NOTE-696: Causality: Models, Reasoning, and Inference

A skim of four excerpts, 46 of the book’s some 480 pages: the contents, the Preface to the Second Edition, §§5.3.2–5.4.1 (pp. 154–163) and the Epilogue (pp. 401–428). Nothing here comes from the rest of the book.

Contribution

In the excerpts read, the book’s contribution is a meaning for causal talk in terms of intervention. A causal model is a set of structural equations, each an autonomous mechanism. Intervening on a variable replaces its equation, and observing it does not. The do() symbol writes the difference into probability notation, so that P(y | do(x)) and P(y | x) are different quantities. The do-calculus has three rules for turning one kind of expression into the other when a diagram licenses it (Epilogue, pp. 421–423).

The Preface to the Second Edition says what the 2009 edition changed (p. xix). It “(1) provides technical corrections, updates, and clarifications in all ten chapters of the original book, (2) adds summaries of new developments and annotated bibliographical references at the end of each chapter, and (3) elucidates subtle issues that readers and reviewers have found perplexing, objectionable, or in need of elaboration. These are assembled into an entirely new chapter (11).” The contents bear this out: a postscript closes chapter 5 (§5.6, “Postscript for the Second Edition”), and chapter 11, “Reflections, Elaborations, and Discussions with Readers”, runs from p. 331 to p. 398.

Key insight

“Intervention amounts to a surgery on equations (guided by a diagram) and causation means predicting the consequences of such a surgery” (p. 417). Seeing and doing are two sentences, not one. Probability theory is the algebra of the first: in P(Rain | Wet) “the vertical bar stands for the phrase: ‘given that we see’” (p. 421). Asked “What is the chance it rained if we make the grass wet?”, it has no syntax for the question. “We know intuitively what the answer should be: P (Rain), because making the grass wet does not change the chance of rain” (pp. 421–422). The do-operator is the new symbol, and the surgery supplies its meaning (“Meaning: surgery + substitution”, slide 44, p. 422).

Assumptions

These are the conditions under which the excerpts’ account of intervention works, as the excerpts state them.

• Autonomy, or modularity. Each equation is “an independent mechanism” that “must be preserved as a separate mathematical sentence” (p. 420). The circuit diagram’s power “comes from what early economists called autonomy. Namely, the gates in this diagram represent independent mechanisms – it is easy to change one without changing the other” (p. 414).
• A diagram that says which equation belongs to which variable. “The diagram tells us which equation is to be deleted when we manipulate Y” (p. 417). Rewrite the equations into an algebraically equivalent pair and “there is no such thing as ‘the equation for Y’”, so the surgery is undefined (“impossible”, slide 33, p. 417).
• A cut between in and out. “If you wish to include the entire universe in the model, causality disappears because interventions disappear – the manipulator and the manipulated lose their distinction” (p. 420). The choice of which variables are exogenous is the investigator’s: “it is the way we carve up the universe that determines the directionality we associate with cause and effect” (p. 420).
• A local intervention. The intervention changes one mechanism and leaves the others alone. Structural parameters are invariant “to local interventions (i.e., changes in specific equations in the system) and not to general changes in the statistics of the variables” (pp. 161–162).
• Nonparametric generality (§5.3.2). In the example model (5.4)–(5.6), x = f₁(u, ε₁), z = f₂(x, ε₂), y = f₃(z, u, ε₃), the f’s are “unknown arbitrary functions” and U, ε₁, ε₂, ε₃ are “mutually independent and arbitrarily distributed” (p. 155).

Key results

The surgery and the do() notation (§5.3.3, pp. 157–159)

• Two covariance-equivalent models with different effects. Model M, (5.7)–(5.9): x = u + ε₁, z = αx + ε₂, y = βz + γu + ε₃. Model M′, (5.12)–(5.14): x = ε₁, z = α′x + ε₂, y = β′z + δx + ε₃. With α′ = α, β′ = β and δ = γ they “yield the same probabilistic predictions”, but “when viewed as data-generating mechanisms, the two models are not equivalent” (p. 157).
• The computation. Substituting X = x into the remaining equations, M′ gives E[Y | do(X = x)] = (β′α′ + δ)x (5.16) and M gives βαx (5.18) (p. 157). The posted PDF prints M′‘s total effect as “βα + δ”, with the δ in red over the printed text, which the text layer still holds as “βα + δγ”. By (5.16) with β′ = β and α′ = α, βα + δ is right, so the red δ reads as the author’s correction of the printed page.
• Why u may not be substituted. “Structural equations are not meant to be treated as immutable mathematical equalities. Rather, they are meant to define a state of equilibrium – one that is violated when the equilibrium is perturbed by outside interventions” (p. 157). Under do(X = x), “the equation x = u + ε₁ … should be overruled and replaced with the equation X = x” (p. 158). So structural equations “stand not for one but for many sets of equations, each corresponding to a subset of equations taken from the original model” (p. 158).
• The notation. E(Y | do(x)) ≜ E[Y | do(X = x)] is “the controlled expectation” (5.19), and E(Y | x) ≜ E(Y | X = x) is “the standard conditional or observational expectation” (5.20) (p. 158).
• Observing is conditioning. “The passive observation X = x should not violate any of the equations, and this is the justification for substituting both (5.7) and (5.8) into (5.9) before taking the expectation” (p. 158). In M, E(Y | do(x)) = αβx, while E(Y | x) = r_YX x = (αβ + γ)x. The page prints the last term as “y”; by (5.11), γ = r_YX − αβ, so γ is meant.
• The nonparametric case. E(Y | do(x)) = E{f₃[f₂(x, ε₂), u, ε₃]}, with f₁ “wiped out” (pp. 158–159). Identification means answering such queries “uniquely, from the data and the graph”, as Definition 3.2.3 (not read) has it (p. 159).

