Causal Identifiability

Causal identifiability asks whether a causal query can be uniquely determined from the available data and a causal model. A query is non-identifiable if multiple causal effects are equally consistent with the observed data, making a unique answer impossible. This is the fundamental challenge before any estimation can occur.

• Example: Without randomization, the effect of a drug on recovery may be non-identifiable due to unmeasured patient health factors.
• The goal is to transform a causal query (e.g., P(Y | do(X))) into an equivalent expression using only observable probabilities.

Identifiability is not a property of data alone; it depends critically on the causal assumptions encoded in a model, typically a causal graph. Key assumptions include:

• Causal Markov Condition: Links graph structure to conditional independencies in the data.
• Causal Faithfulness: Assumes all observed independencies are due to the graph structure, not巧合.
• No Unmeasured Confounding: For a treatment X and outcome Y, there is no common cause not included in the model. This is often the pivotal assumption for identifiability in observational studies.

The primary graphical tool for achieving identifiability from observational data. A set of variables Z satisfies the backdoor criterion for (X, Y) if:

1. Z blocks every backdoor path (non-causal path) between X and Y.
2. No variable in Z is a descendant of X.

If such a set Z exists, the causal effect is identifiable via adjustment: P(Y | do(X)) = Σ_z P(Y | X, Z=z) P(Z=z). This formula allows estimation from observational data by conditioning on the confounders Z.

An alternative identification strategy used when unmeasured confounding between X and Y violates the backdoor criterion. It requires a mediator variable M that:

1. Intercepts all directed paths from X to Y.
2. Has no unmeasured confounding with X.
3. Has no unmeasured confounding with Y, conditional on X.

The effect is then identified by a two-step formula: P(Y | do(X)) = Σ_m P(M=m | X) Σ_x' P(Y | X=x', M=m) P(X=x'). This creatively uses the mediator as a surrogate for randomization.

Do-calculus is a complete set of symbolic rules for transforming causal expressions. Its three rules allow the systematic removal of the do-operator from a query like P(Y | do(X)), replacing it with observational probabilities—if such a transformation is possible.

• Rule 1: Insert/delete observations.
• Rule 2: Exchange actions and observations.
• Rule 3: Insert/delete actions. Algorithms like ID and IDC use these rules to automatically determine if a query is identifiable for a given graph and, if so, output the estimand formula.

A method for identification when treatment X and outcome Y suffer from unmeasured confounding, and no backdoor or frontdoor set exists. An instrumental variable Z must satisfy:

1. Relevance: Z is correlated with X.
2. Exclusion Restriction: Z affects Y only through X (no direct path).
3. Exchangeability: Z shares no common causes with Y (is unconfounded).

Under these assumptions, the causal effect can be bounded or point-identified (e.g., in linear models: β = Cov(Z, Y) / Cov(Z, X)). This is a cornerstone of econometrics and quasi-experimental design.

A Structural Causal Model (SCM) is the formal mathematical framework that defines the system within which identifiability is assessed. It consists of:

• A set of structural equations specifying how each variable is generated from its direct causes and independent noise.
• An associated causal graph (a Directed Acyclic Graph) visualizing these relationships.
• The causal query (e.g., Average Treatment Effect) whose identifiability is in question. Identifiability is meaningless without a well-specified SCM; it asks whether the query can be uniquely computed from the observational distribution implied by the model.

Do-calculus is the complete set of symbolic rules for determining identifiability and computing causal effects. Developed by Judea Pearl, its three rules allow the transformation of probabilistic expressions containing the do-operator (representing an intervention) into equivalent expressions using only observational probabilities.

• If do-calculus can reduce a causal query to a statistical estimand using the graph structure, the effect is identifiable.
• It provides a systematic, graphical method to answer identifiability questions without simulating all possible data-generating models.

The backdoor criterion is the primary graphical test for identifiability of a treatment effect from observational data. A set of variables Z satisfies the backdoor criterion for a treatment X and outcome Y if:

1. Z blocks every backdoor path (a non-causal path connecting X and Y that starts with an arrow into X).
2. No variable in Z is a descendant of X. If such a set Z exists, the causal effect is identifiable via adjustment: P(Y | do(X)) = Σ_z P(Y | X, Z=z) P(Z=z). This is the most common identifiability condition in practice.

The frontdoor criterion provides an identifiability condition when unmeasured confounding blocks the backdoor approach. It requires a mediator variable M such that:

• X affects M and there is no unblocked backdoor path between them.
• M affects Y and all backdoor paths from M to Y are blocked by X.
• X does not affect Y directly, only through M. If satisfied, the effect is identifiable by a two-step formula: P(Y|do(X)) = Σ_m P(M=m|X) Σ_x' P(Y|X=x', M=m) P(X=x'). This demonstrates identifiability even with unobserved confounders.

An instrumental variable (IV) is a tool for achieving identifiability under severe unmeasured confounding. A variable Z is a valid instrument for estimating the effect of X on Y if:

• Relevance: Z is correlated with X.
• Exclusion Restriction: Z affects Y only through X (no direct path).
• Exchangeability: Z shares no common causes with Y (is unconfounded). When these conditions hold, the causal effect is locally identifiable (for compilers), often estimated via Two-Stage Least Squares. IV methods trade stronger assumptions for identifiability where adjustment is impossible.

Causal discovery is the process of learning the causal graph itself from data, which is a prerequisite for assessing identifiability in many real-world settings. Algorithms like PC, FCI, and GES:

• Test for conditional independencies to infer graph structure.
• Can identify equivalence classes of graphs (Partial Ancestral Graphs) when full identifiability is not possible.
• The output graph then informs which identifiability criteria (backdoor, frontdoor, IV) apply. Causal discovery addresses the challenge of not knowing the model structure needed to judge identifiability.