3 Graphical Identification and the Do-Calculus
How to read this chapter. Section 3.1–Section 3.3 are the core material: intervention graphs, back-door identification, and the do-calculus. Section 3.4 previews the front-door strategy, which Chapter 8 develops in full, and Section 3.5 explains completeness and nonidentification, with the supporting graph machinery in Appendix A.
3.1 Intervention and Modified Graphs
In Chapter 2 we learned to read conditional independence from a DAG using d-separation. This chapter takes the next step: translating the graphical language into identification formulas — expressions that write the interventional distribution \(P(y \mid \doop(T{=}t))\) entirely in terms of quantities observable from data.
3.1.1 Conditioning versus Intervention
Before introducing any graph machinery, it is worth being precise about what the \(\doop(\cdot)\) operator means and why it is not the same as conditioning. The conditional distribution \(P(Y \mid T{=}t)\) describes the subpopulation of units for whom \(T\) was observed to equal \(t\). Because treatment assignment may be influenced by confounders, this subpopulation is not representative of the full population. The interventional distribution \(P(Y \mid \doop(T{=}t))\), by contrast, describes the population that would result if \(T\) were externally set to \(t\) for everyone — removing the natural process that determines \(T\) and replacing it with the forced value.
The two distributions coincide in special cases — under randomized treatment assignment, or more generally when the causal graph contains no open back-door path from \(T\) to \(Y\) (a notion made precise in Section 3.2). Note that merely measuring common causes does not make the unadjusted conditional \(P(Y \mid T{=}t)\) equal to \(P(Y \mid \doop(T{=}t))\): measured confounders still must be adjusted for, as the remainder of this chapter develops.
The structural basis of the do-operator. In the SEM framework of Chapter 1, every variable is generated by a structural equation. For the treatment node: \[T \;=\; f_T\!\bigl(\Pa(T),\, U_T\bigr),\] where \(\Pa(T)\) are the causal parents of \(T\) and \(U_T\) is exogenous noise. An intervention \(\doop(T{=}t)\) replaces this entire equation with the constant \(T = t\). The replacement has two consequences:
- \(T\) is no longer influenced by its parents — its dependence on \(\Pa(T)\) and \(U_T\) is severed.
- All other structural equations remain unchanged — the rest of the causal mechanism is unaffected. This is the modularity assumption of Chapter 1: intervening on \(T\) leaves the mechanisms generating every other variable intact.
This is the sense in which an intervention is a surgical operation on the model: it cuts out one equation and replaces it with a constant, leaving the remainder of the system intact.
Graph surgery as the graphical realization. Because \(\Pa(T)\) no longer affects \(T\) after the intervention, every arrow pointing into \(T\) is removed in the post-intervention graph. The resulting graph — the intervention graph \(\Gcal_{\overline{T}}\) — represents the post-intervention world. D-separation in \(\Gcal_{\overline{T}}\) therefore implies conditional independence in \(P(\,\cdot \mid \doop(T{=}t))\), not in the original observational distribution \(P\). This is the key link that allows graphical reasoning to answer causal questions.
Truncated factorization. Suppose the observational distribution factorizes according to \(\Gcal\) as \[p(v) \;=\; \prod_{j=1}^{p} p\{v_j \mid \Pa(v_j)\}.\] Under the intervention \(\doop(X{=}x)\), the factors for the intervened nodes are removed and all remaining mechanisms are preserved: \[p\bigl(v \setminus x \mid \doop(X{=}x)\bigr) \;=\; \left. \prod_{V_j \notin X} p\{v_j \mid \Pa(v_j)\} \right|_{X = x}.\] Thus deleting arrows into \(X\) at the graph level corresponds to deleting the conditional factors for \(X\) at the distributional level.
3.1.2 The Graphs \(\Gcal_{\overline{X}}\) and \(\Gcal_{\underline{X}}\)
Both operations modify the DAG by selectively removing edges.
