true for $k+1$. Note that $Ca(Ca^{k+1}(j))\subset\cup_{n=0}^{k}Ca^{n}(t)$ by definition. Then, by Axiom 3, $i\perp_{\mathcal{K}}Ca^{k+1}(j)|Ca(i)\cup\cup_{n=0}^{k}Ca^{n}(j)$. Moreover, according to the inductive hypothesis, $i\perp_{\mathcal{K}}\cup_{n=0}^{k}Ca^{n}(j)|Ca(i)$. Thus, $i\perp_{\mathcal{K}}(\cup_{n=0}^{k}Ca^{n}(j)\cup Ca(j)^{k+1})|Ca(i)$. Since $Ca(j)=\cup_{n=0}^{\infty}Ca(j)^{n}$, this completes the proof. ∎

As a corollary of the Lemma 3, we obtain the Markov property $i\perp_{\mathcal{K}}j|Ca(i)\setminus\mathcal{K}$ whenever $i\notin ICa(j)$. Indeed, by the intersection property of conditional independence, $i\perp_{\mathcal{K}}j|Ca(i)\cup Ca(j)$ and $i\perp_{\mathcal{K}}Ca(j)|Ca(i)$ implies $i\perp_{\mathcal{K}}j|Ca(i)$.

Theorem 1.

Let $\bar{\succ}$ satisfy Assumption 1. The following are equivalent:

• •

Axioms 1 and 3 hold;

• •

$(\exists G)$* such that $G$ is a DAG and represents $\bar{\succ}$.*

Furthermore, if $G$ represents $\bar{\succ}$, then $G=G(\bar{\succ})$.

Proof.

The uniqueness claim is proven in Proposition 1.

We now prove that the axioms imply the existence of a representation. Without loss of generality, label the variables such that $j<i$ implies $j\in ND(i)$. Construct $G$ by setting $Pa(i)=Ca(i)$. By Axiom 1, $G$ is acyclic. Indeed, if for some length $k\in\mathbb{N}$, there were a cycle $e=((i_{1},i_{2}),(i_{2},i_{3}),...,(i_{k},i_{1}))$, then $i_{1}$ would be an indirect cause of itself. Choose any set $\mathcal{K}\subset\mathcal{N}$ and any realization $x_{\mathcal{K}}\in X_{\mathcal{K}}$. We need to show that $\mu_{x_{\mathcal{K}}}(x_{-\mathcal{K}})=\Pi_{i\notin\mathcal{K}}\mu_{x_{\mathcal{K}}}(x_{i}|Ca(i)\setminus\mathcal{K})$. By our enumeration, $\{j\notin\mathcal{K}:j<i\}\subset\{j\in\mathcal{N}:i\text{ is not an indirect cause of }j\}$. Through Lemma 3, Axiom 3 implies that $\mu_{x_{\mathcal{K}}}(x_{i}|x_{\{j\notin\mathcal{K}:j<i\}})=\mu_{x_{\mathcal{K}}}(x_{i}|x_{Ca(i)\setminus\mathcal{K}})$. By the chain rule, we know that $\mu_{x_{\mathcal{K}}}(x_{-\mathcal{K}})=\Pi_{i=1,i\notin\mathcal{K}}^{N}\mu_{x_{\mathcal{K}}}(x_{i}|x_{\{j\notin\mathcal{K}:j<i\}})$. Combining the last two claims, $\mu_{x_{\mathcal{K}}}(x_{-\mathcal{K}})=\Pi_{i=1,i\notin\mathcal{K}}^{N}\mu_{x_{\mathcal{K}}}(x_{i}|x_{Ca(i)\setminus\mathcal{K}})$, which is what we wanted to prove. We now prove the minimality of $Ca(i)$. Assume that $\mathcal{J}\subsetneq Ca(i)$. Axiom 2 states that $i\not\perp_{\mathcal{K}}(Ca(i)\setminus\mathcal{K}\setminus\mathcal{J})|\mathcal{J}$. Thus, the factorization formula is minimal.

