Causal Inference (C-inf) -- closed form worst case typical phase transitions

TL;DR

This paper rigorously connects causal inference with low-rank recovery, deriving explicit phase transitions that delineate when causal inference is feasible, supported by theoretical analysis and numerical validation.

Contribution

It introduces a novel mathematical framework using Random Duality Theory and free probability to precisely characterize phase transitions in causal inference via low-rank recovery.

Findings

Explicit phase transition formulas derived

Theoretical predictions match numerical experiments

Relations between low rankness and treatment time established

Abstract

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.