Wasserstein-type Gaussian Process Regressions for Input Measurement Uncertainty
This work addresses uncertainty quantification in regression for applications where input measurements are noisy, offering a more reliable method compared to standard approaches.
The paper tackles the problem of Gaussian process regression when inputs have measurement errors, which can cause overly narrow uncertainty estimates and biased decisions, by introducing a new kernel based on Wasserstein distances that handles input noise transparently and robustly without latent variables or Monte Carlo methods.
Gaussian process (GP) regression is widely used for uncertainty quantification, yet the standard formulation assumes noise-free covariates. When inputs are measured with error, this errors-in-variables (EIV) setting can lead to optimistically narrow posterior intervals and biased decisions. We study GP regression under input measurement uncertainty by representing each noisy input as a probability measure and defining covariance through Wasserstein distances between these measures. Building on this perspective, we instantiate a deterministic projected Wasserstein ARD (PWA) kernel whose one-dimensional components admit closed-form expressions and whose product structure yields a scalable, positive-definite kernel on distributions. Unlike latent-input GP models, PWA-based GPs (\PWAGPs) handle input noise without introducing unobserved covariates or Monte Carlo projections, making uncertainty quantification more transparent and robust.