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Entropy 1999, 1, 69-98 entropy ISSN 1099-4300 /entropy/ A Bayesian Reflection on Surfaces David R. Wolf PO 8308, Austin, TX 78713-8308, USA E-mail: drwolf@ Received: 20 September 1999 / Accepted 20 October 1999 / Published: 30 October 1999 Abstract: 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 in- formative 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 al- lows 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 con- tinuous surfaces from measurements (for example, camera image data), is presented. Keywords: Bayesian inference; Generalized Kalman filter; Kalman filter; Kullback-Leibler distance; Maximally informative statistical inference; Knowledge representation; Mini- mumDescription Length; Sufficient s