可实现单调性与逆概率变换
Realizable monotonicity and inverse probability transform
摘要：
系统（P_a：在由另一个指标集A所索引的公共状态空间S上的概率测度的（A中的a）可以由在某个概率空间上的S值随机变量系统（X_a：A中的a）以这样的方式"实现“，即每个X_a被分布为P_a。假设A和S都是偏序的，我们可能会问，系统（P_a：a in A）何时可以由系统（X_a：a in A）实现，而系统（X_a：a in A）具有单调性，即当a <= b时，几乎必然X_a <= X_b。当这样的实现是可能的，我们称系统（P_a：a in A）为可实现单调的。这样的系统必然是随机单调的，即当a <= b时，满足随机序P_a <= P_b。一般来说，随机单调性对于可实现单调性是不够的。然而，对于有限状态下某些特定的偏序选择，这两个单调性概念是等价的。本文对一类广义偏序集S提出了一个逆概率变换，并利用它显式地构造了一个系统（X_a：a in A），当两个单调性概念等价时，该系统实现了随机单调系统的单调性.
英文摘要：
A system (P_a: a in A) of probability measures on a common state space S indexed by another index set A can be ``realized'' by a system (X_a: a in A) of S-valued random variables on some probability space in such a way that each X_a is distributed as P_a. Assuming that A and S are both partially ordered, we may ask when the system (P_a: a in A) can be realized by a system (X_a: a in A) with the monotonicity property that X_a <= X_b almost surely whenever a <= b. When such a realization is possible, we call the system (P_a: a in A) ``realizably monotone.'' Such a system necessarily is stochastically monotone, that is, satisfies P_a <= P_b in stochastic ordering whenever a <= b. In general, stochastic monotonicity is not sufficient for realizable monotonicity. However, for some particular choices of partial orderings in a finite state setting, these two notions of monotonicity are equivalent. We develop an inverse probability transform for a certain broad class of posets S, and use it to explicitly construct a system (X_a: a in A) realizing the monotonicity of a stochastically monotone system when the two notions of monotonicity are equivalent.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