Now, suppose that $G$ is a DAG that represents $\bar{\succ}$. By our uniqueness claim, without loss of generality, $G$ is such that $Pa(i)=Ca(i)$. By contrapositive, that $G$ is acyclic implies Axiom 1 holds. If Axiom 1 did not hold, there would exist $i$ and a sequence $(i,i_{1},...,i_{T},i)$ such that $i\in Ca(i_{1})$, for all $t\in\{1,...,T-1\}$, $i_{t}\in Ca(i_{t+1})$, and $i_{T}\in Ca(i)$. Thus, $((i,i_{1}),...,(i_{t-1},i_{t}),...,(i_{T},i))$ is a cycle in $G$. Axiom 2 holds by the minimality requirement in the definition of representation. Indeed, if there were $\mathcal{K}$, $\mathcal{J}$ such that $i\perp_{\mathcal{K}}(Ca(i)\setminus\mathcal{K}\setminus\mathcal{J})|\mathcal{J}$, then the representation would not be minimal. To see that Axiom 3 holds, note that $Ca(i)\cup Ca(j)\setminus\mathcal{K}$ blocks all paths from $i$ to $j$ in $G_{\mathcal{K}}$. Indeed, assume that $p$ is an undirected path from $i$ to $j$ that is not blocked, and enumerate $p=(i,i_{0},...,i_{T},j)$. Because $p$ is not blocked, $i_{0}\notin Ca(i)$ and $i_{T}\notin Ca(j)$. Indeed, if either $i_{0}\in Ca(i)$ or $i_{T}\in Ca(j)$, then either $i_{0}$ is not a collider or $i_{T}$ is not a collider, thus implying that $p$ is blocked by $Ca(i)\cup Ca(j)$. Therefore, $p$ has a collider. Let $n$ be the smallest number such that $i_{n}$ is a collider and $m$ be the largest number such that $i_{m}$ is a collider (possibly $n=m$). Note that because $G$ is acyclic, $i_{n}\notin Ca(i)$ and $i_{m}\notin Ca(j)$. Because of this, and because $p$ is not blocked, the following must be true:

• •

(i) $i_{n}\in Ca(j)$,

• •

(ii) $i_{m}\in Ca(i)$.

Then, the directed path that goes from $i$ to $i_{n}$, jumps to $j$, returns to $i_{m}$, and skips back to $i$ is a cycle. 666Formally, this is the path $q=(i,i_{0},...i_{n},j,i_{T},i_{T-1},...,i_{m},i)$ This constitutes a contradiction. Thus, every path $p$ from $i$ to $j$ is blocked by $Ca(i)\cup Ca(j)$. Thus, Axiom 3 holds. Similarly, $Ca(i)\cup\{j\}\setminus\mathcal{K}$ blocks all paths from $i$ to $Ca(j)\setminus\mathcal{K}$ in $G_{\mathcal{K}}$. Indeed, let $p$ be an undirected path from $i$ to $Ca(j)$, and assume that $p$ is not blocked by $Ca(i)\cup\{j\}$. Enumerate $p=(i,i_{0},...,i_{T},k)$, where $k\in Ca(j)$. Because $j\in ND(i)$, then $p$ cannot be directed from $i$ to $k$. If $i_{0}\in Ca(i)$, then $i_{0}$ blocks $p$. If $i_{0}\notin Ca(i)$, since $p$ is not directed, $p$ has a collider. Let $n$ be the smallest number such that $i_{n}$ is a collider. First, note that $i_{n}\notin Ca(i)$ because this would constitute a cycle. Second, if $i_{n}=j$, then $j$ would be a descendant of $i$, a contradiction. Thus, $i_{n}\notin Ca(i)\cup\{j\}$, and hence, $p$ is blocked. Thus, Axiom 3 holds. ∎

Theorem 2.

Let $\bar{\succ}$ satisfy Assumption 1, and let $(\mu_{x_{\mathcal{I}}})_{\mathcal{I}\subset\mathcal{N}}$ be the subjective beliefs elicited from $\bar{\succ}$. The following are equivalent:

• •

Axioms 1 through 4 hold;

• •

$\exists$* a Markov representation of $\mu$, $(G,(h_{1},...,h_{N}),(\varepsilon_{1},...,\varepsilon_{N}))$, such that*

• –

$(\forall\mathcal{J}\subset\mathcal{N})$, $(\forall x_{\mathcal{J}}\in X_{\mathcal{J}})$; $\mu_{x_{\mathcal{J}}}=\mu(\cdot|do(x_{\mathcal{J}}))\in\Delta(X_{-\mathcal{J}})$; and

• –

$G$* represents $\bar{\succ}$.*

Furthermore, if $G$ represents $\bar{\succ}$, then $G=G(\bar{\succ})$.

