Stochastic Monotonicity and Realizable Monotonicity

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. Short title. Stochastic and realizable monotonicity.

Realizable monotonicity for continuous-time Markov processes

“…For those two posets, the algorithm above ensures that G mon = G r.mon holds. Note that this result is known to be false in discrete-time, see for instance examples 1.1 and 4.5 in [FM01].…”

“…We recall that in [FM01] a more general definition of stochastic monotonicity and realizable monotonicity for a system of probability measures is considered: Definition 1.3. Let A and S be two partially ordered sets.…”

“…It turns out that if in a poset S stochastic monotonicity implies realizable monotonicity in discrete-time, then the same holds true in continuous-time (see Corollary 2.8). The converse is not true, however; for the posets whose Hasse diagram is represented in Figure 1 (the diamond and the bowtie, following the terminology in [FM01]) equivalence between stochastic monotonicity and realizable monotonicity holds in continuous-time but not in discrete-time.…”

Layered multishift coupling for use in perfect sampling algorithms (with a primer on CFTP)

“…Rather than produce a whole random map which is both monotone and marginalizes to a given Markov chain, it is sufficient to specify how any two given states may be updated together in a monotone fashion. Fill and Machida (1998) show that there are Markov chains with a pairwise monotone update rule but with no monotone randomizing operation, but so far there haven't been any such examples where someone wanted to sample from the steady state distribution.…”

Extension of Fill’s perfect rejection sampling algorithm to general chains (Extended abstract)

“…It is a common misbelief that, conversely, stochastic monotonicity for K implies realizable monotonicity. This myth is annihilated by Fill and Machida [13] and Machida [30], even for the case of a finite poset X , for which it is shown that every stochastically monotone kernel is realizably monotone (i.e., admits a monotone transition rule) if and only if the cover graph of X (i.e., its Hasse diagram regarded as an undirected graph) is acyclic.…”