Causal Inference (C-inf) -- closed form worst case typical phase transitions
This work provides foundational mathematical insights for determining the applicability of causal inference methods, though it is incremental in building on existing Random Duality Theory concepts.
The paper establishes a rigorous connection between causal inference and low-rank recovery, deriving exact closed-form phase transitions that separate feasible from infeasible scenarios, with numerical experiments confirming theoretical predictions for small sample sizes.
In this paper we establish a mathematically rigorous connection between Causal inference (C-inf) and the low-rank recovery (LRR). Using Random Duality Theory (RDT) concepts developed in [46,48,50] and novel mathematical strategies related to free probability theory, we obtain the exact explicit typical (and achievable) worst case phase transitions (PT). These PT precisely separate scenarios where causal inference via LRR is possible from those where it is not. We supplement our mathematical analysis with numerical experiments that confirm the theoretical predictions of PT phenomena, and further show that the two closely match for fairly small sample sizes. We obtain simple closed form representations for the resulting PTs, which highlight direct relations between the low rankness of the target C-inf matrix and the time of the treatment. Hence, our results can be used to determine the range of C-inf's typical applicability.