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18 January 1988 Maximum Cramer-Rao Bound. Applications To: The Estimation Of Prior Probabilities, Image Restoration, And The Generation Of Quantum Mechanics
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Abstract
In this paper, we propose using maximum Cramer-Rao bound (MCRB) as a criterion for the estimation of a prior probability law p(x). This criterion has a useful interpretation: It is known that any estimate of x must have a mean-square error greater than or equal to the CRB. Hence, if one demands that p(x) have maximum CRB, he is demanding that p(x) be so non-informative, i.e., x be so random, that even the best estimate of x (attaining the CRB) will have maximal mean-square error. Hence, p(x) is made to describe a minimally knowable x. Because of the current interest in maximum entropy (ME) as an alternative approach, we compare results throughout this paper with those by ME. For example, if only the mean-value of x is known as prior information, the ME answer for p(x) is an exponential law, a smooth function. By contrast, MCRB gives the square of an Airy function, in fact an even smoother function. Comparisons are also made between ME and MCRB restorations, whereby p(x) is modeled to be the unknown object radiance distribution. Finally, MCRB may be applied to estimating the probability law describing particle position in a known potential energy field. With a constraint on average kinetic energy, the result is the Schroedinger wave equation.
© (1988) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only.
B. Roy Frieden "Maximum Cramer-Rao Bound. Applications To: The Estimation Of Prior Probabilities, Image Restoration, And The Generation Of Quantum Mechanics", Proc. SPIE 0829, Applications of Digital Image Processing X, (18 January 1988); https://doi.org/10.1117/12.942100
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