Exchangeability, Positivity, Consistency

In the previous chapters, we established that causal effects are defined in terms of potential outcomes—the outcomes we would observe under different treatment scenarios. We also recognized the fundamental problem of causal inference: we can never observe both potential outcomes for the same individual.

This raises a critical question: Under what conditions can we use observational data to estimate causal effects? The answer lies in three foundational assumptions collectively known as the identifiability conditions.

"The three conditions for identifiability of causal effects from observational data are exchangeability, positivity, and consistency." — What If, Chapter 3

The Three Pillars of Causal Identification

Assumption	Formal Statement	Intuition
Exchangeability	$Y^a \perp\!\!\!\perp A \mid L$	No unmeasured confounding
Positivity	$P(A=a \mid L=l) > 0$ for all $a, l$	Everyone has a chance of each treatment
Consistency	If $A=a$, then $Y = Y^a$	Treatment is well-defined

When all three conditions hold, we can make the crucial leap from observational association to causal inference:

$E[Y^a] = \sum_l E[Y \mid A=a, L=l] \cdot P(L=l)$

This equation tells us that the mean counterfactual outcome (left side, which is causal) equals a weighted average of conditional means (right side, which we can estimate from data). This is the mathematical bridge between what we want (causal effects) and what we have (observational data).

Why These Assumptions Matter

Without these assumptions, our observational estimates may be:

• Biased due to confounding (exchangeability violation)
• Undefined due to lack of comparable groups (positivity violation)
• Ambiguous due to multiple treatment versions (consistency violation)

Understanding these assumptions—their meaning, testability, and common violations—is essential for any causal analysis.