Conditioning vs intervening
- Conditioning \(P(Y \mid X = x)\): sub-population with \(X = x\), graph intact
- Intervening \(P(Y \mid do(X = x))\): \(X\) forced to \(x\) for everyone
- Confounder \(Z\): association through \(X \leftarrow Z \to Y\) only in the first case
\[
P(Y \mid X = x) \neq P(Y \mid do(X = x)) \quad \text{in general}
\]
Modularity
Each mechanism \(f_j\), or conditional \(P(v_j \mid v_{pa(j)})\), is an autonomous module: an intervention on one variable leaves all other modules unchanged.
- \(do(X = x)\): equation of \(X\) replaced by \(X := x\)
- Graph surgery: arrows into \(X\) deleted → \(G_{\overline X}\)
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Truncated factorisation
Under modularity, the interventional law is the observational factorisation with the factor of the intervened variable removed.
Causal effect on \(Y\): marginalisation of the truncated product over \(V \setminus \{X, Y\}\).
Confounder: adjustment by \(Z\)
DAG \(Z \to X\), \(Z \to Y\), \(X \to Y\):
\[
\begin{aligned}
P(y \mid do(x)) &= \textstyle\sum_z P(z)\, P(y \mid x, z) \\
P(y \mid x) &= \textstyle\sum_z P(z \mid x)\, P(y \mid x, z)
\end{aligned}
\]
Difference: weights \(P(z)\) vs \(P(z \mid x)\), exactly the mechanism of Simpson’s paradox.
Kidney stones: effect under intervention
\[
\begin{aligned}
P(R \mid do(\text{new})) &= 0.51 \times 0.93 + 0.49 \times 0.73 = 0.83 \\
P(R \mid do(\text{standard})) &= 0.51 \times 0.87 + 0.49 \times 0.69 = 0.78
\end{aligned}
\]
- Conditioning, weights \(P(s \mid t)\): \(0.78\) vs \(0.83\), difference \(-0.05\)
- Intervention, weights \(P(s)\), \(P(\text{small}) = 357/700\): \(0.83\) vs \(0.78\), difference \(+0.05\)
Simpson’s paradox resolved Same stratified rates, other weights: +5 points for the new drug.
Linear example: regression slope vs causal effect
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Theory: \(\mathbb E[Y \mid X = x] = 3x + \mathbb E[Z \mid X = x] = 3.4\,x\), \(\;\mathbb E[Y \mid do(X = x)] = 3x\).
Identification by adjustment
Backdoor path
- Path between \(X\) and \(Y\) starting with an arrow into \(X\): \(X \leftarrow \cdots\; Y\)
- Carrier of non-causal association
- Causal paths: directed paths \(X \to \cdots \to Y\)
Backdoor criterion
A set \(Z\) satisfies the backdoor criterion relative to \((X, Y)\) when it contains no descendant of \(X\) and blocks every backdoor path between \(X\) and \(Y\).
Backdoor adjustment formula \[
P\big(y \mid do(x)\big) = \sum_z P(y \mid x, z)\, P(z)
\]
Several confounders
- Treatment \(C\), outcome \(Y\), mediator \(M\)
- Backdoor paths \(C \leftarrow W_k \to Y\)
- Valid set: \(\{W_1, W_2, W_3\}\)
- \(M\): descendant of \(C\), never adjusted
\[
P(y \mid do(c)) = \sum_{w} P(y \mid c, w)\, P(w)
\]
Bad controls
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- Mediator \(M\): part of the effect removed
- Collider \(M\) (M-bias): adjustment opens \(X \leftarrow A \to M \leftarrow B \to Y\)
- Rule: adjustment set chosen from the graph, not from correlations
Frontdoor criterion
Unobserved confounder \(U\), mediator \(M\) intercepting all causal paths.
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- \(M\) intercepts every directed path \(X \to Y\)
- No unblocked backdoor path \(X \to M\)
- Every backdoor path \(M \to Y\) blocked by \(X\)
