  Inference of fields defined in space and time from observational data is a
core discipline in many scientific areas. This work approaches the problem in a
Bayesian framework. The proposed method is based on statistically homogeneous
random fields defined in space and time and demonstrates how to reconstruct the
field together with its prior correlation structure from data. The prior model
of the correlation structure is described in a non-parametric fashion and
solely builds on fundamental physical assumptions such as space-time
homogeneity, locality, and causality. These assumptions are sufficient to
successfully infer the field and its prior correlation structure from noisy and
incomplete data of a single realization of the process as demonstrated via
multiple numerical examples.