3.2 Back-Door Identification
The back-door criterion is the most widely used identification strategy. It gives a simple graphical condition under which conditioning on an observed set \(\mathbf{S}\) is sufficient to identify the causal effect of \(T\) on \(Y\). The formula it produces — standardization over \(\mathbf{S}\) — is computable directly from the observed data distribution; Section 3.3.2 supplies the do-calculus proof.
3.2.1 The Back-Door Criterion
A back-door path from \(T\) to \(Y\) is a path that begins with an arrow pointing into \(T\) (a path beginning \(T \leftarrow \cdots\)). Such paths are the graphical routes along which confounding travels: when open, they can transmit association between \(T\) and \(Y\) that reflects no causal effect of \(T\).
If an observed pretreatment set blocks every back-door path from \(T\) to \(Y\), identification by adjustment is straightforward, subject to positivity. Measuring the relevant common causes often provides such a set, but the graphical criterion is path blocking, not an instruction to condition indiscriminately on every measured variable. The interesting case — and the one the back-door criterion is designed for — is when some confounders are unobserved, yet an observed set \(\mathbf{S}\) can still block every spurious path.
Consider the following DAG. A drug \(T\) affects recovery \(Y\). An unobserved variable \(U\) (socioeconomic status) is a cause of \(T\) (\(U \to T\)) and affects \(Y\) only indirectly, through an observed intermediate variable \(W\) on the confounding path (e.g., access to healthcare) with \(U \to W\) and \(W \to Y\). \(U\) does not act on \(Y\) through any other pathway.
The only back-door path is \(T \leftarrow U \to W \to Y\). Although \(U\) is unobserved and cannot be adjusted for directly, \(W\) sits on this path and is observed. Conditioning on \(W\) blocks the path at the chain node \(W\), and \(W\) is not a descendant of \(T\), so \(\mathbf{S} = \{W\}\) satisfies the back-door criterion. The causal effect is therefore identified even though \(U\) is never measured.
This is the key insight: the criterion does not require adjusting for the confounders themselves, only for an observed set that intercepts every back-door path.
3.2.2 The Adjustment Formula and Positivity
Throughout the identification theorems of this chapter, let \(\mathcal{M}\) be a structural causal model compatible with \(\Gcal\), with interventions defined by modular replacement of structural equations as in Section 3.1. The theorems are statements about such models, not about arbitrary distributions that merely factorize according to \(\Gcal\).
Theorem 3.1 (Back-Door Adjustment Formula (Pearl 1993)) If \(\mathbf{S}\) satisfies the back-door criterion for the effect of \(T\) on \(Y\) in \(\Gcal\) and, for discrete \(T\), the positivity condition \[P(T{=}t \mid \mathbf{S}{=}\mathbf{s}) \;>\; 0 \quad \text{for } P_{\mathbf{S}}\text{-almost every } \mathbf{s} \tag{3.1}\] holds (for continuous \(T\), replace Equation 3.1 by the corresponding conditional-support requirement — \(t\) must lie in the support of \(T \mid \mathbf{S}{=}\mathbf{s}\) for \(P_{\mathbf{S}}\)-almost every \(\mathbf{s}\) — together with the regularity conditions needed to define the conditional law at \(t\); see Chapter 4), then: \[P\!\left(y \mid \doop(T{=}t)\right) \;=\; \int P(y \mid T{=}t,\;\mathbf{S}{=}\mathbf{s})\,dP(\mathbf{s}). \tag{3.2}\] Equivalently, for any measurable function \(h\) for which the displayed expectations are well-defined and finite, \[\E\!\left[h(Y) \mid \doop(T{=}t)\right] \;=\; \E_{\mathbf{S}}\!\Bigl[\E\bigl[h(Y) \mid T{=}t,\;\mathbf{S}\bigr]\Bigr]. \tag{3.3}\] When \(\mathbf{S}\) is discrete, the integral reduces to \(\sum_{\mathbf{s}} P(y \mid t, \mathbf{s})\,P(\mathbf{s})\).
