A covariate-conditioned Ornstein-Uhlenbeck bridge model for reconstructing probability field in sparse domain

A natural strategy for inferring the latent state at an unobserved location in space, especially when target observations are sparse, is to use information from neighboring target observations and its auxiliary information. We consider this problem in a setting where target observations are sparse, while auxiliary covariate information is available throughout the space domain. In order to solve the problem, we propose a covariate-conditioned Ornstein-Uhlenbeck (COU) bridge model for reconstructing probability field at unobserved location. The proposed model decomposes the probability field into a deterministic component induced by global covariates and a stochastic residual modeled by an OU process. The local decorrelation structure of the residual is modulated by the covariates, allowing the model to reflect spatial heterogeneity and structural discontinuities. Given observations at two flanking locations, the resulting COU bridge provides probabilistic inference at interior locations. We first perform experiment on spatial data, and then for more generalization we evaluate the model on multiple spatio-temporal datasets. The results show that the proposed model is effective when sparse observations are complemented by informative covariates and the underlying dependence structure is consistent with the bridge formulation.