We characterize Martin-Löf randomness and Schnorr randomness in terms of the merging of opinions, along the lines of the Blackwell-Dubins Theorem. After setting up a general framework for defining notions of merging randomness, we focus on finite horizon events, that is, on weak merging in the sense of Kalai-Lehrer. In contrast to Blackwell-Dubins and Kalai-Lehrer, we consider not only the total variational distance but also the Hellinger distance and the Kullback-Leibler divergence. Our main result is a characterization of Martin-Löf randomness and Schnorr randomness in terms of weak merging and the summable Kullback-Leibler divergence. The main proof idea is that the Kullback-Leibler divergence between $μ$ and $ν$, at a given stage of the learning process, is exactly the incremental growth, at that stage, of the predictable process of the Doob decomposition of the $ν$-submartingale $L(σ)=-\ln \frac{μ(σ)}{ν(σ)}$. These characterizations of algorithmic randomness notions in terms of the Kullback-Leibler divergence can be viewed as global analogues of Vovk's theorem on what transpires locally with individual Martin-Löf $μ$- and $ν$-random points and the Hellinger distance between $μ,ν$.

翻译：我们依据Blackwell-Dubins定理的思想，通过观点合并来刻画Martin-Löf随机性和Schnorr随机性。在建立了定义合并随机性概念的一般框架后，我们聚焦于有限视界事件，即Kalai-Lehrer意义上的弱合并。与Blackwell-Dubins和Kalai-Lehrer不同，我们不仅考虑全变差距离，还考虑了Hellinger距离和Kullback-Leibler散度。我们的主要结果是用弱合并与可和的Kullback-Leibler散度来刻画Martin-Löf随机性和Schnorr随机性。证明的核心思想是：在学习过程的给定阶段，$μ$与$ν$之间的Kullback-Leibler散度恰好等于$ν$-下鞅$L(σ)=-\ln \frac{μ(σ)}{ν(σ)}$的Doob分解中可预测过程在该阶段的增量增长。这些基于Kullback-Leibler散度的算法随机性刻画，可视为Vovk定理的全局类比——该定理在局部层面描述了单个Martin-Löf $μ$-随机点与$ν$-随机点以及$μ,ν$间Hellinger距离的关系。