  Refining low-resolution (LR) spatial fields with high-resolution (HR)
information, often known as statistical downscaling, is challenging as the
diversity of spatial datasets often prevents direct matching of observations.
Yet, when LR samples are modeled as aggregate conditional means of HR samples
with respect to a mediating variable that is globally observed, the recovery of
the underlying fine-grained field can be framed as taking an "inverse" of the
conditional expectation, namely a deconditioning problem. In this work, we
propose a Bayesian formulation of deconditioning which naturally recovers the
initial reproducing kernel Hilbert space formulation from Hsu and Ramos (2019).
We extend deconditioning to a downscaling setup and devise efficient
conditional mean embedding estimator for multiresolution data. By treating
conditional expectations as inter-domain features of the underlying field, a
posterior for the latent field can be established as a solution to the
deconditioning problem. Furthermore, we show that this solution can be viewed
as a two-staged vector-valued kernel ridge regressor and show that it has a
minimax optimal convergence rate under mild assumptions. Lastly, we demonstrate
its proficiency in a synthetic and a real-world atmospheric field downscaling
problem, showing substantial improvements over existing methods.
