CausalBiome-I: Second-Order Invariance for Causal Interaction Discovery in Microbiome Data

Microbiome effects on host phenotypes are often non-additive. Pairs or groups of taxa can exhibit synergistic or antagonistic interactions that are invisible to marginal feature-importance methods. We introduce CausalBiome-I, a second-order extension of the CausalBiome framework that discovers causal interactions between microbial features. Starting from a second-order Taylor expansion of the empirical risk we show that the interaction (synergy) between two features is captured by a Hessian-weighted cross-term involving the loss curvature and the prediction perturbations induced by feature knock-out. We apply the same invariance principle as CausalBiome by computing both the mean synergy magnitude and synergy stability across random data partitions. This yields a unified interaction importance score that ranks feature pairs by causal interaction strength and robustness. To avoid the quadratic cost of exhaustive pairwise testing we introduce a tree-path co-occurrence screen that restricts evaluation to the pairs that actually co-occur on root-to-leaf paths in the trained ensemble. For typical gradient boosting configurations this reduces runtime by orders of magnitude. On synthetic microbiome data with planted marginal and interaction effects CausalBiome-I recovers ground-truth interactions with high precision while standard permutation importance and marginal-only CausalBiome miss them.