Causal Graphs

Let $V$ be a finite set. A directed graph on $V$ is a pair $G=\left(V,E\right)$ where $E\subseteq V\times V$ is a set of ordered pairs. The elements of $V$ are the vertices or nodes of $G$ and the elements of $E$ are its directed edges. The edge $\left(X,Y\right)\in E$ is written $X\rightarrow Y.$

Let $G=\left(V,E\right)$ be a directed graph and $X,Y\in V.$ The vertex $X$ is a parent of $Y,$ and $Y$ is a child of $X,$ if and only if $X\rightarrow Y\in E.$ The set of parents of $Y$ is written $\mathrm{pa}\left(Y\right).$

Let $G=\left(V,E\right)$ be a directed graph. A path in $G$ is a sequence of distinct vertices $V_1,\ldots,V_k$ such that consecutive vertices are joined by an edge in either direction. A directed path from $V_1$ to $V_k$ is a path in which every edge points from $V_j$ to $V_{j+1}.$ A directed cycle is a directed path from a vertex to itself.

Let $G=\left(V,E\right)$ be a directed graph and $X,Y\in V.$ The vertex $X$ is an ancestor of $Y,$ and $Y$ is a descendant of $X,$ if and only if there is a directed path from $X$ to $Y.$ The set of descendants of $X$ is written $\mathrm{de}\left(X\right).$

A directed acyclic graph (DAG) is a directed graph with no directed cycles. In a DAG the vertices can be ordered so that every edge points from an earlier vertex to a later one.

Let $G$ be a DAG and $V_{j-1}\rightarrow V_j\rightarrow V_{j+1}$ or $V_{j-1}\leftarrow V_j\leftarrow V_{j+1}$ be consecutive vertices on a path. The vertex $V_j$ is a chain on the path. Let $V_{j-1}\leftarrow V_j\rightarrow V_{j+1}$ be consecutive vertices on a path. The vertex $V_j$ is a fork on the path. Let $V_{j-1}\rightarrow V_j\leftarrow V_{j+1}$ be consecutive vertices on a path. The vertex $V_j$ is a collider on the path. Every non-endpoint vertex of a path is exactly one of a chain, a fork or a collider on that path.

Let $V=\left\{V_1,\ldots,V_p\right\}$ be a set of endogenous variables, $U=\left\{U_1,\ldots,U_p\right\}$ be a set of exogenous variables with a joint probability distribution $\mathbb{P}_U,$ and for each $j$ let $\mathrm{pa}_j\subseteq V\setminus\{V_j\}$ and $f_j$ be a function. A structural causal model (SCM) is the collection of structural equations that determine each endogenous variable from its parents and its exogenous variable, together with $\mathbb{P}_U.$

$$V_j:=f_j\left(\mathrm{pa}_j,U_j\right)\,\,\forall\,j=1,\ldots,p$$

The structural equations are assignments rather than algebraic equalities: the left-hand side is determined by the right-hand side and not conversely.

Let $\mathcal{M}$ be a structural causal model. The causal graph of $\mathcal{M}$ is the directed graph on $V$ with an edge from each element of $\mathrm{pa}_j$ to $V_j.$ The model $\mathcal{M}$ is recursive if and only if its causal graph is a DAG. A recursive structural causal model has the following properties:

• the exogenous variables determine the endogenous variables uniquely, so $\mathbb{P}_U$ induces a joint distribution $\mathbb{P}_V$ of the endogenous variables, the observational distribution;
• if the exogenous variables are mutually independent then the model is Markovian, and otherwise it is semi-Markovian.

Let $G$ be a DAG on $V=\left\{V_1,\ldots,V_p\right\}$ and $\mathbb{P}$ be a joint distribution of $V$ with pmf or pdf $p.$ The distribution $\mathbb{P}$ is Markov with respect to $G$ if and only if it factorises into the conditional distributions of each variable given its parents.

$$p\left(v_1,\ldots,v_p\right)=\prod_{j=1}^p p\left(v_j\mid\mathrm{pa}_j\right)$$

The observational distribution of a Markovian structural causal model is Markov with respect to its causal graph.

The causal Markov condition is the assumption that every variable is independent of its non-descendants given its parents in the causal graph. The causal Markov condition is equivalent to the Markov factorisation.

Let $G$ be a DAG, $\pi$ be a path in $G$ and $Z\subseteq V$ be a set of vertices. The path $\pi$ is blocked by $Z$ if and only if at least one of the following holds:

• $\pi$ contains a chain or a fork whose middle vertex is in $Z;$
• $\pi$ contains a collider such that neither the collider nor any of its descendants is in $Z.$

A path that is not blocked by $Z$ is open given $Z.$

Let $G$ be a DAG and $X,Y,Z\subseteq V$ be disjoint sets of vertices. The sets $X$ and $Y$ are d-separated by $Z$ in $G,$ written $X\perp_G Y\mid Z,$ if and only if every path from a vertex in $X$ to a vertex in $Y$ is blocked by $Z.$

Let $G$ be a DAG and $\mathbb{P}$ be Markov with respect to $G.$ The global Markov property states that d-separation implies conditional independence.

$$X\perp_G Y\mid Z\Rightarrow X\perp\!\!\!\perp Y\mid Z$$

The converse need not hold: a distribution can have conditional independences that are not implied by d-separation. Conditioning on a collider, or on a descendant of a collider, opens a path between its parents and can create dependence between variables that are marginally independent.