What structural equations mean (§5.4.1, pp. 159–163)

• The paradox. If β in y = βx + ε is the change in E(Y) per unit of X, rewriting gives x = (y − ε)/β, and 1/β “ought” to be the effect of Y on X. That conflicts with intuition and with the model (p. 159). Pearl rejects both textbook escapes: the purely statistical reading, and the “isolation” condition cov(X, ε) = 0, which every bivariate normal pair meets in both directions (pp. 159–160).
• Definition 5.4.1 (structural equations). “An equation y = βx + ε is said to be structural if it is to be interpreted as follows: In an ideal experiment where we control X to x and any other set Z of variables (not containing X or Y) to z, the value y of Y is given by βx + ε, where ε is not a function of the settings x and z” (p. 160).
• The equality sign is asymmetric. Structural equations “act symmetrically in relating observations on X and Y (e.g., observing Y = 0 implies βx = −ε), but they act asymmetrically when it comes to interventions (e.g., setting Y to zero tells us nothing about the relation between x and ε)” (p. 160).
• The empirical claim of a missing variable. Excluding a variable from the right-hand side is a testable invariance, P(y | do(x), do(z)) = P(y | do(x)) for all Z disjoint from {X ∪ Y} (5.23). “In contrast, regression equations make no empirical claims whatsoever” (p. 160). The invariance “holds relative to manipulations, not observations, of Z”: P(y | do(x), z) “would certainly depend on z if the measurement were taken on a consequence (i.e., descendant) of Y” (p. 161).
• The structural parameter. β = ∂E[Y | do(x)]/∂x (5.24), “the rate of change (relative to x) of the expectation of Y in an experiment where X is held at x by external control”. It “has nothing to do with the regression coefficient r_YX” (p. 161).
• The error term. ε = y − E[Y | do(x)] (5.25), the deviation from the controlled expectation, not from E[Y | x] (p. 162). Error correlations are E[ε_Y ε_X] = E[YX | do(pa_Y, pa_X)] − E[Y | do(pa_Y)]E[X | do(pa_X)] (5.26), and ε_Y, ε_X are uncorrelated iff E[Y | x, do(s_XY)] = E[Y | do(x), do(s_XY)] (5.27) (p. 162).
• But think in omitted factors. The operational definition “prescribes how errors are measured, not how they originate”. Judging whether errors correlate needs the omitted-factors conception, because the assessments (5.26) asks for “are cognitively unfeasible” (p. 163). Bidirected arcs “should be assumed to exist, by default, between any two nodes”, and deleted only with justification (p. 163).
• Counterfactual content. Footnote 17 adds that an equation also makes “a dynamic or counterfactual claim: If we were to control X to x′ instead of x, then Y would attain the value βx′ + ε”, testable only if ε stays unaltered, and “such counterfactual claims constitute the empirical content of every scientific law” (pp. 160–161).

The Epilogue (pp. 401–428)

