Wefindalarge range of behavior that can be precisely characterized by a new measure ofconfounding strength.

However, because they are not identifiable without any assumptions, various assumptions have been utilized to evaluate the joint probabilities of potential outcomes, e.g., the assumption of monotonicity (Pearl, 2009; Tian and Pearl, 2000), the independence between potential outcomes (Robins and Richardson, 2011), the condition of gain equality (Li and Pearl, 2019), and the specific functional relationshipsbetween cause and effect (Pearl, 2009). Unlike existing identification conditions, in order to evaluate the joint probabilities of potential outcomeswithoutsuch assumptions,this paper proposestwo types of novel identification conditions using covariate information. In addition, when the joint probabilities of potential outcomes are identifiable through the proposed conditions, the estimation problem of the joint probabilities of potential outcomes reduces to that of singular models and thus they can not be evaluated by standard statistical estimation methods. To solve the problem,this paper proposes a new statisticalestimationmethod based on the augmented Lagrangianmethod and shows the asymptoticnormality of the proposed estimators. Given space constraints, the proofs, the details on the statistical estimationmethod, some numerical experiments, and the case study are provided in the supplementary material.

"sufficiency", and "necessity and sufficiency", which are important concepts In practical science, it is crucial to evaluate the likelihood of one event causing another event. For example, epidemiologists pay attention to determining the likelihood of a particular exposure being the cause of a particular disease.

We study the problem of learning causal models from observational data through the lens of interpolation and its counterpart -- regularization. A large volume of recent theoretical, as well as empirical work, suggests that, in highly complex model classes, interpolating estimators can have good statistical generalization properties and can even be optimal for statistical learning. Motivated by an analogy between statistical and causal learning recently highlighted by Janzing (2019), we investigate whether interpolating estimators can also learn good causal models. To this end, we consider a simple linearly confounded model and derive precise asymptotics for the *causal risk* of the min-norm interpolator and ridge-regularized regressors in the high-dimensional regime. Under the principle of independent causal mechanisms, a standard assumption in causal learning, we find that interpolators cannot be optimal and causal learning requires stronger regularization than statistical learning. This resolves a recent conjecture in Janzing (2019). Beyond this assumption, we find a larger range of behavior that can be precisely characterized with a new measure of *confounding strength*. If the confounding strength is negative, causal learning requires weaker regularization than statistical learning, interpolators can be optimal, and the optimal regularization can even be negative. If the confounding strength is large, the optimal regularization is infinite, and learning from observational data is actively harmful.