Pearl's Do-Calculus Foundations

• Pearl’s do-calculus is a set of syntactic rules that convert interventional distributions into purely observational expressions in directed acyclic graphs using graph surgery and d-separation.
• It provides necessary and sufficient criteria to identify causal effects even in the presence of unobserved confounders, underpinning applications in mediation analysis, transportability, and meta-synthesis.
• The sequential application of its three rules enables researchers to decide whether causal queries, like P(y|do(x)), can be computed from observational data, streamlining causal inference.

Pearl’s do-calculus is a formal system of syntactic rules for manipulating interventional distributions in graphical causal models. Originating in the mid-1990s, it underpins the modern theory of nonparametric causal identification, providing necessary and sufficient criteria—expressed as transformations of probability expressions—to decide whether a causal query (such as $P(y \mid \mathrm{do}(x))$) can be computed from purely observational data in a given directed acyclic graph (DAG), even in the presence of unobserved confounding. The rules of do-calculus, their soundness, and their completeness with respect to nonparametric identification, establish it as the foundational algebra of causal inference in both finite and infinite models, and underlie key developments in mediation analysis, transportability, and multi-source meta-synthesis.

1. Formal Statement: Three Rules of Do-Calculus

Let $G$ be a causal DAG over nodes $V$, and let $X,Y,Z,W$ denote disjoint subsets of $V$. The effect of an intervention $\mathrm{do}(X=x)$ is defined by replacing the structural equations for the nodes in $X$ by $X:=x$ and “surgically” deleting all incoming edges into $X$ in $G$. This yields the post-intervention (mutilated) graph, denoted $G_{\bar{x}}$. Do-calculus provides rules for transforming interventional expressions under specific conditional independence (d-separation) conditions in such mutilated graphs.

The three rules are:

1. Insertion/Deletion of Observations

$$P(y \mid \mathrm{do}(x),z,w) = P(y \mid \mathrm{do}(x),w) \quad\text{if}\quad (Y \perp Z \mid X,W)_{G_{\bar{x}}}$$

If $Y$ is d-separated from $Z$ given $X,W$ in the graph where incoming edges to $X$ are removed, observation on $Z$ can be inserted or deleted from the conditional.

2. Action/Observation Exchange

$$P(y \mid \mathrm{do}(x),\mathrm{do}(z),w) = P(y \mid \mathrm{do}(x),z,w) \quad\text{if}\quad (Y \perp Z \mid X,W)_{G_{x}}$$

If $Y$ is d-separated from $Z$ given $X,W$ in the graph where incoming edges to $X$ and outgoing edges from $Z$ are deleted, intervening on $Z$ is equivalent to conditioning on $Z$.

3. Insertion/Deletion of Actions

$$P(y \mid \mathrm{do}(x),\mathrm{do}(z),w) = P(y \mid \mathrm{do}(x),w) \quad\text{if}\quad (Y \perp Z \mid X,W)_{G_{x\bar{z}(WY)}}$$

Here $G_{x\bar{z}(WY)}$ is obtained by deleting incoming arrows into $X$ and outgoing arrows from those nodes in $Z$ that are not ancestors of any node in $W\cup Y$ in $G_x$.

These rules are applied recursively, in conjunction with standard probabilistic manipulations (law of total probability, marginalization), to eliminate “do” operators from a query, culminating in a purely observational estimand when possible.

2. Completeness and Identification

Do-calculus is both sound and complete for the task of identifying causal queries in any semi-Markovian model (allowing for arbitrary unobserved confounders). This means that any interventional probability that is expressible in terms of $P(v)$ (where $v$ are observable nodes) under the assumed causal graph can be derived using only these three rules and standard probability manipulations. This was established by constructively linking the do-calculus to the c-component decomposition method of Tian & Pearl (2002) and the identification algorithms formalized by Huang & Valtorta (2006) and Shpitser & Pearl (2006). If the application of rules 1–3 cannot reduce $P(y \mid \mathrm{do}(x))$ to an observational expression, then the query is non-identifiable within the specified model (Pearl, 2012).

The completeness proof proceeds by induction and is centered on showing that for any identifiable query:

• The identification problem can be decomposed into c-components (districts connected in the latent projection of the DAG).
• Each component can be handled recursively, either reducing the number of interventions or variables until “atomic” base cases—solvable by back-door or front-door adjustment—are reached.

3. Application: Example of Identification

Consider the canonical front-door model with hidden confounder $U$ between $X$ and $Y$, and mediator $M$:

$U \to X \to M \to Y \leftarrow U$

The identification of $P(y \mid \mathrm{do}(x))$ proceeds:

• In $G_{\bar{M}}$ (outgoing edges from $M$ cut), $(Y \perp X \mid M)_{G_{\bar{M}}}$ holds (Rule 1), yielding:

$$P(y \mid \mathrm{do}(x)) = \sum_m P(y \mid m, \mathrm{do}(x)) P(m \mid \mathrm{do}(x))$$

• In $G_x$ (incoming into $X$ deleted), $X \perp M$, so $P(m \mid \mathrm{do}(x))=P(m \mid x)$ (Rule 2).
• In $G_{\bar{x}}$ (outgoing from $X$ cut), $X \perp Y \mid M$, so $P(y \mid m, \mathrm{do}(x))=P(y \mid m,x)$ (Rule 1).