The derivation uses Rules 2 and 3 of the do-calculus; the complete three-step proof is given in Section 3.3.2.
Equation Equation 3.2 is also known as the standardization formula or, in the epidemiological literature, the g-formula (Robins 1986). Standardization is the point-treatment (single-time-point) special case of Robins’s g-formula, which was developed for general time-varying treatments and covariates; the multi-period version underlies the theory of dynamic treatment regimes.
The interpretation of the formula is important. Within each stratum \(\mathbf{s}\), all back-door paths are blocked, so the conditional and interventional distributions coincide: \(P(y \mid \doop(t), \mathbf{s}) = P(y \mid t, \mathbf{s})\) (proved as Step 3 in Section 3.3.2). The outer integration then weights the strata by the population marginal distribution \(P(\mathbf{S})\), rather than the treatment-conditional distribution \(P(\mathbf{S} \mid T{=}t)\) that a naive comparison implicitly uses — this re-weighting, made explicit by the iterated-expectation form Equation 3.3, is what removes the confounding.
3.2.3 Invalid Adjustment Variables
Two failure modes deserve explicit warnings.
3.3 The Do-Calculus
General theory. Having developed back-door identification as a concrete strategy, we now introduce Pearl’s three rules of do-calculus, the general transformation system of which it is one instance (Pearl 1995). Each rule uses a d-separation condition in a specified modified graph to license a transformation of an expression involving the do-operator, bringing it one step closer to a purely observational formula. Section 3.3.2 derives the back-door formula as a short sequence of these rules, and Section 3.5 presents the completeness theorem of Shpitser and Pearl (2006) and Huang and Valtorta (2006), which shows that the three rules identify every identifiable unconditional interventional distribution \(P(y \mid \doop(t))\) — and thereby delimits what can be identified from observational data alone.
3.3.1 The Three Transformations
In the rule statements below, the graph subscript records which incoming or outgoing arrows have been deleted before the d-separation condition is checked; the table that follows gives the exact recipe for each subscript.
Recall from Chapter 2 that \(P\) is Markov with respect to \(\Gcal\) if every d-separation in \(\Gcal\) implies the corresponding conditional independence under \(P\). The do-expressions below acquire meaning through the causal semantics of Section 3.1.1: \(\doop(x)\) replaces structural equations, so the post-intervention distribution is Markov with respect to \(\Gcal_{\overline{X}}\).
The table below gives the recipe for each graph subscript. The subscripts should be treated as a reference system rather than memorized at the outset: first determine whether the desired operation concerns an observation, an action–observation exchange, or an action, and then use the table to select the corresponding test graph.
| Subscript | Appears in | Arrows deleted |
|---|---|---|
| \(\Gcal_{\overline{X}}\) | Rule 1 | all arrows into \(X\) |
| \(\Gcal_{\overline{X}\,\underline{Z}}\) | Rule 2 | all arrows into \(X\); all arrows out of \(Z\) |
| \(\Gcal_{\overline{X}\,\overline{Z}}\) | Rule 3 | all arrows into \(X\); all arrows into \(Z\) |
Theorem 3.2 (The Three Rules of Do-Calculus (Pearl 1995)) Let \(\mathcal{M}\) be a structural causal model compatible with the DAG \(\Gcal\) over variables \(\mathcal{V}\), with interventions defined by structural-equation replacement (equivalently, the truncated factorization) as in Section 3.1.1, and let \(X, Y, Z, W \subseteq \mathcal{V}\) be pairwise disjoint sets. The rules hold for every interventional distribution generated by \(\mathcal{M}\). All conditional distributions in the rules below are understood at values for which the relevant regular conditional laws are defined; in the discrete case, the conditioning events are assumed to have positive probability.
