2024/08/20 by Motoya Machida, Machida, Motoya
Biochemistry, Genetics and Molecular Biology · Economics, Econometrics and Finance · Physics and Astronomy · #05C05 #05C38 (Secondary) #60E05 (Primary) 06A06 #Complex Systems and Time Series Analysis #FOS: Mathematics #Gene Regulatory Network Analysis #Opinion Dynamics and Social Influence #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2408.10896
openalex publication_date 2024/08/20 · openalex created_date 2024/10/01 · openalex updated_date 2026/07/28
A system (Pα: α\inA) of probability distributions on a partially ordered set (poset) S indexed by another poset A can be realized by a system of S-valued random variables Xα's marginally distributed as Pα. It is called realizably monotone if Xα≤ Xβ in S whenever α≤β in A. Such a system necessarily is stochastically monotone, that is, it satisfies Pα\preceq Pβ in stochastic ordering whenever α≤ β. It has been known exactly when these notions of monotonicity are equivalent except for a certain subclass of acyclic posets, called Class W. In this paper we introduce inverse probability transforms and synchronizing bijections recursively when S is a poset of Class W and A is synchronizable, and validate monotonicity equivalence by constructing (Xα: α\inA) explicitly. We also show that synchronizability is necessary for monotonicity equivalence when S is in Class W.