• Two riddles (pp. 406–407): how people acquire causal knowledge, given that “correlation does not imply causation”, and what difference it makes to be told that a connection is causal.
• The logic bug (p. 413). Given “If the grass is wet, then it rained” and “If we break this bottle, the grass will get wet”, a program concludes “If we break this bottle, then it rained.”
• The three ideas (p. 414): “treating causation as a summary of behavior under interventions”; “using equations and graphs as a mathematical language”; and “treating interventions as a surgery over equations”.
• Surgery on the multiplier–adder circuit (pp. 416–417, slide 33). Y = 2X, Z = Y + 1: setting Y = 0 deletes Y = 2X and gives Z = 1. The reversed pair X = Y/2, Y = Z − 1 gives X = 0 and leaves Z unconstrained. The equivalent pair 2X − 2Y + Z − 1 = 0, 2X + 2Y − 3Z + 3 = 0 admits no surgery. The definition drawn from it: “Y is a cause of Z if we can change Z by manipulating Y, namely, if after surgically removing the equation for Y, the solution for Z will depend on the new value we substitute for Y” (p. 417). The idea of “wiping out” equations is credited to Herman Wold in 1960 (p. 417).
• Randomized experiments (p. 418). Fisher’s experiment is “randomization and intervention”. Intervention is surgery: “we are severing one functional link and replacing it with another. Fisher’s great insight was that connecting the new link to a random coin flip guarantees that the link we wish to break is actually broken.”
• Policy evaluation (pp. 418–419). The task is “to infer the behavior of this mutilated model from data governed by a nonmutilated model”. Taxation “is an endogenous variable during the model-building phase and turns exogenous only when evaluated.”
• Russell’s enigma (pp. 419–420). The equations of physics are symmetric, but “A causes B” and “B causes A” compare two models, one with A’s equation removed and one with B’s.
• Equational against causal models (pp. 420–421, slide 39). Both use symmetric equations for normal conditions. The causal model adds “(i) a distinction between the in and the out; (ii) an assumption that each equation corresponds to an independent mechanism and hence must be preserved as a separate mathematical sentence; and (iii) interventions that are interpreted as surgeries over those mechanisms.”
• Observation and intervention (p. 421). “Scientific activity, as we know it, consists of two basic components: Observations [40] and interventions [41]. The combination of the two is what we call a laboratory”. Standard algebras, probability among them, “are geared to serve observational sentences but not interventional sentences.”
• The do-calculus (pp. 422–423, slide 45). Three rules, each licensed by a d-separation condition in a modified graph. Rule 1, “Ignoring observations”: P(y | do{x}, z, w) = P(y | do{x}, w) if (Y ⫫ Z | X, W) in G_X̄. Rule 2, “Action/observation exchange”: P(y | do{x}, do{z}, w) = P(y | do{x}, z, w) if (Y ⫫ Z | X, W) in G_X̄Z̲. Rule 3, “Ignoring actions”: P(y | do{x}, do{z}, w) = P(y | do{x}, w) if (Y ⫫ Z | X, W) in G_X̄,Z̄(W). In words: “The first allows us to ignore an irrelevant observation, the third to ignore an irrelevant action; the second allows us to exchange an action with an observation of the same fact” (p. 422). The graph notation is not explained in the Epilogue, and its definitions are in §3.4 (not read).
• Smoking, tar and cancer (pp. 423–424, slide 46). With an unobserved genotype, P(c | do(s)) is “noncomputable”. With tar deposits measured as a mediator it is computable, and the derivation eliminates every “do” to reach “a formula involving no ‘do’ symbols”. Pearl adds that this does not settle the debate, since the model assumes “no direct link between smoking and lung cancer unmediated by tar deposits” (p. 424).
• Simpson’s paradox and the adjustment problem (pp. 425–427): the Berkeley admissions case, “reverse regression”, and a graphical test of whether measurements Z₁, Z₂ suffice for adjustment, shown on slides 51–56 and not reproduced in the text.

Concepts

• Intervention, do(x). The surgery that replaces the equation for X by X = x and solves the rest (pp. 158, 417). Read “given that we do” (p. 421).
• Observation, conditioning. “Given that we see” (p. 421). A passive observation violates no equation (p. 158).
• Structural equation. Definition 5.4.1 (p. 160): a claim about Y under ideal control of X and any other variables. It is distinct from an algebraic or regression equation.
• Autonomy. Mechanisms can be changed one at a time (p. 414).
• Control query and observational query. “What would be the change in the expected value of Y if we were to intervene and change the value of Z from z to z + 1?” against “What would be the difference in the expected value of Y if we were to find Z at level z + 1 instead of level z?” (p. 158).
• Mutilated model. The model after surgery (p. 419).
• Terminology. The excerpts say “structural equation models”, “causal models” and, in the contents, “functional causal models” (§1.4). The phrase “structural causal model” does not occur in them.

Connections

• Wold, Haavelmo, Marschak and Simon. Pearl credits the wiping-out of equations to Wold, 1960 (p. 417), and the policy reading of structural equations to Haavelmo 1943, Marschak 1950 and Simon 1953 (p. 158). The invariance idea behind (5.23) goes back to Marschak and Simon, and is formalised in Hurwicz, Mesarovic, Sims, Cartwright, Hoover and Woodward (footnote 16, p. 160).
• Cartwright’s capacities (pp. 161–162). β is invariant to local interventions, not to “general changes in the statistics of the variables”. So her ratio E(YX)/E(X²) stays constant only when X’s own mechanism is changed.
• Fisher’s randomized trial, read as surgery in which a coin is the new mechanism (p. 418).
• The record’s interventionist vocabulary. TERM-018 defines a framing intervention as “an action that changes the conditions under which an utterance is interpreted”. That is Pearl’s sense of intervention, applied to the variable for what the interpreter is told.

