论文标题
马尔可夫类别和熵
Markov Categories and Entropy
论文作者
论文摘要
马尔可夫类别是描述和处理概率和信息理论中问题的新型框架。 在这项工作中，我们将分类形式主义与传统的熵，相互信息和数据处理不平等的定量概念相结合。我们表明，信息理论的几个定量方面可以通过富集的马尔可夫类别捕获，其中形态的空间配备了分歧甚至指标。 由于在信息理论中是习惯的，因此可以将共同信息定义为衡量联合来源距离其组成部分独立性的距离。 更引人注目的是，马尔可夫类别给出了来源和渠道的确定性概念，我们可以通过测量源或通道距确定性的程度来精确定义熵。这恢复了Shannon和Rényi的熵，以及用于量化多样性的Gini-Simpson指数，可用于给出广义熵的概念定义。
Markov categories are a novel framework to describe and treat problems in probability and information theory. In this work we combine the categorical formalism with the traditional quantitative notions of entropy, mutual information, and data processing inequalities. We show that several quantitative aspects of information theory can be captured by an enriched version of Markov categories, where the spaces of morphisms are equipped with a divergence or even a metric. As it is customary in information theory, mutual information can be defined as a measure of how far a joint source is from displaying independence of its components. More strikingly, Markov categories give a notion of determinism for sources and channels, and we can define entropy exactly by measuring how far a source or channel is from being deterministic. This recovers Shannon and Rényi entropies, as well as the Gini-Simpson index used in ecology to quantify diversity, and it can be used to give a conceptual definition of generalized entropy.
