A Unified Interface for Covariance and Precision Geostatistical Inference

A common geostatistical pipeline can be implemented either with a dense covariance action or with a sparse precision action, depending on whether the prior is specified by a kernel or by a stochastic PDE or graph Laplacian. In practice, switching between these backends often forces one to re-derive formulas and re-implement inference. We present a unified, compositional interface for linear Gaussian geostatistical models that separates (i) where the latent field is represented, (ii) how measurements and targets are assembled from linear operators, and (iii) how inference is performed. The same assembled model can be executed by two interchangeable backends: a covariance-first method that solves in measurement space and a precision-first method that solves in latent space. We show that both compute the same posterior mean and expose the same matrix-free posterior covariance operator, enabling uncertainty quantification without forming dense matrices. We give practical algorithms for marginal variance estimation via Hutchinson trace estimators and for conditional simulation via Lanczos methods. Intrinsic, rankdeficient precisions are handled compositionally using projector constraints and projected preconditioned conjugate gradients. A complexity comparison clarifies the regimes in which each backend is preferable. Mathematics Subject Classification (2020) 18B40 · 62M30 · 65F10 · 62F15