Hence,

$$P(y \mid \mathrm{do}(x)) = \sum_m P(m \mid x) \sum_{x'} P(y \mid m, x')P(x')$$

This exemplifies how sequential rule applications can eliminate all interventions from the target query (Pearl, 2012).

4. Advanced Extensions: Mediation Analysis, Transportability, and Meta-Synthesis

The versatility of do-calculus is evident in its generalizations:

Mediation Analysis

Do-calculus computes both the controlled direct effect (CDE) and the natural direct effect (NDE) by enabling decomposition into terms expressible via observable data under appropriate graphical conditions. For instance, the NDE can be represented as: $$NDE = \sum_w\sum_m\left[ E(Y \mid \mathrm{do}(1,m),w) - E(Y \mid \mathrm{do}(0,m),w) \right] P(m \mid \mathrm{do}(0),w)P(w)$$ This relies on identifying suitable adjustment sets ($W$) for each local subgraph and recursively applying the do-calculus rules.

Transportability

Transportability addresses inference of causal effects in a target domain from data collected in a separate source domain, formalized by adding selection nodes $S_i \to V_i$ to the DAG to mark population-specific mechanisms. A query $P^*(y\mid \mathrm{do}(x))$ is transportable if successive applications of the do-calculus can reduce $P(y \mid \mathrm{do}(x), S=s^*)$ to a function of observable quantities from both domains, with $S$ present only in do-free conditionals (Pearl, 2012, Pearl et al., 2015).

Meta-Synthesis

Meta-synthesis fuses data from multiple studies (possibly with disjoint variable sets or paper-specific interventions) to estimate a target causal effect. Here, do-calculus provides a framework for decomposing the target estimand $R(I^*)$ into subrelations, each of which is transportable from at least one paper population and reconstructs the causal effect in the target domain (Pearl, 2012).

5. Alternatives, Generalizations, and Methodological Context

Categorical Generalization

Recent work frames the syntax of do-calculus within the free Markov category on a DAG, replacing probabilistic conditionals with string diagrams whose equalities correspond to purely causal, diagrammatic versions of the three do-calculus rules. The two nontrivial rules—decomposition and screening-off—together with standard probability theory, suffice to syntactically recover the ordinary do-calculus for causal Bayesian networks (Yin et al., 2022).

Bayesian Integration

It has been demonstrated that in the large-data limit, any interventional distribution identified by do-calculus coincides with the predictive distribution of a Bayesian graphical model that encodes the same structural (invariance-under-intervention) assumptions (Lattimore et al., 2019). This establishes a full equivalence, at the level of asymptotic algebraic expressions, between the symbolic manipulations of do-calculus and parameter-sharing-based Bayesian inference.

Quantum and Topos Extensions

In quantum causal models, the notion of “surgical intervention” requires generalized do-operators as completely positive maps. The CP-do(C)-calculus reformulates the three classical do-calculus rules as operator-algebraic identities, but finds that Rule 2 fails in the presence of indefinite causal order (quantum switch). This forces the introduction of genuinely quantum rules and revised semantics for intervention and counterfactual dependence (Vallverdu, 5 Aug 2025). In topos-theoretic frameworks, an intuitionistic $j$-do-calculus formalizes intervention and conditional independence as local truths on regime-covers; each rule of classical do-calculus lifts to a $j$-stable version, sound in the internal logic of the causal topos (Mahadevan, 20 Oct 2025).

6. Practical Impact and Limitations

Pearl’s do-calculus enables automated and constructive causal identification algorithms: given a structural equation model or causal DAG, one can, in principle, mechanically search for a derivation reducing $P(y \mid \mathrm{do}(x))$ to observable distributions, or establish non-identifiability if this is not possible. Its completeness ensures that no additional inference rules are required for nonparametric causal identification. However, the practical application requires accurate knowledge of the graph structure, and the computational overhead associated with repeated d-separation testing and rule applications can be significant in large graphs or in the presence of extensive latent confounding (Huang et al., 2012).

Do-calculus does not, by itself, quantify finite-sample statistical uncertainty or support model selection. Additionally, it does not extend natively to cyclic or dynamical graphs without further formal elaboration. In quantum and topos settings, modifications and new rules are necessary to accommodate non-classical notions of independence, intervention, and locality.

7. Synthesis

Pearl’s do-calculus forms the algebraic and logical core of modern causal inference. Its three rules, underpinned by graph surgery and d-separation, exhaustively characterize the possibilities and limitations of expressing interventional distributions in terms of observable data, within the assumptions of a structural causal model. Its impact is most evident in mediation analysis, transportability, and data-fusion (meta-synthesis) tasks, as well as in the development of automated causal identification algorithms and the conceptual unification of counterfactual modeling frameworks. Do-calculus continues to illuminate the foundational structure of both classical and generalized causal theories (Pearl, 2012, Yin et al., 2022, Pearl et al., 2015, Vallverdu, 5 Aug 2025, Mahadevan, 20 Oct 2025).