Unified theory for estimating frequency-of-occurrence laws and optical objects

Abstract

What is the optimum estimator for optical objects and other occurrence-rate distributions, such as for droplet size? Any contender must obey positivity: optical objects are nonnegative energy sources, and occurrence rates cannot be negative. Maximum entropy has been a contender, but it occurs in at least four forms—used by Jaynes, Burg, Kullbach, and Kikuchi and Soffer. Is any of these the correct one? And, if so, under what conditions? Our viewpoint is that a higher principle should be used that becomes one of the entropy forms, or some other form, under different conditions of prior knowledge. Such a principle is maximum likelihood (ML). We have derived a general ML estimator for spatially incoherent photon or electron objects, using a conventional statistical-physics approach. Results are also applied to other occurrence-rate problems, in particular that of water-droplet or snowflake size in a cloud. The ML estimator accommodates all physical and statistical prior knowledge about the unknown that may be known to the user. The physical knowledge for the optical problem includes the taking conditions (exposure time, pixel-cell size), coherence conditions, and either the Bose–Einstein or the Fermi–Dirac nature of the quanta involved. Statistical knowledge enters through an object-class law on joint intensities, or on occurrences, across the unknown. Specific forms of statistical knowledge are assumed in turn: strong, empirical, none at all, and negative. The resulting ML estimators are maximum (Shannon) entropy, maximum Kikuchi–Soffer entropy, minimum entropy, maximum or minimum cross entropy, maximum Burg or weighted-Burg entropy, minimum photon-site entropy, maximum empirical entropy, or no estimator at all. Other estimators exist as well, depending on the prior knowledge at hand. It behooves the user to define the prior knowledge that most accurately and completely defines his problem.