Rule 1 (Insertion/Deletion of Observations). Purpose: remove or insert an observation when it becomes irrelevant in the post-intervention graph. \[P(y \mid \doop(x), z, w) \;=\; P(y \mid \doop(x), w) \quad\text{if}\quad (Y \indep Z \mid X, W)_{\Gcal_{\overline{X}}}.\] The graph \(\Gcal_{\overline{X}}\) is obtained by deleting all arrows into \(X\); it represents the world after the intervention \(\doop(x)\). If \(Y\) and \(Z\) are d-separated given \(\{X, W\}\) in this graph, then conditioning on \(Z\) carries no information about \(Y\) beyond what \(W\) already does.
Rule 2 (Action/Observation Exchange). Purpose: replace an intervention by an observation, or vice versa, when the modified graph makes them equivalent. \[P(y \mid \doop(x), \doop(z), w) \;=\; P(y \mid \doop(x), z, w) \quad\text{if}\quad (Y \indep Z \mid X, W)_{\Gcal_{\overline{X}\,\underline{Z}}}.\] The graph \(\Gcal_{\overline{X}\,\underline{Z}}\) is obtained by deleting arrows into \(X\) and arrows out of \(Z\). It is an auxiliary test graph, not a model of the intervention \(\doop(z)\) (which would delete arrows into \(Z\)). When the condition holds, intervening on \(Z\) and conditioning on \(Z\) produce the same distribution of \(Y\); the soundness remark below explains why this deletion pattern isolates exactly the routes on which observation and intervention differ.
Rule 3 (Insertion/Deletion of Actions). Purpose: remove or insert an intervention when it becomes irrelevant in the modified graph. \[P(y \mid \doop(x), \doop(z), w) \;=\; P(y \mid \doop(x), w) \quad\text{if}\quad (Y \indep Z \mid X, W)_{\Gcal_{\overline{X}\,\overline{Z}}},\] stated here for the case — sufficient for every application in this course, and covering \(W = \varnothing\) in particular — in which no member of \(Z\) is an ancestor of any \(W\)-node in \(\Gcal_{\overline{X}}\). When the condition holds, the intervention \(\doop(z)\) can be deleted entirely. The general statement appears in the remark below.
| Rule | Statistical operation | Graphical idea |
|---|---|---|
| 1 | Add or remove an observed conditioning variable | The variable is irrelevant after the intervention |
| 2 | Exchange \(\doop(z)\) and observation \(z\) | Natural causes of \(Z\) no longer create a difference |
| 3 | Add or remove \(\doop(z)\) | Intervening on \(Z\) cannot affect \(Y\) in the modified graph |
3.3.2 A Do-Calculus Derivation of Back-Door Adjustment
With the three rules in hand, we can prove the back-door adjustment formula of Section 3.2.
3.4 Front-Door Identification: A Preview
The back-door criterion requires an observed set that blocks every back-door path. When the confounder is unobserved and no observed variable intercepts the back-door path, adjustment is unavailable — yet the causal effect may still be identified by routing through an observed mediator. The prototypical graph is \(T \to M \to Y\) with an unobserved common cause \(U\) of \(T\) and \(Y\):
The identification logic runs in two stages. The link \(T \to M\) is identified directly: by condition 2, the empty set satisfies the back-door criterion for the effect of \(T\) on \(M\), so \(P(m \mid \doop(t)) = P(m \mid t)\). The link \(M \to Y\) is identified by adjustment for \(T\): by condition 3, \(\{T\}\) satisfies the back-door criterion for the effect of \(M\) on \(Y\) (in the prototypical graph it blocks \(M \leftarrow T \leftarrow U \to Y\)), so \(P(y \mid \doop(m)) = \int P(y \mid t', m)\,dP(t')\). Chaining the two stages — condition 1 ensuring that all of \(T\)’s effect on \(Y\) passes through \(M\) — yields the front-door formula (Pearl 1995): under support conditions made precise in Chapter 8, \[P\!\left(y \mid \doop(T{=}t)\right) \;=\; \sum_m \underbrace{P(m \mid T{=}t)}_{\text{Stage 1}} \;\underbrace{\sum_{t'} P(y \mid T{=}t',\;M{=}m)\,P(t')}_{\text{Stage 2}}, \tag{3.4}\] with integrals replacing sums for continuous variables. Stage 2 is the back-door adjustment formula for \(P(y \mid \doop(M{=}m))\) with \(T\) as the adjustment variable: it averages \(P(y \mid t', m)\) over the marginal \(P(t')\), not the conditional \(P(t' \mid m)\). In this sense the front-door strategy does not remove confounding directly; it routes around it through an observed causal pathway, never directly comparing the treatment groups on \(Y\).
