Abstract: Predictions of spatial processes using large data sets have become an important area of research. Current solutions often include placing strong assumptions on the error process associated with the data. Specifically, it is typically assumed that the data are equal to the spatial process of principal interest plus a mutually independent error process, which avoids modeling confounded cross- covariances between the signal and the noise within an additive model. In this study, we consider an alternative latent-process model, in which we assume that the error process is spatially correlated and correlated with the latent process of interest. We show that such error-process dependencies allow us to obtain precise predictions and prevent confounded error covariances within an expression of the marginal distribution of the data. We refer to these covariances as "nonconfounded discrepancy error covariances." In addition, we develop a "process augmentation" technique as a computation aid. The proposed method is demonstrated using simulated examples and an analysis of a large data set from the U.S. Census Bureau's American Community Survey.

Key words and phrases: Bayesian, low rank, machine learning, mixed effects model, nonresponse, parsimony.