The Causal Description Gap: Information-Theoretic Separations Across Pearl's Hierarchy
For causal inference researchers, the work provides a rigorous information-theoretic measure of the difficulty of moving between causal levels, revealing fundamental limits on compressibility.
The paper quantifies the information-theoretic gaps between rungs of Pearl's causal hierarchy, showing that binary acyclic SCMs can have constant observational description length but quadratic interventional description length, and that this gap persists under approximate descriptions.
Pearl's causal hierarchy shows that observational, interventional, and counterfactual queries are qualitatively distinct. We ask a quantitative version of this question: how many additional bits are needed to specify higher-rung causal answers once lower-rung answers are known? We formalize this via query-class description length, the Kolmogorov complexity of the answer oracle induced by an SCM for a class of queries. Our main construction gives binary acyclic SCMs whose observational distribution has constant description length, while the single-variable interventional answer oracle has description length $Θ(n^2)$. A degree-sensitive upper bound shows that finite-gate-schema SCMs of indegree $d$ have observational-interventional gap at most $O(nd \log(en/d) + n \log n)$, making the quadratic construction order-optimal in the dense regime and a rooted-tree construction order-optimal for bounded indegree. The quadratic separation persists under $\varepsilon$-accurate total-variation descriptions for every fixed $\varepsilon < 1/4$. At the next rung, the full hard-do interventional oracle can still leave a $Θ(n)$ counterfactual description gap. A general ambiguity-to-bits theorem and Shannon analogue show that these gaps equal the logarithm of residual higher-rung ambiguity up to lower-order terms.