GGMPs: Generalized Gaussian Mixture Processes for Repeated-Measure Conditional Density Estimation

Gaussian processes (GPs) offer a principled nonparametric framework for learning conditional densities with calibrated uncertainty, but their Gaussian predictive form cannot represent multimodal or strongly non-Gaussian outputs. We introduce the Generalized Gaussian Mixture Process (GGMP), a GP-based method for learning multimodal conditional densities from repeated-measure or distribution-valued data, where each input carries many output samples or an empirical conditional density. Exact inference in a fully coupled mixture-of-GPs model is exponentially intractable; the GGMP instead fits local Gaussian mixtures, aligns their components across inputs, and trains one heteroscedastic GP per component, yielding a closed-form Gaussian mixture predictive density that remains compatible with standard and scalable GP solvers. We prove that under component separation, smooth component trajectories, accurate local fits, and dense inputs, the alignment stage recovers the true component correspondence, and we show when component collisions make that correspondence unrecoverable for any method. On synthetic and real-world repeated-measure datasets the GGMP is competitive with neural and random-forest conditional density estimators, and controlled ablations map its operating range.