Source
Pearl (2009), LIT-894, Causality: Models, Reasoning, and Inference, second edition, skimmed in NOTE-696. Read: the table of contents, the Preface to the Second Edition (p. xix), §§5.3.2–5.4.1 (pp. 154–163) and the Epilogue (pp. 401–428), each in full. Nothing else in the book was read. NOTE-696 filed no THEORY because the excerpts ground no claim about language or interpretation. This one is about the operator itself, which the excerpts state in full; it is Proposed because it rests on a skim.
What was actually shown
Surgery (pp. 157–158, 417). “Intervention amounts to a surgery on equations (guided by a diagram) and causation means predicting the consequences of such a surgery” (p. 417). Under do(X = x), X’s own equation “should be overruled and replaced with the equation X = x” (p. 158). Structural equations “define a state of equilibrium – one that is violated when the equilibrium is perturbed by outside interventions” (p. 157), so they stand for every submodel an intervention can produce.
Seeing is conditioning (pp. 158, 421). “The passive observation X = x should not violate any of the equations” (p. 158). E(Y | x) is “the standard conditional or observational expectation”, E(Y | do(x)) “the controlled expectation”. In P(Rain | Wet) “the vertical bar stands for the phrase: ‘given that we see’” (p. 421).
Two models, one distribution (§5.3.3, p. 157). Model M (x = u + ε₁, z = αx + ε₂, y = βz + γu + ε₃) and M′ (x = ε₁, z = α′x + ε₂, y = β′z + δx + ε₃), with α′ = α, β′ = β and δ = γ, “yield the same probabilistic predictions”. But E[Y | do(x)] is βαx in M and (β′α′ + δ)x in M′. In M, E(Y | do(x)) = αβx while E(Y | x) = (αβ + γ)x.
What a structural equation claims (§5.4.1, pp. 160–162). Definition 5.4.1: y = βx + ε is structural if, “in an ideal experiment where we control X to x and any other set Z of variables … to z”, Y is βx + ε with ε not a function of the settings. So β = ∂E[Y | do(x)]/∂x (5.24), which “has nothing to do with the regression coefficient r_YX” (p. 161). Leaving a variable out of an equation is a testable invariance under manipulation (5.23), while “regression equations make no empirical claims whatsoever” (p. 160).
What could have come out otherwise. The two models could have been told apart by their observational distribution. They cannot. Only the interventional query separates them.
What this does not say
- That surgery is always defined. It needs autonomy (each equation an independent mechanism, p. 414) and a diagram assigning each equation to its variable (p. 417). Rewrite the equations into an algebraically equivalent set and “there is no such thing as ‘the equation for Y’”. Causality also needs a cut between modelled and unmodelled: “If you wish to include the entire universe in the model, causality disappears” (p. 420).
- That every do-query can be answered from data. Surgery is well defined even for unknown functions, but identification, “uniquely, from the data and the graph”, is a further question (p. 159). With an unobserved genotype, P(c | do(s)) is “noncomputable” (p. 423).
- That an assigned input needs the operator. For a parentless, unconfounded variable, as in an experiment that assigns it, seeing and doing coincide (NOTE-696’s inference from Rule 2 and p. 418, not a sentence of the excerpts).
- That there is a third operation besides observing and intervening. The Epilogue names two components of science, observations and interventions (p. 421).
- That the book states a three-level ladder. None appears in the pages read, and the rest was not checked.
- That the do-calculus is derived here. Its three rules are stated (pp. 422–423), with graph notation defined in §3.4, which was not read.
Connections
- Woodward’s interventionism (THEORY-138) defines cause by the possibility of an intervention and constrains what counts as one (M1–M4). This account supplies the formal semantics of the intervention in a structural model. The two are compatible; neither is filed as relying on the other.
- CLAIM-127 and CLAIM-130 cite the book for the definition of do(), and CLAIM-127’s point that observing and conditioning are one operation is stated here.
- Randomisation as surgery, with a coin as the new mechanism (p. 418), bears on the arguments weighed in THEORY-491.
Note of 2026-10-20: the two consequences, derived
Derived for the promotion audit (batch 054, ADR-047 v2), taking the surgery and conditioning definitions as stated on pp. 157–158, 417 and 421.
- Same observations, different interventions. In M (x = u + ε₁, z = αx + ε₂, y = βz + γu + ε₃, with independent zero-mean Gaussian u and ε’s), put k = σ_u²/(σ_u² + σ₁²) and ε₃′ = γ(u − kx) + ε₃. Cov(u − kx, x) = 0, so ε₃′ is independent of x and of ε₂. Then M′ (x exogenous, z = αx + ε₂, y = βz + γk·x + ε₃′) is a Markovian model with the same joint distribution of (x, z, y). Surgery on x gives E[Y | do(x)] = αβx in M and (αβ + γk)x in M′, which differ whenever γ ≠ 0 and σ_u > 0. A simulation (α = 0.7, β = 1.3, γ = 0.9, σ_u = 1, σ₁ = 0.8, 2·10⁶ draws) reproduced matching covariances and do-slopes 0.91 and 1.46.
- The structural coefficient. If y = βx + ε is structural (Definition 5.4.1), then under do(X = x), Y = βx + ε with ε’s distribution unchanged, so E[Y | do(x)] = βx + E[ε] and its derivative is β. The regression slope is β + Cov(X, ε)/Var(X), which is β exactly when X and ε are uncorrelated.
- A correction to “What was actually shown”. The quoted agreement of M and M′ with α′ = α, β′ = β and δ = γ, and E(Y | x) = (αβ + γ)x in M, hold only when σ₁ = 0, that is when x = u. For σ₁ > 0 the regression slope is αβ + γk, and the matching M′ needs δ = γk and Var(ε₃′) = γ²σ_u²(1 − k) + σ₃². The conclusion needs only one such pair, so it stands.
Whether the book defines intervention the same way in the chapters not read is QUESTION-226.