identification-theory

Identification Theory

Comprehensive framework for causal identification in statistical methodology

Use this skill when working on: causal identification, mediation analysis identification, DAG-based reasoning, potential outcomes, identification assumptions, partial identification, sensitivity analysis, or deriving identification formulas.

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Core Concepts

What is Identification?

A causal parameter $\psi$ is identified if it can be uniquely determined from the observed data distribution $P(O)$.

Formally: $\psi$ is identified if $P_1(O) = P_2(O) \Rightarrow \psi_1 = \psi_2$.

Why Identification Matters

Causal Question → Target Estimand → Identification → Estimation → Inference
     ↓                  ↓                ↓               ↓            ↓
  "Does A           E[Y(1)-Y(0)]     Express in      Statistical   Confidence
   cause Y?"                         terms of P(O)    methods      intervals

Without identification, no amount of data can answer causal questions.

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Two Frameworks

1. Potential Outcomes (Rubin/Neyman)

Primitives:

• $Y(a)$ = potential outcome under treatment $a$
• Only $Y = Y(A)$ is observed (consistency)
• Fundamental problem: never observe both $Y(0)$ and $Y(1)$ for same unit

Advantages:

• Clear definition of causal effects
• Natural for experimental reasoning
• Connects to missing data theory

2. Structural Causal Models (Pearl)

Primitives:

• Directed Acyclic Graph (DAG) encoding causal structure
• Structural equations: $Y := f_Y(PA_Y, U_Y)$
• Interventions via do-operator: $P(Y | do(A=a))$

Advantages:

• Visual representation of assumptions
• Systematic identification algorithms
• Clear separation of statistical and causal assumptions

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DAG Framework

Directed Acyclic Graphs (DAGs)

A DAG $\mathcal{G} = (V, E)$ consists of:

• Vertices $V$: Random variables
• Directed edges $E$: Direct causal relationships
• Acyclic: No directed cycles

Key DAG Terminology