熵：马尔可夫排序方法
Entropy: The Markov Ordering Approach
摘要：
本文的重点是熵和马尔可夫过程。我们研究了单调变换不变的泛函的性质，并分析了两个不变的“可加性”性质：（i）存在单调变换，使函数可加性相对于独立系统的加入和（ii）存在单调变换，使函数可加性相对于状态空间的划分。导出了具有性质（i）和（ii）的马尔可夫链的所有李雅普诺夫泛函。我们描述了分布空间的最一般的排序，相对于所有连续时间马尔可夫过程是单调的（{ \em马尔可夫顺序}）。该解与熵增长不等式给出的排序有很大不同。对于推理，这种方法会导致一个凸紧集的条件“最随机”的分布。
英文摘要：
The focus of this article is on entropy and Markov processes. We study the properties of functionals which are invariant with respect to monotonic transformations and analyze two invariant "additivity" properties: (i) existence of a monotonic transformation which makes the functional additive with respect to the joining of independent systems and (ii) existence of a monotonic transformation which makes the functional additive with respect to the partitioning of the space of states. All Lyapunov functionals for Markov chains which have properties (i) and (ii) are derived. We describe the most general ordering of the distribution space, with respect to which all continuous-time Markov processes are monotonic (the {\em Markov order}). The solution differs significantly from the ordering given by the inequality of entropy growth. For inference, this approach results in a convex compact set of conditionally "most random" distributions.
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