Algorithmic theories of randomness can be related to theories of probabilistic sequence prediction through the notion of a predictor, defined as a function which supplies lower bounds on initial-segment probabilities of infinite sequences. An infinite binary sequence $z$ is called unpredictable iff its initial-segment "redundancy" $n+\log p(z(n))$ remains sufficiently low relative to every effective predictor $p$. A predictor which maximizes the initial-segment redundancy of a sequence is called optimal for that sequence. It turns out that a sequence is random iff it is unpredictable. More generally, a sequence is random relative to an arbitrary computable distribution iff the distribution is itself an optimal predictor for the sequence. Here "random" can be taken in the sense of Martin-L\"{o}f by using weak criteria of effectiveness, or in the sense of Schnorr by using stronger criteria of effectiveness. Under the weaker criteria of effectiveness it is possible to construct a universal predictor which is optimal for all infinite sequences. This predictor assigns nonvanishing limit probabilities precisely to the recursive sequences. Under the stronger criteria of effectiveness it is possible to establish a law of large numbers for sequences random relative to a computable distribution, which may be useful as a criterion of "rationality" for methods of probabilistic prediction. A remarkable feature of effective predictors is the fact that they are expressible in the special form first proposed by Solomonoff. In this form sequence prediction reduces to assigning high probabilities to initial segments with short and/or numerous encodings. This fact provides the link between theories of randomness and Solomonoff's theory of prediction.

翻译：算法随机性理论可通过预测器的概念与概率序列预测理论相关联，预测器定义为对无穷序列初始段概率提供下界的函数。无穷二进制序列 $z$ 被称为不可预测的，当且仅当其初始段"冗余度" $n+\log p(z(n))$ 相对于每个有效预测器 $p$ 始终足够低。能使序列初始段冗余度最大化的预测器称为该序列的最优预测器。结果表明：一个序列是随机的当且仅当它是不可预测的。更一般地，一个序列相对于任意可计算分布是随机的，当且仅当该分布本身是该序列的最优预测器。这里"随机"可按使用弱有效性准则的Martin-Löf意义理解，或按使用强有效性准则的Schnorr意义理解。在较弱有效性准则下，可以构造对所有无穷序列均最优的通用预测器，该预测器恰好将非零极限概率赋予递归序列。在较强有效性准则下，可建立相对于可计算分布随机序列的强大数定律，这有助于作为概率预测方法"合理性"的判据。有效预测器的一个显著特征是它们均可表达为Solomonoff首次提出的特殊形式。在此形式下，序列预测归结为对具有短编码和/或多重编码的初始段赋予高概率。这一事实建立了随机性理论与Solomonoff预测理论之间的关联。