 # Causality

Book about analysis of causation by Judea Pearl

Scott Mueller (lvl 17)
Chapter 1

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

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Aug 9, 2020

## Cards(12)

Chapter 1

(8 cards)

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

Front

$$\forall P_x \in P_*$$

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 $$V_i$$.
Back

Probability function P is Markov relative to DAG G

Front

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

relative to G

Back

Markovian model

Front

Causal diagram is acyclic and error terms are jointly independent

Back

3-step procedure to calculate counterfactuals

Front
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 $$P(Y = y)$$
Back

Semi-Markovian model

Front

Causal diagram is acyclic

Back

Graphoid axioms

Front
• 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: $$(X \perp \!\!\! \perp Y | ZW)\ \&\ (X \perp \!\!\! \perp W | ZY) \implies (X \perp \!\!\! \perp YW | Z)$$
Back

Causal Bayesian network $$\implies$$ 2 properties

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

Truncated factorization of causal Bayesian network

Front

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

for all $$v$$ consistent with $$x$$

Back

Chapter 2

(4 cards)

do Calculus rule 3

Front

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 $$G_{\overline{X}}$$

Back

do Calculus rule 2

Front

Action/observation exchange:

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

Back

do Calculus rule 1

Front

Insertion/deletion of observations:

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

Back

IC (inductive causation) algorithm

Front

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.
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