  The topic of this paper is a novel Bayesian continuous-basis field
representation and inference framework. Within this paper several problems are
solved: The maximally informative inference of continuous-basis fields, that is
where the basis for the field is itself a continuous object and not
representable in a finite manner; the tradeoff between accuracy of
representation in terms of information learned, and memory or storage capacity
in bits; the approximation of probability distributions so that a maximal
amount of information about the object being inferred is preserved; an
information theoretic justification for multigrid methodology. The maximally
informative field inference framework is described in full generality and
denoted the Generalized Kalman Filter. The Generalized Kalman Filter allows the
update of field knowledge from previous knowledge at any scale, and new data,
to new knowledge at any other scale. An application example instance, the
inference of continuous surfaces from measurements (for example, camera image
data), is presented.