Proof.

The uniqueness claim was proven in 1.

We first show that the axioms imply the representation. By Theorem 1, Axioms 1 and 2 imply that there exists a DAG $G$ such that $G$ represents $\bar{\succ}$. For each $i\in\mathcal{N}$ let $Pa(i)$ be the set of parents of $i$ in $G$. Note that $Pa(i)=Ca(i)$ by the uniqueness claim. For each $i\in\mathcal{N}$, let $\varepsilon_{i}\sim U[0,1]$. For each realization $x_{i}\in X_{i}$ and each $x_{Pa(i)}\in X_{Pa(i)}$, let $I(x_{i},x_{Pa(i)})\subset[0,1]$ be an interval of length $\mu_{x_{Pa(i)}}(x_{i})$. Because $\sum_{x_{i}\in X_{i}}\mu_{x_{Pa(i)}}(x_{i})=1$ for each $x_{Pa(i)}$, then $I(\cdot,x_{Pa(i)})$ can be chosen to form a partition of $[0,1]$. Fix any variable $i\in\mathcal{N}$, and let $h_{i}(x_{Pa(i)},\varepsilon_{i})=\sum_{x_{i}\in X_{i}}x_{i}\mathbbm{1}_{I(x_{i},x_{Pa(i)})}(\varepsilon_{i})$. By construction, $(G,(h_{1},...,h_{N}),(\varepsilon_{1},...,\varepsilon_{N}))$ is a Markov representation of the beliefs elicited from $\bar{\succ}$. Choose any $\mathcal{J}\subset\mathcal{N}$ and any $i\notin\mathcal{J}$. By Axiom 4, for each $x_{i}\in X_{i}$ and each $x_{Ca(i)\cup\mathcal{J}}\in X_{Ca(i)\cup\mathcal{J}}$, we obtain

[TABLE]

Our Markov representation implies

[TABLE]

By 9 and 10, $\mu_{x_{\mathcal{J}}}(x_{i}|x_{Ca(i)\setminus\mathcal{J}})=\mu(x_{i}|do(x_{\mathcal{J}}),x_{Ca(i)\setminus\mathcal{J}})$. Because $G$ represents $\bar{\succ}$, for each $x\in X$,

[TABLE]

Thus, $\mu_{x_{\mathcal{J}}}(\cdot)=\mu(\cdot|do(x_{\mathcal{J}}))\in\Delta(X_{-\mathcal{J}})$.

We now show that the representation implies the axioms. If there exists a DAG $G$ that represents $\bar{\succ}$, then that Axioms 1 and 2 hold is proven in Theorem 1. Let $i\in\mathcal{N}$, $\mathcal{J}\subset\mathcal{N}\setminus\{i\}$, $f,g\in\mathbb{R}^{X_{i}}$, $x_{\mathcal{J}}\in X_{\mathcal{J}}$ and $x_{Ca(i)\setminus\mathcal{J}}\in X_{Ca(i)\setminus\mathcal{J}}$ be arbitrarily selected. We know from the Markov representation that for each $x_{i}\in X_{i}$, $\mu^{i}_{x_{\mathcal{J}}}(x_{i}|x_{Ca(i)\setminus\mathcal{J}})=\mu^{i}(x_{i}|x_{Ca(i)})$, where $\mu^{i}$ denotes the marginal of $\mu$ on $X_{i}$. Thus, Axiom 4 holds. ∎

Proposition 2 is a direct consequence of Theorem 2 and Theorem 3, which is stated and proven below.

Theorem 3**.**

Let $\bar{\mu}=\{\mu_{p}:p\in\mathcal{P}\}$ be a collection of intervention beliefs, and let $G$ be a DAG that represents $\bar{\mu}$. If equations 11 and 12 hold, then Axiom 4 holds.

