Intervention and Conditioning in Causal Bayesian Networks
Sainyam Galhotra, Joseph Y. Halpern
TL;DR
The paper addresses the difficulty of assigning and computing probabilities for interventional and counterfactual queries in Causal Bayesian Networks. It introduces probabilistic independence of causal mechanisms, formalizes semantics for interventional formulas via complete/constrained combinations of conditional events, and shows how CBNs can be converted to i-compatible causal models to ensure unique identifiability of interventional probabilities. It derives practical, observational-data–driven methods for counterfactual quantities like the probability of sufficiency, probability of necessity, and their combination, with explicit formulas and complexity bounds. The results enable reliable causal reasoning in settings where controlled experiments are impractical, by linking CBN CPTs to causal-model contexts and showing when observational data suffice for estimation.
Abstract
Causal models are crucial for understanding complex systems and identifying causal relationships among variables. Even though causal models are extremely popular, conditional probability calculation of formulas involving interventions pose significant challenges. In case of Causal Bayesian Networks (CBNs), Pearl assumes autonomy of mechanisms that determine interventions to calculate a range of probabilities. We show that by making simple yet often realistic independence assumptions, it is possible to uniquely estimate the probability of an interventional formula (including the well-studied notions of probability of sufficiency and necessity). We discuss when these assumptions are appropriate. Importantly, in many cases of interest, when the assumptions are appropriate, these probability estimates can be evaluated using observational data, which carries immense significance in scenarios where conducting experiments is impractical or unfeasible.