Bearing on the record

The record cites the book for one thing: the definition of do(). It does so in CLAIM-127 (version 2) and CLAIM-130.

• “Observing a value and conditioning on it are one operation” (CLAIM-127, version 2 history). The excerpts support this. The conditional bar is “given that we see” (p. 421), E(Y | x) is “the standard conditional or observational expectation” (p. 158), and the Epilogue’s two components are observations and interventions (p. 421). There is no third operation in the excerpts. The review’s triad of “observation, conditioning, and intervention” is not Pearl’s.
• The gloss of do(). The LIT’s key takeaway (“replacing X’s structural equation, the other equations left in place”) matches pp. 158 and 417 in substance. Two qualifications from the excerpts sharpen it.
  • The variable must have an equation. Surgery deletes “the equation for” the variable, which presupposes autonomy and a diagram that assigns each equation to its variable (pp. 414, 417, 420). In A129’s model as CLAIM-127 quotes it, R′ = f_R(R, Z, I, ε_R) and Y = f_Y(R′, Z, C, ε_Y), the framing variable I is “externally supplied” and has no equation. So do(I = a) sets an exogenous input. CLAIM-130’s “do(I = a) replaces I’s” equation is accurate only once how I is chosen is modelled.
  • For an exogenous, unconfounded variable, seeing and doing coincide. This is my inference, not a sentence of the excerpts. It applies Rule 2 (p. 423), whose condition holds trivially for a parentless I that shares no cause with the rest of the model, and it is the sense of p. 418, where a coin-assigned treatment “guarantees that the link we wish to break is actually broken”. So in an experiment that assigns I, P(Y | I = a) = P(Y | do(I = a)). The do() notation for I separates something only when I is not assigned, as in corpora where framing co-varies with the situation. That leaves CLAIM-127’s structural point (index the empirical model by a, do not add a to the cover) and CLAIM-130’s distinction (do(I) against do(S), a difference of variable) untouched. It narrows where the operator itself does work.
• “Nor does it say Pearl’s calculus applies as it stands” (CLAIM-127). The excerpts agree that identification is the hard part. The surgery is “well defined” even for unknown f’s (p. 158), but answering do-queries “uniquely, from the data and the graph” is a further question (p. 159), and the smoking example shows a do-query that is “noncomputable” (p. 423).
• Elicitation as measurement within an intervention. §5.4.1 separates conditioning on a measurement taken under an intervention, P(y | do(x), z), from intervening on it, P(y | do(x), do(z)), and notes the first depends on z when Z is a descendant (p. 161). That is the shape of CLAIM-127’s e_C^a = P(Y_C | do(a)) when C is an elicitation and the question is whether eliciting one judgement is itself a do() on the interpreter’s state.
• No THEORY is filed or touched. A skim of definitional excerpts grounds no claim about language or interpretation.
• No instruction for machine-learning practice, so nothing for the Anthology.
• Tags. causality first. probabilistic-modeling for the conditional probability and the graphical machinery. philosophy-of-science for the Epilogue’s account of causation, Hume and Russell, and for §5.4.1 on what the equations of a science claim.

Limitations

• 46 pages of some 480. None of chapters 1–4 or 6–11 was read. The definitions the Epilogue and §5 rely on (Definition 3.2.3, the back-door and front-door criteria, the graph notation of the three rules, the counterfactual semantics of chapter 7) were seen only by name.
• The Epilogue is a 1996 lecture written for a general audience. It states results, does not prove them, and relies on slides some of which are not reproduced in the text (51–56).
• §5.3.2–5.4.1 is about linear structural equation models in social science and economics. Its general claims are illustrated on three-variable linear examples.
• Extraction was imperfect. Equations were read from page images where the text layer failed, and p. 154 is overlaid by an author’s note.
• No hierarchy of association, intervention and counterfactual appears in these pages. Whether the book states one elsewhere was not checked.

Open questions

• Sequential interventions. CLAIM-127 notes that the composition K_bK_a needs time-indexed states. The contents list §4.4, “The Identification of Dynamic Plans”, with a “Sequential Back-Door Criterion” (§4.4.3, p. 121). Does it treat a sequence of interventions on one interpreter, or only plans over distinct variables?
• Is telling a do()? The contents list §11.4.3, “Can do(x) Represent Practical Experiments?”, §11.4.4, “Is the do(x) Operator Universal?”, §11.4.5, “Causation without Manipulation!!!” and §11.4.7, “The Illusion of Nonmodularity”. These look like the places where the book says when an instruction to a subject counts as surgery.
• Utterances. §7.2.3 is titled “Causal Explanations, Utterances, and Their Interpretation” (p. 221). What it says about interpreting causal utterances bears directly on the record’s interpretive programme and was not read.