Here the mediator is used as an identification bridge for the total effect of \(T\) on \(Y\); no natural direct or indirect effect is being defined. Those decomposition estimands require different counterfactual objects and are introduced in Chapter 8. The formula demonstrates that identification need not proceed through direct adjustment for the treatment–outcome confounders. Chapter 8 returns to the front-door strategy, derives the formula formally via the do-calculus, and examines its relationship to mediation analysis.
3.5 Identification and Nonidentification
General theory (continued). This section is included to explain the scope and limits of the do-calculus. Students need not master the ID algorithm procedurally on a first reading; the main takeaway is that identifiability is a structural property of the graph, not merely a matter of algebraic cleverness. For a first reading, treat the bow-graph example and its two-model witness as the core of this section; the ADMG and semi-Markovian terminology records the formal scope of the completeness theorem.
A natural question after seeing the three rules is: are they enough? Could there be a graph where the causal effect is identifiable in principle, but none of the rules — applied in any sequence — can derive a purely observational expression? Shpitser and Pearl (2006) and, independently, Huang and Valtorta (2006) answered this definitively.
The completeness theorem is stated for semi-Markovian causal models represented by acyclic directed mixed graphs (ADMGs), in which a bidirected edge \(X \leftrightarrow Y\) compactly encodes a latent common cause of \(X\) and \(Y\) (in place of the explicit unobserved nodes, such as \(U \to T\) and \(U \to Y\), drawn so far); Appendix A gives the relevant graph terminology and an outline of the ID algorithm. The IV DAG and front-door graphs of this course are both special cases.
Theorem 3.3 (Completeness of the Do-Calculus (Shpitser and Pearl 2006; Huang and Valtorta 2006)) Let \(\Gcal\) be the ADMG of a semi-Markovian causal model. An unconditional interventional distribution \(P(y \mid \doop(t))\) is identifiable from the observed joint distribution \(P\) relative to \(\Gcal\) if and only if it can be reduced to a purely observational expression by the three rules of do-calculus together with standard probability calculus.
The proof is beyond the scope of this chapter; a proof sketch — the ID algorithm, districts, and the hedge obstruction — is given in Appendix A for reference and independent reading. We ground the theorem in the simplest possible nonidentifiable graph.
Example 3.1 (The Bow Graph — The Simplest Non-Identifiable Structure) The bow graph is the ADMG with one directed edge \(T \to Y\) and one bidirected edge \(T \leftrightarrow Y\) (representing an unobserved \(U\) with \(U \to T\) and \(U \to Y\)):
The single back-door path is \(T \leftarrow U \to Y\). Because \(U\) is unobserved and there are no other variables in the graph, no observed set can block this path, so the back-door criterion cannot be satisfied. The front-door criterion also fails: there is no observed mediator on any directed path from \(T\) to \(Y\). We now show directly that \(P(y \mid \doop(t))\) is not determined by the observed distribution \(P(T, Y)\) alone, by constructing two models that agree on \(P(T, Y)\) but disagree on \(P(y \mid \doop(t))\).
Following the convention for nonidentification witnesses stated in Appendix A, a model is compatible with \(\Gcal\) if each structural equation depends on a subset of the displayed parents, so a displayed arrow may carry a null effect. Let \(T, Y, U \in \{0, 1\}\) with \(U \sim \mathrm{Bernoulli}(1/2)\).
