Monotonicity Requirements for Efficient Exact Sampling with Markov Chains

Pawel Lorek, Piotr Markowski

Abstract:

We recall three methods for exact sampling from a stationary dis-
tribution of a Markov chain: the coupling from the past (CFTP) algorithm, a
method based on strong stationary duality (SSD), and Fill’s rejection algorithm.
Each method, to be applied efficiently, requires a different notion of monotonicity,
which is defined with respect to a partial ordering of the state space, namely
realizable monotonicity, Möbius monotonicity, and stochastic monotonicity. We
show full relations between monotonicities. The applicability of the CFTP algo-
rithm implies the applicability of Fill’s rejection algorithm, but does not imply
that of the SSD-based method. We also state one open problem related to these
monotonicities.