Let $\mathcal{M}$ be a structural causal model, $X\in V$ and $x$ be a value of $X.$ The intervention $\mathrm{do}\left(X=x\right)$ is the operation that replaces the structural equation of $X$ by the constant assignment $X:=x$ and leaves every other structural equation and $\mathbb{P}_U$ unchanged. The intervened model is written $\mathcal{M}_{x}$ and its causal graph is $G$ with every edge into $X$ removed.

Let $\mathcal{M}$ be a structural causal model and $Y\in V.$ The interventional distribution of $Y$ under $\mathrm{do}\left(X=x\right),$ written $\mathbb{P}\left(Y\mid\mathrm{do}\left(X=x\right)\right)$ or $p\left(y\mid\mathrm{do}\left(x\right)\right),$ is the distribution of $Y$ in the intervened model $\mathcal{M}_{x}.$ Interventional distributions have the following properties:

• conditioning on $X=x$ selects the units with $X=x,$ whereas intervening with $\mathrm{do}\left(X=x\right)$ sets $X=x$ for every unit, so $p\left(y\mid\mathrm{do}\left(x\right)\right)\neq p\left(y\mid x\right)$ in general;
• the interventional distribution is Markov with respect to the intervened graph;
• the potential outcome $Y(x)$ is the value of $Y$ in $\mathcal{M}_x,$ so $\mathbb{P}\left(Y(x)=y\right)=p\left(y\mid\mathrm{do}\left(x\right)\right)$ and $\mathbb{E}\left[Y(x)\right]=\mathbb{E}\left[Y\mid\mathrm{do}\left(x\right)\right].$

Let $\mathcal{M}$ be a Markovian structural causal model with observational pmf or pdf $p$ and $X=V_k.$ The truncated factorisation states that the interventional distribution of the remaining variables is the Markov factorisation with the factor of $X$ removed and $X$ fixed at $x.$

$$p\left(v\setminus\{v_k\}\mid\mathrm{do}\left(x\right)\right)=\prod_{j\neq k}p\left(v_j\mid\mathrm{pa}_j\right)\Big|_{v_k=x}$$

Let $\mathcal{M}$ be a structural causal model, $X,Y\in V$ and $x,x'$ be values of $X.$ The average causal effect of changing $X$ from $x'$ to $x$ on $Y$ is the difference of the interventional means.

$$\mathbb{E}\left[Y\mid\mathrm{do}\left(X=x\right)\right]-\mathbb{E}\left[Y\mid\mathrm{do}\left(X=x'\right)\right]$$

For a binary treatment $A$ this is the average treatment effect $\mathbb{E}\left[Y(1)\right]-\mathbb{E}\left[Y(0)\right].$

Let $\mathcal{M}$ be a structural causal model with exogenous variables $U,$ and $E$ be an observed event. The counterfactual distribution of $Y(x)$ given $E$ is the distribution of $Y$ in $\mathcal{M}_x$ with $\mathbb{P}_U$ replaced by the conditional distribution of $U$ given $E.$ The counterfactual distribution is computed in three steps:

• abduction: update the distribution of $U$ to its conditional distribution given $E;$
• action: replace the structural equation of $X$ by $X:=x;$
• prediction: compute the distribution of $Y$ in the modified model.

Counterfactual quantities such as $\mathbb{P}\left(Y(0)=y\mid A=1,Y=y'\right)$ depend on the structural equations and not only on the interventional distributions.

Let $G$ be a causal graph with treatment $A$ and outcome $Y.$ A confounder of $A$ and $Y$ is a common cause of $A$ and $Y,$ that is, an ancestor of both. A mediator of $A$ and $Y$ is a vertex on a directed path from $A$ to $Y.$ A collider between $A$ and $Y$ is a common effect of $A$ and $Y,$ or of their causes.

Let $G$ be a causal graph with treatment $A$ and outcome $Y.$ A backdoor path from $A$ to $Y$ is a path from $A$ to $Y$ that starts with an edge into $A.$ Backdoor paths carry the association between $A$ and $Y$ that is not due to the causal effect of $A$ on $Y.$ Confounding is the presence of an open backdoor path.

Selection is conditioning on a variable, by design or by the way the data are collected. Collider bias is the spurious association between $A$ and $Y$ created by selection on a collider or on a descendant of a collider between them. Conditioning on a mediator blocks part of the causal effect, and conditioning on a confounder blocks a backdoor path.