Model \(\mathcal{M}_1\) (pure confounding; no causal effect of \(T\)). \[T \;=\; U, \qquad Y \;=\; U.\] Both \(T\) and \(Y\) are driven entirely by \(U\); the arrow \(T \to Y\) carries no causal influence. Since \(T = U\) and \(Y = U\), the observed joint distribution is \(P(T{=}0, Y{=}0) = P(T{=}1, Y{=}1) = 1/2\) and \(P(T \neq Y) = 0\).
Under either intervention, the equation \(T = U\) is replaced by a constant while \(Y = U\) is unchanged, so \[P_1\!\left(Y{=}1 \mid \doop(T{=}1)\right) \;=\; P_1\!\left(Y{=}1 \mid \doop(T{=}0)\right) \;=\; P(U{=}1) \;=\; \tfrac{1}{2},\] and the average treatment effect is \(0\).
Model \(\mathcal{M}_2\) (full causal effect; \(U\) acts on \(Y\) only through \(T\)). \[T \;=\; U, \qquad Y \;=\; T.\] \(Y\) is determined entirely by \(T\); \(U\) affects \(Y\) only via \(T\). Since \(T = U\) again, we have \(Y = T = U\), so the observed joint distribution is again \(P(T{=}0, Y{=}0) = P(T{=}1, Y{=}1) = 1/2\) and \(P(T \neq Y) = 0\) — identical to \(\mathcal{M}_1\).
Under either intervention, the equation \(T = U\) is replaced by a constant, and \(Y = T\) deterministically, so \[P_2\!\left(Y{=}1 \mid \doop(T{=}1)\right) \;=\; 1, \qquad P_2\!\left(Y{=}1 \mid \doop(T{=}0)\right) \;=\; 0,\] and the average treatment effect is \(1\).
Conclusion. The two models produce identical observed distributions \(P(T,Y)\), so no observational data set, however large, can distinguish \(\mathcal{M}_1\) from \(\mathcal{M}_2\). Yet they assign different values to \(P(Y{=}1 \mid \doop(T{=}1))\) (\(1/2\) versus \(1\)) and to \(P(Y{=}1 \mid \doop(T{=}0))\) (\(1/2\) versus \(0\)), hence different average treatment effects (\(0\) versus \(1\)). Therefore \(P(y \mid \doop(t))\) is not identified from \(P(T, Y)\) in the bow graph. The completeness theorem tells us that no identification method — the do-calculus or otherwise — can overcome this obstruction: the bow graph contains a hedge, and the ID algorithm returns FAIL.
3.6 Summary
Two graph operations underpin the do-calculus: \(\Gcal_{\overline{X}}\) deletes arrows into \(X\) and models the intervention \(\doop(x)\); \(\Gcal_{\underline{X}}\) deletes arrows out of \(X\) and is used as a technical device in graphical conditions for certain do-calculus steps — it does not model conditioning on \(X\).
The back-door criterion identifies \(P(y \mid \doop(t))\) when an observed set \(\mathbf{S}\) blocks all back-door paths and contains no descendant of \(T\). The general formula, valid under the stated causal model and support conditions, is \(\int P(y \mid t, \mathbf{s})\,dP(\mathbf{s})\) or, in iterated-expectation form, \(\E_{\mathbf{S}}\!\left[\E[h(Y)\mid T{=}t,\mathbf{S}]\right]\) (standardization; the point-treatment g-formula).
The front-door criterion identifies \(P(y \mid \doop(t))\) via an observed mediator \(M\) intercepting all causal paths from \(T\) to \(Y\), even when the confounder \(U\) is entirely unobserved, provided the three front-door conditions hold.
The do-calculus has three rules, each licensed by a d-separation condition in an appropriate intervention graph. Rule 1 adds/removes observations; Rule 2 exchanges actions for observations; Rule 3 inserts or deletes interventions. The back-door formula is derivable as a short sequence of these rules (Section 3.3.2); Chapter 8 derives the front-door formula the same way.
