Unveiling the Structure of Do-Calculus Reasoning via Derivation Graphs

The do-calculus defines a general system of inference for interventional queries, allowing causal quantities to be transformed through successive applications of its rules. This process induces a rich space of equivalent interventional expressions, but combining and ordering these rules remains challenging. In this work, we introduce derivation graphs, which represent how do-calculus rules are applied and combined, and characterize the full space of observational and interventional probabilities which are equivalent under the do-calculus. The structure of these graphs yields a simple procedure that uses at most four applications of do-calculus rules. Finally, we show how applying identification algorithms to equivalent causal queries produces multiple valid estimands for the same causal quantity, eventually yielding more efficient estimators.

Understanding cause-and-effect relationships is essential in science. However, we often only have observational data (where we passively observe what happens) rather than experimental data (where we actively intervene). Pearl’s do-calculus provides mathematical rules to reason about interventions using such observational data. In this work, we introduce derivation graphs, a new tool that maps out all the different but mathematically equivalent ways to express a causal question using these rules. We prove that any two equivalent expressions are connected by at most four simple steps, making it easier to explore the space of possible solutions. This helps researchers find multiple methods to estimate the same causal effect from data. Surprisingly, we show that while these methods are mathematically equivalent, they can give more or less precise answers in practice.