Kolmogorov Complexity, Statistical Regularization of Inverse Problems, and Birkhoff\u27s Formalization Of Beauty

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 (e.g., for an inverse problem, on the set of all functions) and does not allow us to divide functions into random (possible) and non-random ( impossible ) ones. This distinction was made possible by the idea of algorithmic randomness, introduced by Kolmogorov and his student Martin-Lof 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 correspondingly 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