Completeness. The do-calculus is complete for identification from observational data relative to the assumed graph (Shpitser and Pearl 2006; Huang and Valtorta 2006): an unconditional interventional distribution of the form \(P(y \mid \doop(t))\) is identifiable if and only if the do-calculus can derive a purely observational expression for it. Operationally: if the complete ID procedure fails for the target, then no functional of the observed joint distribution identifies it relative to the assumed graph, and no purely observational method can recover it without additional assumptions or new data. Failure to find a do-calculus derivation by hand, however, proves nothing by itself, as the completeness-versus-search remark in Section 3.5 explains.
This chapter completes the graphical foundation for point identification: causal assumptions (the graph) are converted into an identification functional \(\Psi(P_{\mathrm{obs}})\). Chapters 4–9 apply the machinery to specific designs — potential outcomes and randomization (Chapters 4–5), propensity scores (Chapter 6), instrumental variables (Chapter 7), mediation and the front-door strategy (Chapter 8), and sensitivity analysis (Chapter 9) — and Part III turns to the estimation of \(\Psi\).
3.7 Problems
1. Warm-up: graph operations. Let \(\Gcal\) be the DAG with edges \(Z \to T\), \(T \to Y\), \(U \to T\), \(U \to Y\) (the IV DAG, with \(U\) unobserved).
- Draw \(\Gcal_{\overline{T}}\) and identify all paths from \(Z\) to \(Y\) that remain open.
- Draw \(\Gcal_{\underline{T}}\) and identify all paths from \(Z\) to \(Y\) that remain. For each path, classify each intermediate node as a collider or a non-collider (chain or fork), and state whether the path is open or closed by default (without conditioning on any variable).
- Show that \((Z \indep Y)_{\Gcal_{\underline{T}}}\) holds in this four-node graph (part (b) has already classified the relevant path). Explain how this conclusion combines two graphical features: every directed path from \(Z\) to \(Y\) passes through \(T\) (the exclusion restriction), and \(Z\) has no open back-door path to \(Y\) (instrument exogeneity). Why would merely omitting the direct edge \(Z \to Y\) be insufficient in a richer graph?
- Use Rule 1 to show that \(P(y \mid \doop(t), z) = P(y \mid \doop(t))\) in this graph: state the required d-separation condition, verify it in \(\Gcal_{\overline{T}}\) by tracing paths, and interpret the result (once \(T\) is set externally, the instrument carries no further information about \(Y\)).
2. Back-door practice. Consider the DAG: \(X \to T\), \(X \to Y\), \(T \to M\), \(M \to Y\), \(T \to Y\), where \(X\) is observed. Parts (a)–(f) are core; part (g) is additional practice; part (h) is retrieval from Chapter 2.
- List all back-door paths from \(T\) to \(Y\).
- Does \(\{X\}\) satisfy the back-door criterion for the effect of \(T\) on \(Y\)? Write the resulting adjustment formula.
- Does \(\{M\}\) satisfy the back-door criterion? Explain why or why not.
- Does \(\{X, M\}\) satisfy the back-door criterion? Identify which condition of the criterion \(M\) violates, and explain why this rules it out as an adjustment variable.
- Even if the formal criterion issue in (d) were somehow set aside, explain why conditioning on \(M\) blocks the indirect causal pathway \(T \to M \to Y\) and therefore changes the target away from the total effect. Why does the back-door criterion deliberately rule out this operation?
- In a finite sample from this model you observe no treated units in the stratum \(X = x_0\), where \(P(X{=}x_0) > 0\). Distinguish two explanations: (i) a sample empty cell with \(P(T{=}1 \mid X{=}x_0) > 0\), and (ii) population nonpositivity, \(P(T{=}1 \mid X{=}x_0) = 0\). Which is an identification failure for \(P(y \mid \doop(T{=}1))\), and which is an estimation problem?
