Causality

DAG \(G\) is a causal Bayesian network compatible with \(\bm{P*}\) \(\iff\) 3 conditions

\(\forall P_x \in P_*\)

1. \(P_x(v)\) is Markov relative to \(G\);
2. \(P_x(v_i) = 1\) \(\forall V_i \in X\) whenever \(v_i\) is consistent with \(X = x\);
3. \(P_x(v_i | pa_i) = P(v_i | pa_i)\) \(\forall V_i \notin X\) whenever \(pa_i\) is consistent with \(X = x\), i.e., each \(P(v_i | pa_i)\) remains invariant to interventions not involving \(V_i\).

Probability function P is Markov relative to DAG G

P admits the factorization

$$P(x_1, \ldots, x_n) = \prod_{i=1}^n P(x_i | pa_i)$$

relative to G

Markovian model

Causal diagram is acyclic and error terms are jointly independent

3-step procedure to calculate counterfactuals

1. Abduction: Update \(P(u)\) to obtain \(P(u|e)\)
2. Action: Replace equations corresponding to \(X\) with \(X = x\)
3. Prediction: Use modified model to compute \(P(Y = y)\)

Semi-Markovian model

Causal diagram is acyclic

Graphoid axioms

• Symmetry: \((X \perp \!\!\! \perp Y | Z) \implies (Y \perp \!\!\! \perp X | Z)\)
• Decomposition: \((X \perp \!\!\! \perp YW | Z) \implies (X \perp \!\!\! \perp Y | Z)\)
• Weak union: \((X \perp \!\!\! \perp YW | Z) \implies (X \perp \!\!\! \perp Y | ZW)\)
• Contraction: \((X \perp \!\!\! \perp Y | Z)\ \&\ (X \perp \!\!\! \perp W | ZY) \implies (X \perp \!\!\! \perp YW | Z)\)
• Intersection: \((X \perp \!\!\! \perp Y | ZW)\ \&\ (X \perp \!\!\! \perp W | ZY) \implies (X \perp \!\!\! \perp YW | Z)\)

Causal Bayesian network \(\implies\) 2 properties

1. \(\forall i\\P(v_i | pa_i) = P_{pa_i}(v_i)\)
2. \(\forall i, S \subseteq V\setminus\{V_i, PA_i\}\\P_{pa_i, s}(v_i) = P_{pa_i}(v_i)\)

Truncated factorization of causal Bayesian network

$$P_x(v) = \prod_{\{i | V_i \notin X \}} P(v_i | pa_i)$$

for all \(v\) consistent with \(x\)

do Calculus rule 3

Insertion/deletion of actions:

\(P(y | \hat{x}, \hat{z}, w) = P(y | \hat{x}, w)\), if \((Y \perp \!\!\! \perp Z | X, W)_{G_{\overline{X},\overline{Z(W)}}}\), where \(Z(W)\) is the set of \(Z\)-nodes that are not ancestors of any \(W\)-node in \(G_{\overline{X}}\)

do Calculus rule 2

Action/observation exchange:

\(P(y | \hat{x}, \hat{z}, w) = P(y | \hat{x}, z, w)\), if \((Y \perp \!\!\! \perp Z | X, W)_{G_{\overline{X}\underline{Z}}}\)

do Calculus rule 1

Insertion/deletion of observations:

\(P(y | \hat{x}, z, w) = P(y | \hat{x}, w)\), if \((Y \perp \!\!\! \perp Z | X, W)_{G_{\overline{X}}}\)

IC (inductive causation) algorithm

Input: \(\hat{P}(\bm{v})\)

Output: Pattern compatible with \(\hat{P}\)

1. For each pair of variables \(a\) and \(b\) in \(V\), search for a set \(S_{ab}\) such that \((a \perp \!\!\! \perp b | S_{ab})\) holds in \(\hat{P}\). Construct an undirected graph \(G\) such that vertices \(a\) and \(b\) are connected with an edge \(\iff\) no set \(S_{ab}\) can be found.
2. For each pair of nonadjacent variables \(a\) and \(b\) with a common neighbor \(c\), check if \(c \in S_{ab}\). If it is not, then add arrowheads \(a \rightarrow c \leftarrow b\).
3. In the partially directed graph that results, orient as many of the undirected edges as possible subject to two conditions:
   1. Any alternative orientation would yield a new \(v\)-structure; or
   2. Any alternative orientation would yield a directed cycle.