22 September 1998 Kolmogorov complexity, statistical regularization of inverse problems, and Birkhoff's formalization of beauty
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Abstract
Most practical applications of statistical methods are based on the implicit assumption that if an event has a very small probability, then it cannot occur. For example, the probability that a kettle placed on a cold stove would start boiling by itself is not 0, it is positive, but it is so small, that physicists conclude that such an event is simply impossible. This assumption is difficult to formalize in traditional probability theory, because this theory only describes measures on sets and does not allow us to divide functions into 'random' and non-random ones. This distinction was made possible by the idea of algorithmic randomness, introduce by Kolmogorov and his student Martin- Loef in the 1960s. We show that this idea can also be used for inverse problems. In particular, we prove that for every probability measure, the corresponding set of random functions is compact, and, therefore, the corresponding restricted inverse problem is well-defined. The resulting techniques turns out to be interestingly related with the qualitative esthetic measure introduced by G. Birkhoff as order/complexity.
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Vladik Kreinovich, Vladik Kreinovich, Luc Longpre, Luc Longpre, Misha Koshelev, Misha Koshelev, } "Kolmogorov complexity, statistical regularization of inverse problems, and Birkhoff's formalization of beauty", Proc. SPIE 3459, Bayesian Inference for Inverse Problems, (22 September 1998); doi: 10.1117/12.323795; https://doi.org/10.1117/12.323795
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