What we talk about when we talk about fields

In astronomical and cosmological studies one often wishes to infer some properties of an infinite-dimensional field indexed within a finite-dimensional metric space given only a finite collection of noisy observational data. Bayesian inference offers an increasingly-popular strategy to overcome the inherent ill-posedness of this signal reconstruction challenge. However, there remains a great deal of confusion within the astronomical community regarding the appropriate mathematical devices for framing such analyses and the diversity of available computational procedures for recovering posterior functionals. In this brief research note I will attempt to clarify both these issues from an "applied statistics" perpective, with insights garnered from my post-astronomy experiences as a computational Bayesian / epidemiological geostatistician.