Proof.

Let $\bar{\mu}$ and $G$ be as in the theorem. Let $i\in\mathcal{N}$ and $\mathcal{J}\subset\mathcal{N}\setminus\{i\}$. We want to show that $\mu(x_{i}|x_{Ca(i)})=\mu_{x_{\mathcal{J}}}(x_{i}|x_{Ca(i)\setminus\mathcal{J}})$. Let $\mathcal{J}^{*}\equiv\mathcal{J}\cap Ca(i)$; that is, $\mathcal{J}^{*}$ are those variables in $\mathcal{J}$ that are direct causes of $i$. Thus, we need to show that $\mu(x_{i}|x_{Ca(i)})=\mu_{x_{\mathcal{J}}}(x_{i}|x_{Ca(i)\setminus\mathcal{J}^{*}})$; we do this in two steps.

First, we show that $\mu(x_{i}|x_{Ca(i)})=\mu_{x_{\mathcal{J}^{*}}}(x_{i}|x_{Ca(i)\setminus\mathcal{J}^{*}})$. To see this, note that $Ca(i)\setminus\mathcal{J}^{*}$ blocks any path from $\{i\}$ to $\mathcal{J}^{*}$ in graph $G_{(\mathcal{J}\setminus\mathcal{J}^{*})^{in},(\mathcal{J}^{*})^{out}}$. Indeed, let $p$ be any path from $i$ to some $j\in\mathcal{J}^{*}$ in graph $G_{(\mathcal{J}\setminus\mathcal{J}^{*})^{in},(\mathcal{J}^{*})^{out}}$. Write $p=(i_{0},...,i_{T})$, where $i_{0}=i$ and $i_{T}=j$. Because $j\in Ca(i)$, then $p$ cannot be a directed path from $i$ to $j$; otherwise, $G$ would have a cycle. Similarly, $p$ cannot be a directed path from $j$ to $i$ since $G_{(\mathcal{J}\setminus\mathcal{J}^{*})^{in},(\mathcal{J}^{*})^{out}}$ has no arrows emerging from $j$. Therefore, $p$ has a collider or a tail-to-tail node. Let $w$ be the first node that is either a collider or a tail-to-tail node. First, assume $w$ is tail to tail. Then, $p$ is of the form $i\leftarrow i_{1}(...)\leftarrow w\rightarrow(...)j$. Then, $i_{1}\in Ca(i)\setminus\mathcal{J}^{*}$: indeed, $i_{1}\in Ca(i)$ and $i_{1}\notin\mathcal{J}^{*}$ (since there are no arrows emerging from nodes in $\mathcal{J}^{*}$). Furthermore, $i_{1}$ is not a collider. Then, $i_{1}$ blocks $p$. Now, assume that $w$ is a collider rather than tail to tail. Then, $p$ is of the form $i\rightarrow i_{1}(...)\rightarrow w\leftarrow(...)j$. Then, $w$ is a descendant of $i$, so neither $w$ nor any $w$ descendant is in $Ca(i)$. A fortori, neither $w$ nor any descendant of $w$ is in $Ca(i)\setminus\mathcal{J}^{*}$. Thus, $w$ blocks $p$. Therefore, by formula 11, we have $\mu(x_{i}|x_{Ca(i)})=\mu_{x_{\mathcal{J}^{*}}}(x_{i}|x_{Ca(i)\setminus\mathcal{J}^{*}})$.

