Do-calculus is a sound and complete symbolic calculus for identifying interventional distributions from a causal graph. It manipulates actions and observations by applying conditional-independence statements in surgically modified graphs.
Graph surgery notation
Definition. \(G_{\overline X}\) deletes arrows into \(X\); \(G_{\underline X}\) deletes arrows out of \(X\); and \(G_{\overline X,\underline Z}\) performs both operations. For \(Z\) and conditioning set \(W\), let \[ Z(W)=Z\setminus\operatorname{An}(W)_{G_{\overline X}} \] when stating Rule 3.
Rule 1: insertion and deletion of observations
Rule 1. If \[ Y\perp_d Z\mid X,W \quad\text{in }G_{\overline X}, \] then \[ p(y\mid\operatorname{do}(x),z,w) = p(y\mid\operatorname{do}(x),w). \]
Proof. In the postintervention model represented by \(G_{\overline X}\), the global Markov theorem gives \(Y\perp Z\mid X,W\). Conditioning on the fixed intervention value \(x\) yields the equality.
Rule 2: action-observation exchange
Rule 2. If \[ Y\perp_d Z\mid X,W \quad\text{in }G_{\overline X,\underline Z}, \] then \[ p(y\mid\operatorname{do}(x),\operatorname{do}(z),w) = p(y\mid\operatorname{do}(x),z,w). \]
Proof outline. Removing outgoing arrows from \(Z\) neutralizes its downstream causal influence while testing whether the mechanism assigning \(Z\) carries additional information about \(Y\). The d-separation condition makes the conditional law of \(Y\) invariant to whether the value \(z\) was observed or externally set, after \(X,W\) are fixed.
Rule 3: insertion and deletion of actions
Rule 3. If \[ Y\perp_d Z\mid X,W \quad\text{in }G_{\overline X,\overline{Z(W)}}, \] then \[ p(y\mid\operatorname{do}(x),\operatorname{do}(z),w) = p(y\mid\operatorname{do}(x),w). \]
Proof outline. The modified graph removes incoming arrows to those \(Z\)-nodes that are not ancestors of the conditioning variables after intervention on \(X\). The stated separation means their intervention cannot change the conditional distribution of \(Y\) given \(X,W\). Nodes ancestral to \(W\) are excluded from deletion because intervening on them can alter selection into \(W=w\).
Back-door adjustment from the rules
Proposition. If \(Z\) satisfies the back-door criterion, \[ p(y\mid\operatorname{do}(x)) =\sum_zp(y\mid x,z)p(z). \]
Proof. Expand \[ p(y\mid\operatorname{do}(x)) =\sum_zp(y\mid\operatorname{do}(x),z) p(z\mid\operatorname{do}(x)). \] Rule 2 exchanges action \(x\) for observation \(x\) in the first factor because \(Z\) blocks back-door paths. Rule 3 removes the action from the second factor because \(Z\) has no descendants of \(X\). This gives \(p(y\mid x,z)p(z)\).
Front-door identification as a derivation
Theorem. Under the front-door graphical criterion for mediator \(M\), \[ p(y\mid\operatorname{do}(x)) = \sum_m p(m\mid x) \sum_{x'}p(y\mid m,x')p(x'). \]
Proof outline. First, \(p(m\mid\operatorname{do}(x))=p(m\mid x)\) by Rule 2 because there is no unblocked back-door path from \(X\) to \(M\). Second, all directed \(X\to Y\) paths pass through \(M\), so after intervening on \(M\), Rule 3 removes the action on \(X\). Finally, back-door paths from \(M\) to \(Y\) are blocked by \(X\), so \[ p(y\mid\operatorname{do}(m)) =\sum_{x'}p(y\mid m,x')p(x'). \] Sum over the interventional mediator distribution.
Identification and the ID algorithm
In semi-Markovian models with latent confounding represented by bidirected edges, the ID algorithm recursively decomposes interventional queries into districts, applies ancestral restriction, and detects hedge structures. It returns an observed-data functional exactly when the query is identifiable.
Definition. A hedge for \(P_x(y)\) is a pair of nested confounded forests satisfying specific root and intervention conditions. Its presence witnesses two causal models with the same observational law but different target intervention laws.
Completeness
Theorem. For causal effects in semi-Markovian graphical models, if a query is identifiable from the observational distribution, then it can be derived by do-calculus together with ordinary probability manipulations.
Proof outline. The ID algorithm is complete: failure occurs exactly when it finds a hedge, which constructs observationally equivalent models disagreeing on the query. Every successful recursive ID step can be represented as an application of a do-calculus rule and marginalization or conditioning. Therefore every identifiable query has a do-calculus derivation.
Conditional effects and selection bias
Identifying \(p(y\mid\operatorname{do}(x),z)\) is not equivalent to identifying the joint \(p(y,z\mid\operatorname{do}(x))\) and dividing if \(z\) is post-treatment or affected by selection. The IDC algorithm first uses Rule 2 to move eligible conditioned variables into interventions and then invokes ID. Similar extensions handle transportability by adding selection nodes.
What do-calculus does not provide
The calculus answers whether and how a population interventional distribution is a functional of observed distributions. It does not choose a finite-sample estimator, guarantee positivity, control nuisance estimation, or validate the graph. Identification is the first layer; estimation and sensitivity are separate layers.