Spatio-Causal Patterns of Sample Growth

Statistical samples under (1) Unconfounded growth preserve estimators' ability to determine the independent effects of their individual variables on any measurements (and lead, therefore, to fair and interpretable black-box predictions). Samples under (2) Externally-Valid growth preserve their ability to make predictions that generalize across out-of-sample variation. The first promotes predictions that generalize over populations, the second over their shared exogeneous factors. We illustrate these theoretic patterns in the full, and spatially-localized, American census from 1840 to 1940, and samples ranging from the street-level all the way to the national for 60 thousand different locations. The resulting Binomial-Exponential sample size requirements for generalizability over space reveals connections among the Shapley value, U-Statistics (Unbiased Statistics), and Hyperbolic Geometry in spatial systems.