Kolmogorov complexity, statistical regularization of inverse problems, and Birkhoff's formalization of beauty
Kolmogorov complexity, statistical regularization of inverse problems, and Birkhoff's formalization of beauty
复制标题
柯尔莫哥洛夫复杂性、反问题的统计正则化以及伯克霍夫的美形式化
DOI：
10.1117/12.323795
复制
发表时间：
1998
期刊：
影响因子：
--
通讯作者：
M. Koshelev
中科院分区：
文献类型：
--
作者：
V. Kreinovich;L. Longpré;M. Koshelev
统计方法的大多数实际应用都是基于这样一个隐含的假设：如果一个事件的概率很小，那么它就不可能发生。例如，一个放在冷炉子上的水壶自己开始沸腾的概率不是0，它是正的，但它是如此之小，以至于物理学家得出结论，这样的事件根本不可能发生。这种假设在传统概率论中很难形式化，因为该理论仅描述集合上的度量，不允许我们将函数分为“随机”和非随机函数。这种区别是由算法随机性的想法，介绍了科尔莫戈洛夫和他的学生马丁-洛夫在20世纪60年代。我们表明这个想法也可以用于逆问题。特别是，我们证明了，对于每个概率测度，相应的随机函数集是紧的，因此，相应的限制反问题是定义良好的。由此产生的技术原来是有趣的是与G。伯克霍夫的秩序/复杂性。
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.