Kolmogorov结构函数与模型选择
Kolmogorov's Structure Functions and Model Selection
摘要：
1974年，Kolmogorov提出了一种非概率的统计方法和模型选择。设数据是有限的二进制字符串，模型是有限的二进制字符串集合。考虑由给定最大（Kolmogorov）复杂度的模型组成的模型类。给定数据的“结构函数”表达了模型类的复杂性水平约束与包含数据的类中模型的最小对数基数之间的关系。我们表明，结构函数决定了所有的随机属性的数据：对于每个约束模型类，它确定了类中的个人最佳拟合模型，而不管“真”模型是否在模型类中考虑或不考虑。在这种情况下，这发生{ \em确定性}，而不是像经典情况那样具有高概率。我们精确地量化了个体模型相对于个体数据的拟合优度。我们表明，-在明显的约束-每个图实现的一些数据的结构功能。我们确定的（非）可计算性性能的各种功能的预期和``算法的最小充分统计。''
英文摘要：
In 1974 Kolmogorov proposed a non-probabilistic approach to statistics and model selection. Let data be finite binary strings and models be finite sets of binary strings. Consider model classes consisting of models of given maximal (Kolmogorov) complexity. The ``structure function'' of the given data expresses the relation between the complexity level constraint on a model class and the least log-cardinality of a model in the class containing the data. We show that the structure function determines all stochastic properties of the data: for every constrained model class it determines the individual best-fitting model in the class irrespective of whether the ``true'' model is in the model class considered or not. In this setting, this happens {\em with certainty}, rather than with high probability as is in the classical case. We precisely quantify the goodness-of-fit of an individual model with respect to individual data. We show that--within the obvious constraints--every graph is realized by the structure function of some data. We determine the (un)computability properties of the various functions contemplated and of the ``algorithmic minimal sufficient statistic.''
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