Attribute-adaptive statistical inference for finite populations under distribution shift

Parameters of sub-populations can be more relevant than super-population ones. For example, a healthcare provider might be interested in the effect of a treatment plan on a subset of their patients; a video subscription service might be interested in the satisfaction of its current customers; or policymakers might care about the impact of a policy in a state with a given population. In these cases, one is interested in a finite population, as opposed to an infinite super-population. Such finite population can be characterized by fixing some attributes that are intrinsic to them, leaving unexplained variation like measurement error as random. More generally, inference for a population with fixed attributes can be modeled as inferring parameters of a conditional distribution given these attributes. It is desirable that confidence intervals are conditionally valid for the realized population, instead of marginalizing over many draws of such populations. We provide a statistical inference framework for parameters of populations with fixed attributes. By leveraging the attribute information, we derive estimators and confidence intervals that are closely related to a specific finite population. When the data is from the population of interest, our confidence intervals attain asymptotic conditional validity given the attributes, and are typically shorter than those for super-population inference. In addition, we develop procedures to infer parameters of new populations with differing covariate distributions; the confidence intervals are also conditionally valid under mild conditions. Our methods extend to situations where the fixed information has weaker structure or is only partially observed. We demonstrate the validity and applicability of our methods on simulated and real-world data.