- Now consider a different DAG with observed \(C_1\), \(C_2\), unobserved \(U\), and edges \(C_1 \to T\), \(C_1 \to Y\), \(U \to T\), \(U \to C_2\), \(C_2 \to Y\), \(T \to Y\). List the two back-door paths from \(T\) to \(Y\), determine for each of \(\{C_1\}\), \(\{C_2\}\), and \(\{C_1, C_2\}\) whether the back-door criterion is satisfied, and write the adjustment formula for any valid set. What does the validity of \(\{C_1, C_2\}\), despite \(U\) being unobserved, show about the role of a variable’s position on a back-door path?
- (Retrieval.) The education–earnings DAG of Chapter 2 has edges \(N \to E\), \(B \to E\), \(B \to Y\), \(E \to Y\) (\(N\) = neighborhood, \(B\) = background, \(E\) = education, \(Y\) = earnings), all observed. Which of \(\{B\}\), \(\{N\}\), and \(\{N, B\}\) satisfy the back-door criterion for the effect of \(E\) on \(Y\)? Write the adjustment formula for one valid set.
3. Back-door formula: proof and uniqueness.
(Second pass.) Let \(\mathcal{M}\) be a structural causal model compatible with \(\Gcal\), and let \(\mathbf{S}\) satisfy the back-door criterion for \((T, Y)\) in \(\Gcal\) with positivity. Prove that \[P(y \mid \doop(t)) \;=\; \int P(y \mid t,\, \mathbf{s})\,dP(\mathbf{s}).\] (Hint: begin with \(P(y \mid \doop(t)) = \int P(y \mid \doop(t), \mathbf{s})\,dP\{\mathbf{s} \mid \doop(t)\}\). Apply Rule 3 with \(X = \varnothing\), \(Z = T\), \(W = \varnothing\): its graphical condition \((\mathbf{S} \indep T)_{\Gcal_{\overline{T}}}\) follows from back-door condition 1, and Rule 3 yields \(P(\mathbf{s} \mid \doop(t)) = P(\mathbf{s})\). Then apply Rule 2 with \(X = \varnothing\), \(Z = T\), \(W = \mathbf{S}\): its graphical condition \((Y \indep T \mid \mathbf{S})_{\Gcal_{\underline{T}}}\) follows from back-door condition 2, and Rule 2 yields \(P(y \mid \doop(t), \mathbf{s}) = P(y \mid t, \mathbf{s})\).)
Suppose \(\mathbf{S}_1\) and \(\mathbf{S}_2\) both satisfy the back-door criterion for \((T, Y)\) in \(\Gcal\) (with positivity). Deduce from part (a) that \[\int P(y \mid t,\, \mathbf{s}_1)\,dP(\mathbf{s}_1) \;=\; \int P(y \mid t,\, \mathbf{s}_2)\,dP(\mathbf{s}_2).\] Interpret this result: the identification target \(P(y \mid \doop(t))\) is unique even when the adjustment set is not.
4. Nonidentification in the bow graph. Consider the bow graph of Example 3.1: \(T \to Y\) together with \(T \leftrightarrow Y\) (an unobserved \(U\) with \(U \to T\) and \(U \to Y\)).
- Show that no observed set satisfies the back-door criterion for the effect of \(T\) on \(Y\), and that no observed mediator satisfies the front-door criterion.
- Verify that the two models \(\mathcal{M}_1\) and \(\mathcal{M}_2\) of Example 3.1 are compatible with the graph under the stated convention. Confirm that they induce the same observed distribution \(P(T, Y)\), but compute different values of both \(P(Y{=}1 \mid \doop(T{=}1))\) and \(P(Y{=}1 \mid \doop(T{=}0))\); hence compute the average treatment effect under each model.
- Explain why part (b) establishes that both the interventional distribution \(P(y \mid \doop(t))\) and the average treatment effect are not identified from \(P(T, Y)\). What does the completeness theorem add beyond this particular two-model witness?