Nonnegativity Shapes Predictive Uncertainty: Complementary Model Fits and Quantile Construction

Abstract

Knowing that a response cannot be negative can inform predictive uncertainty, including extreme upper quantiles. This knowledge matters especially when a black-box system returns long vector responses at few covariate settings, leaving large uncertainty across many entries. We develop a framework that assembles predictive distributions feature by feature from distributional fits that incorporate nonnegativity, optionally around any given mean predictor. The fits come from a latent factor model that represents response intensity through nonnegative patterns whose amplitudes follow rectified Gaussian processes over covariates. A variational algorithm jointly learns the factorization and the latent posterior. Varying a conditional-dispersion parameter yields a spectrum of fits that show complementary strengths in estimating distributional shape features. An orthogonal quantile expansion motivates using L-moments or their ratios as features. We select a fit separately for each using scores computable from a single validation observation per entry, then assemble the selected estimates through the expansion. In simulations, this construction reduces extreme upper-quantile errors by 17-22% relative to the respective best single fit, despite the mode-seeking tendency of our variational inference. In a streamflow demonstration across agricultural catchments, using an external point simulation to supply the mean reduces miss rates of rare shortage by 86-91%.