Second, we show that $\mu_{x_{\mathcal{J}^{*}}}(x_{i}|x_{Ca(i)\setminus\mathcal{J}^{*}})=\mu_{x_{\mathcal{J}^{*}\cup(\mathcal{J}\setminus\mathcal{J}^{*})}}(x_{i}|x_{Ca(i)\setminus\mathcal{J}^{*}})$. This is because $Ca(i)$ blocks all paths between $i$ and $\mathcal{J}\setminus\mathcal{J}^{*}$ in graph $G_{{\mathcal{J}\setminus\mathcal{J}^{*}(Ca(i)\setminus\mathcal{J}^{*})}^{in}}$. Note that if $\mathcal{J}\setminus\mathcal{J}^{*}$ contains only nondescendants of $i$, then the result is a direct consequence of lemma 1. Let $p$ be a path (not necessarily directed) between $i$ and $j\in\mathcal{J}\setminus\mathcal{J}^{*}$. By contradiction, assume that $j\in\mathcal{J}\setminus\mathcal{J}^{*}$ is a descendant of $i$. Then, $j\notin Ca(i)$, and $j$ is not an ancestor of any node in $Ca(i)$. Therefore, $j\in\mathcal{J}\setminus\mathcal{J}^{*}(Ca(i)\setminus\mathcal{J}^{*})$, so there are no arrows in $j$. Therefore, no path from $i$ to $j$ can be directed in any direction, so there is at least one collider or tail-to-tail node. Let $w$ be the first such node, and assume thats $w$ is a collider. Then, $p$ is of the form $i\rightarrow(...)\rightarrow w\leftarrow(...)\leftarrow j$. Then, neither $w$ nor any descendant of $w$ can be in $Ca(i)$, so $p$ is blocked by $Ca(i)$. Alternatively, $w$ is a tail-to-tail node. Then, $p$ is of the form $i\leftarrow i_{1}(...)\leftarrow w\rightarrow(...)\leftarrow j$ (with possibly $w=i_{1}$). Then, $i_{1}\in Ca(i)$, and $i_{1}$ is not a collider. Thus, $Ca(i)=\mathcal{J}^{*}\cup(Ca(i)\setminus\mathcal{J}^{*})$ blocks $p$. Thus, by formula 12, $\mu_{x_{\mathcal{J}^{*}}}(x_{i}|x_{Ca(i)\setminus\mathcal{J}^{*}})=\mu_{x_{\mathcal{J}^{*}\cup(\mathcal{J}\setminus\mathcal{J}^{*})}}(x_{i}|x_{Ca(i)\setminus\mathcal{J}^{*}})=\mu_{x_{\mathcal{J}}}(x_{i}|x_{Ca(i)\setminus\mathcal{J}})$.

Combining this with the first step, we conclude that $\mu(x_{i}|x_{Ca(i)})=\mu_{x_{\mathcal{J}}}(x_{i}|x_{Ca(i)\setminus\mathcal{J}})$ as desired.

∎

Appendix B The Rules of Causal Calculus

Theorem 2 in Section 7.2 shows a formalism (do-probability) that is useful for identifying causal effects. Under such a formalism, Pearl [14] shows the following two results. First, if a DAG satisfies certain properties, then intervening on a variable and conditioning on a variable generate the same distribution – that is, $p(x_{i}|x_{j},do(x_{k}))=p(x_{i}|x_{j},x_{k})$. Second, if a DAG satisfies certain other properties, then one can drop interventions altogether – that is, $p(x_{i}|x_{j},do(x_{k}))=p(x_{i}|x_{j})$. These two results are generally known as the “rules of causal calculus”. Huang and Valtorta [8]) go a step further: they show that these rules are complete. That is, any identification theorem that is true can be proven by iterative application of these rules of causal calculus.

However, there remains an open question: are do-probabilities the only model of causality under which these rules apply, or are these rules consistent with other models of causality? Using our framework, we prove that the do-probability model is the only model of causal effects where these two rules are complete.

Explaining and proving the result above requires two definitions: we need to define specific truncations of a DAG, and we need the definition of a blocked path. We provide these definitions and then formally state the result. Appendix B discusses the intuition for why we need these definitions.

Definition 10**.**

Given $G$ and three disjoint sets of variables $\mathcal{I},\mathcal{J},\mathcal{K}\subset\mathcal{N}$, the truncated DAGs $G_{\mathcal{I}^{in}}$, $G_{\mathcal{J}^{out}}$, and $G_{\mathcal{I}^{in},\mathcal{J}(\mathcal{K})^{out}}$ are defined as follows:

• 1

$G_{\mathcal{I}^{in}}$* is obtained from $G$ by eliminating all arrows pointing to nodes in $\mathcal{I}$,*

