Exact sampling with coupled Markov chains and applications to statistical mechanics

Abstract

For many applications it is useful to sample from a finite set of objects in accordance with some particular distribution. One approach is to run an ergodic (i.e., irreducible aperiodic) Markov chain whose stationary distribution is the desired distribution on this set; after the Markov chain has run for M steps, with M sufficiently large, the distribution governing the state of the chain approximates the desired distribution. Unfortunately, it can be difficult to determine how large M needs to be. We describe a simple variant of this method that determines on its own when to stop and that outputs samples in exact accordance with the desired distribution. The method uses couplings which have also played a role in other sampling schemes; however, rather than running the coupled chains from the present into the future, one runs from a distant point in the past up until the present, where the distance into the past that one needs to go is determined during the running of the algorithm itself. If the state space has a partial order that is preserved under the moves of the Markov chain, then the coupling is often particularly efficient. Using our approach, one can sample from the Gibbs distributions associated with various statistical mechanics models (including Ising, random-cluster, ice, and dimer) or choose uniformly at random from the elements of a finite distributive lattice.

Parallel Markov chain Monte Carlo Simulation by Pre-Fetching

“…Since the seminal work of Propp and Wilson (1996) appeared, a lot of effort has gone into this area, but so far, no practical general-purpose scheme for typical applied Bayesian analysis problems has been developed. If it were possible to obtain these perfect samples, then the parallel chains approach would clearly be ideal.…”

Perfect sampling with aggregated envelopes

“…While increasing t by 1 at each loop would lead to a quadratic cost in τ e , doubling it keeps the complexity linear. This was already observed in [1].…”

Approximate zero-one laws and sharpness of the percolation transition in a class of models including two-dimensional Ising percolation

“…Also, note that we can extend the collection of Y variables to negative t and that for all integers s, t with s ≤ t, and all configurations ω ∈ {−1, +1} Z 2 , we can define σ ω (s, t) = (σ ω v (s, t), v ∈ Z 2 ) as the configuration at time t for the system that starts at time s with configuration ω and evolves as described above. Analogously as in [6] (which was partly inspired by the perfect simulation ideas in [24]), we observe that if t < 0 and σ + v (t, 0) = σ − v (t, 0), then (by obvious monotonicity) σ ω v (s, 0) = σ ω ′ v (s, 0) for all s ≤ t and all ω, ω ′ . From this observation, (7) and standard arguments, it follows that if we define…”