Stochastic monotonicity and realizable monotonicity

TL;DR

This paper investigates the relationship between stochastic and realizable monotonicity for probability measures on finite posets, providing conditions for their equivalence and implications for perfect sampling algorithms.

Contribution

It establishes necessary and sufficient conditions for the equivalence of stochastic and realizable monotonicity on finite posets, especially when the indexing poset equals the underlying poset.

Findings

Counterexamples showing non-equivalence in general

Cycle-free cover graphs are necessary and sufficient for equivalence when A=S

Implications for perfect sampling algorithms of Propp and Wilson

Abstract

We explore and relate two notions of monotonicity, stochastic and realizable, for a system of probability measures on a common finite partially ordered set (poset) S when the measures are indexed by another poset A. We give counterexamples to show that the two notions are not always equivalent, but for various large classes of S we also present conditions on the poset A that are necessary and sufficient for equivalence. When A = S, the condition that the cover graph of S have no cycles is necessary and sufficient for equivalence. This case arises in comparing applicability of the perfect sampling algorithms of Propp and Wilson and the first author of the present paper.