• 2

$G_{\mathcal{I}^{in},\mathcal{J}^{out}}$* is obtained from $G$ by eliminating all arrows emerging from nodes in $\mathcal{J}$ and all arrows pointing to nodes in $\mathcal{I}$,*

• 3

$G_{\mathcal{I}^{in},\mathcal{J}(\mathcal{K})^{in}}$* is obtained by eliminating all arrows pointing to nodes in $\mathcal{J}(\mathcal{K})$ and $\mathcal{I}$, where $\mathcal{J}(\mathcal{K})$ is the set of $\mathcal{J}$ nodes that are not ancestors of any $\mathcal{K}$ nodes in $G_{\mathcal{I}^{in}}$.*

The following figures show the base DAG, $G$, and its corresponding truncations. In all cases, $\mathcal{J}=\{J_{0},J_{1}\}$, $\mathcal{I}=\{I\}$, $\mathcal{K}=\{K\}$.

For the following definition, suppose that $Q$ is an undirected path between two nodes, i.e., a collection of nodes, regardless of directionality, and that $q$ is a node on $Q$. For example, Figure 12 shows an undirected path $Q=(J_{1},I,J_{0},K)$ from $J_{1}$ to $K$. We say that $Q$ has converging arrows at $q$ if there exist nodes $q_{0}$ and $q_{1}$ that are adjacent to $q$ in $Q$ such that $q_{0}\rightarrow q\leftarrow q_{1}$. For example, path $Q=(J_{1},I,J_{0},K)$ has converging arrows at $I$. We say that $Q$ does not have converging arrows at $q$ if for all nodes $q_{0}$ and $q_{1}$ that are adjacent to $q$ in $Q$, either $q_{0}\rightarrow q\rightarrow q_{1}$ or $q_{0}\leftarrow q\rightarrow q_{1}$ holds. For example, $Q=(J_{1},I,J_{0},K)$ does not have converging arrows at $J_{0}$.

Definition 11**.**

Let $\mathcal{I},\mathcal{J},\mathcal{K}$ be three disjoint sets of variables, and let $Q$ be an undirected path between a node in $\mathcal{I}$ and a node in $\mathcal{J}$. We say that $\mathcal{K}$ blocks $Q$ if there exists a node $q$ on $Q$ such that one of the following conditions holds:

• •

$Q$* has converging arrows at $q$, and neither $q$ nor any of its descendants is in $\mathcal{K}$, and*

• •

$Q$* does not have converging arrows at $q$ and $q\in\mathcal{K}$.*

Below, we state the two rules of causal calculus (in the language of our model) and Proposition 2.

Rule 1**.**

(Exchanging intervention and observation.) Let $\mathcal{I}_{0},\mathcal{I}_{1},\mathcal{I}_{2},\mathcal{I}_{3}$ be disjoint sets of variables. If $\mathcal{I}_{1}\cup\mathcal{I}_{3}$ block all paths from $\mathcal{I}_{0}$ to $\mathcal{I}_{2}$ in graph $G_{\mathcal{I}_{1}^{in},\mathcal{I}_{2}^{out}}$, then

[TABLE]

Rule 2**.**

(Eliminating interventions.) Let $\mathcal{I}_{0},\mathcal{I}_{1},\mathcal{I}_{2},\mathcal{I}_{3}$ be disjoint sets of variables. If $\mathcal{I}_{1}\cup\mathcal{I}_{3}$ block all paths from $\mathcal{I}_{0}$ to $\mathcal{I}_{2}$ in graph $G_{\mathcal{I}_{1}^{in},\mathcal{I}_{2}(\mathcal{I}_{3})^{in}}$, then

[TABLE]

Proposition 2**.**

Let $\bar{\succ}$ satisfy Axioms 1 through 3, let $G$ represent $\bar{\succ}$, and let $\{\mu_{p}:p\in\mathcal{P}\}$ be the DM’s intervention beliefs. Then, the following statements are equivalent.

• •

$\bar{\succ}$* satisfies Axiom 4.*

• •

Rules 1 and 2 hold. Furthermore, if $\mu_{p}$ is identified for some $p\in\mathcal{P}$, then the identification is obtained by iterative application of these two rules.

