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On Positivity Condition for Causal Inference

ICML 2024 · Probabilistic and scientific modeling › Causal discovery · approach: Causal effect identification

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Derives sufficient positivity conditions (beyond strict positivity) for graphical causal effect identification via do-calculus, relaxed generalized Q-decomposition, and an algorithm (IDENTIFY+) that outputs a formula plus its positivity condition.

Problem

Causal effect identification formulas are usually derived assuming strict positivity of P(V), which is often violated in practice; the positivity conditions required for identification formulas to be well-defined under a general causal graph are not well understood.

Approach

Relaxes do-calculus rules and generalized Q-decomposition to hold under weaker positivity (e.g., P(x,z)>0 or conditions on c-factors), propagates positivity constraints through identification steps, and embeds this into the IDENTIFY algorithm to return a formula together with a positivity condition.

Key results

• Relaxed generalized Q-decomposition (Thm. 5.1) shows a c-factor can be expressed as a product of fractions when a specific c-factor is positive, and is identified as zero under another condition.
• Lemma 4.1 restates do-calculus rules with positivity conditions only on P(z,w) or P(w) under interventions instead of strict positivity.
• IDENTIFY+ returns an identification formula together with a positivity condition that is provably correct (Thm. 6.1).

Surprising findings

• Different identification formulas for the same effect (e.g., backdoor sets {Z1} vs {Z2}) can imply different positivity conditions, so one formula may be usable on data where another is not.
• The topological order chosen can change whether the relaxed Q-decomposition applies, so identifiability certificates depend on ordering.

Limitations

• Discrete case only; continuous case deferred to appendix.
• Positivity-aware identification algorithm is discussed but not fully developed; IDENTIFY+ is data-agnostic.
• Non-identifiability when Thm. 5.1 is inapplicable in any topological order is left open; post-hoc analysis validity only conjectured in general.

Experimental setup

Builds on: do-calculus IDENTIFY algorithm ID algorithm generalized Q-decomposition backdoor criterion

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