可计算泛先验的存在性与收敛性
On the Existence and Convergence Computable Universal Priors
摘要：
所罗门诺夫将奥卡姆剃刀和伊壁鸠鲁的多重解释原则统一为一个优雅的、形式化的、普遍的归纳推理理论，开创了算法信息论的领域。他的中心结果是，后他的普遍半措施M收敛迅速的真实序列产生后亩，如果后者是可计算的。因此，在未知μ的情况下，M有资格作为通用预测因子。我们调查的存在性和收敛性的可计算的普遍（半）措施的层次结构的可计算性类：可计算的，可估计的，可计算的，可逼近的。例如，已知M是可计算的，但不是可计算的，并且支配所有可计算的半测度。我们定义了七类（半）措施的基础上，这四个可计算性的概念。每个类可能包含或可能不包含一个（半）措施，其中占主导地位的所有元素的另一个类。对这49个案例的分析可归纳为四个基本案例，其中两个是新案例。结果保持离散和连续半测度。我们还调查更密切的类型的收敛，可能隐含的普遍性：在差异和比例，概率为1，平均和，马丁-勒夫随机序列。我们引入了一个广义的概念，个别序列的随机性，并使用它来表现出这些问题的困难。
英文摘要：
Solomonoff unified Occam's razor and Epicurus' principle of multiple explanations to one elegant, formal, universal theory of inductive inference, which initiated the field of algorithmic information theory. His central result is that the posterior of his universal semimeasure M converges rapidly to the true sequence generating posterior mu, if the latter is computable. Hence, M is eligible as a universal predictor in case of unknown mu. We investigate the existence and convergence of computable universal (semi)measures for a hierarchy of computability classes: finitely computable, estimable, enumerable, and approximable. For instance, M is known to be enumerable, but not finitely computable, and to dominate all enumerable semimeasures. We define seven classes of (semi)measures based on these four computability concepts. Each class may or may not contain a (semi)measure which dominates all elements of another class. The analysis of these 49 cases can be reduced to four basic cases, two of them being new. The results hold for discrete and continuous semimeasures. We also investigate more closely the types of convergence, possibly implied by universality: in difference and in ratio, with probability 1, in mean sum, and for Martin-Loef random sequences. We introduce a generalized concept of randomness for individual sequences and use it to exhibit difficulties regarding these issues.
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