0.2.1 Directed Acyclic Graphs (DAGs)

A DAG is a graph with directed edges, and no cycles. (you cannot return to the same starting node travelling only forward along the arrows.)

DAGs are defined by a set of vertices and (directed) edges.

We show the directions of edges by writing them as arrows.

For nodes $X,Y\in \mathbf{V}$ we write $X\to Y$ to mean there is a directed edge joining them.

0.3.1 Local Markov assumption

Given its parents in the DAG, a node is independent of all its non-descendants.

With four variable example, the chain rule of probability tells us that we can factorize any $P$ thus $$P(x_1,x_2,x_3,x_4)=P\left(x_1\right)P(x_2\mid x_1)P(x_3\mid x_2,x_1)P(x_4\mid x_3,x_2,x_1)\text{. }$$

If $P$ is Markov with respect to the above graph then we can simplify the last factor: $$P(x_1,x_2,x_3,x_4)=P\left(x_1\right)P(x_2\mid x_1)P(x_3\mid x_2,x_1)P(x_4\mid x_3).$$

If we further remove edges, removing $X_1\to X_2$ and $X_2\to X_3$ as the below figure,

we can further simplify the factorization of $P$ : $$P(x_1,x_2,x_3,x_4)=P\left(x_1\right)P\left(x_2\right)P(x_3\mid x_1)P(x_4\mid x_3)$$

0.3.2 Bayesian Network Factorization

Given a probability distribution $P$ and a DAG $G$, we say $P$ factorizes according to $G$ if $$P(x_1,\dots ,x_n)=\prod _{i}P(x_i\mid \mathrm{parents}(X_i)).$$

0.3.3 Minimality

1. Given its parents in the DAG, a node X is independent of all its non-descendants
2. Adjacent nodes in the DAG are dependent.

0.4.1 Causal Edges

In a directed graph, every parent is a direct cause of all its children.

$Y$ “directly causing“ $X$ means that $X=f(\mathrm{parent}(X),\omega )$ is a (stochastic) function of some parent set which includes $Y,$ and some independent noise.

0.4.2 Causal Bayesian Networks

Causal Edges + Local Markov

0.5.1 Chains

$$P(a,b,c)=P(a)P(b|a)P(c|b)$$

We assert that, conditional on B, A and C are independent: $$\begin{gathered} A\perp \perp C|B \\ \iff \\ P(a,c|b)=P(a|b)P(c|b) \end{gathered}$$

In slow motion, $$\begin{array}{rl}P(a,b,c)& =P(a)P(b|a)P(c|b)\\ P(a,c|b)& =\frac{P(a)P(b|a)P(c|b)}{P(b)}\\ & =P(c|b)\frac{P(a)P(b|a)}{P(b)}\\ & =P(c|b)\frac{P(a,b)}{P(b)}\\ & =P(c|b)P(a|b)\end{array}$$

0.5.2 Forks

$$P(a,b,c)=P(b)P(a|b)P(c|b)$$

We assert that, conditional on B, A and C are independent: $$\begin{gathered} A\perp \perp C|B \\ \iff \\ P(a,c|b)=P(a|b)P(c|b) \end{gathered}$$ In slow motion, $$\begin{array}{rl}P(a,b,c)& =P(b)P(a|b)P(c|b)\\ P(a,c|b)& =\frac{P(b)P(a|b)P(c|b)}{P(b)}\\ & =P(a|b)P(c|b)\end{array}$$

0.5.3 Immoralities

(Colliders when I grew up.)

$$P(a,b,c)=P(b)P(c)P(a|b,c)$$

We assert that, conditional on B, A and C are not in general independent: $$\begin{gathered} A\cancel{\perp \perp }C|B \\ \iff \\ P(a,c|b)=P(a|b)P(c|b) \end{gathered}$$

Proof that this never factorizes?

0.5.4 Blocked paths

A path between nodes $X$ and $Y$ is blocked by a (potentially empty) conditioning set $Z$ if either of the following is true:

1. Along the path, there is a chain $\cdots \to W\to \cdots$ or a fork $\cdots \leftarrow W\to \cdots$, where $W$ is conditioned on $(W\in Z)$.
2. There is a collider $W$ on the path that is not conditioned on $(W\notin Z)$ and none of its descendants are conditioned on $(\mathrm{descendants}(W)⊈Z)$.

0.5.5 d-separation

Two (sets of) nodes $\mathit{X}$ and $\mathit{Y}$ are $d$-separated by a set of nodes $\mathit{Z}$ if all of the paths between (any node in) $\mathit{X}$ and (any node in) $\mathit{Y}$ are blocked by $\mathit{Z}$.

0.5.6 d-separation in Bayesian networks

We use the notation $X{\perp \perp }_{G}Y\mid Z$ to denote that $X$ and $Y$ are d-separated in the graph $G$ when conditioning on $Z$. Similarly, we use the notation $X{\perp \perp }_{P}Y\mid Z$ to denote that $X$ and $Y$ are independent in the distribution $P$ when conditioning on $Z$.

Given that $P$ is Markov with respect to $G$ if $X$ and $Y$ are d-separated in $G$ conditioned on $Z$, then $X$ and $Y$ are independent in $P$ conditioned on $Z$.

$$X\underset{G}{\perp \perp }Y|Z⟹X\underset{P}{\perp \perp }Y|Z.$$

1.1 Recommended reading

• Mohan and Pearl’s 2012 tutorial
• Elwert’s intro (Elwert 2013)
• d-separation without tears is an interactive version of Pearl’s original based on daggity.
• Likewise, the ggdag bias structure vignette shows off the useful explanation diagrams available in ggdag and is also a good introduction to selection bias and causal dags themselves

1.2 References