With Proposition 2, we can refer back to Example 3 in Section 7.2 and obtain the identification result by applying Rules 1 and 2. In Rule 2, set $\mathcal{I}_{0}=\{L\}$, $\mathcal{I}_{1}=\emptyset$, $\mathcal{I}_{2}=\{E\}$, and $\mathcal{I}_{3}=\{A\}$. The corresponding truncated DAG is $G$ itself. In $G$, $A$ blocks the unique path from $E$ to $L$ since no converging arrows exist at $A$. Thus, $\mu_{e}(l|a)=\mu(l|a)$. Similarly, in Rule 1, set $\mathcal{I}_{0}=\{A\}$, $\mathcal{I}_{1}=\emptyset$, $\mathcal{I}_{2}=\{E\}$, and $\mathcal{I}_{3}=\emptyset$. In the truncated graph that results, $E$ is isolated from all other variables, so any path from $E$ to $A$ is blocked; thus, $\mu_{e}(a)=\mu(a|e)$. These two conclusions yield the identification of $\mu_{e}(l)=\sum_{a}\mu(l|a)\mu(a|e)$.

B.1 On the relevance of blocks

In what follows, we intuitively explain why the notion of a block is relevant for analyzing conditional independence. Furthermore, we provide intuition for why the truncations in Figure 13 are the relevant truncations for identifying intervention beliefs.

To illustrate the notion of a block, see Figure 13 below. The singleton $\{K\}$ blocks all paths from $J_{1}$ to $J_{0}$. Indeed, one such path is $J_{1}\rightarrow K\rightarrow J_{0}$. This path is blocked by $\{K\}$ because (i) the path has no converging arrows at $K$, and (ii) $K\in\{K\}$. The other path from $J_{1}$ to $J_{0}$ is $J_{1}\rightarrow I\leftarrow J_{0}$. This path is blocked by $\{K\}$ because $I$ is a node along the path such that there are converging arrows at $I$, but neither $I$ nor any of its descendants are in $\{K\}$.

The notion of a block is a graphical depiction of conditional independence. Indeed, that a path exists between two sets of variables, $\mathcal{I}$ and $\mathcal{J}$, implies that $\mathcal{I}$ and $\mathcal{J}$ are (a priori) statistically dependent: any variable $w$ present on a path from $\mathcal{I}$ to $\mathcal{J}$ may potentially act as a correlating device between $\mathcal{I}$ and $\mathcal{J}$.

In particular, the position of a variable $w$ on a path between $\mathcal{I}$ and $\mathcal{J}$ is relevant to the way in which $w$ correlates these variables. Say that there is a path $i\rightarrow w\leftarrow j$, where $i\in\mathcal{I}$ and $j\in\mathcal{J}$; i.e., there is a path joining $\mathcal{I}$ and $\mathcal{J}$ that has converging arrows at $w$. This implies that observations of $w$ (and its descendants) are informative about $i$ and $j$ simultaneously. However, interventions of $w$ are useless for the purposes of predicting the value of either $i$ or $j$ since neither $w$ nor any of its descendants are a cause of either $i$ or $j$. By contrast, if there is a path of the from $i\rightarrow w\rightarrow j$ or $i\leftarrow w\rightarrow j$ (i.e., a path with nonconverging arrows), then we know that both observations of and interventions on $w$ are useful for predicting the values of $i$ and $j$, although in different ways. In the case in which $i\leftarrow w\rightarrow j$, observing or intervening on $w$ provides the same joint information about $i$ and $j$ since $w$ is a common direct cause of $j$ and $i$. However, if $i\rightarrow w\rightarrow j$, intervening on $w$ provides information about $j$ (since $w$ is a direct cause of $j$) but provides no information about $i$ (since $w$ is neither a direct nor an indirect cause of $i$). In this case, intervening on $w$ breaks down the statistical dependence of $i$ and $j$ in a way that is different from simply conditioning on observations of $w$. This sparks a natural question: can the structure of the graph tell us something about the conditional independence properties of the underlying conditional and do-probability distributions? This is the object of study in Dawid ([1]), Geiger, Pearl, and Verma ([3]), Lauritzen et al ([12]) and others. The rules of causal calculus are a particular way in which the structure of the graph is informative about intervention beliefs.