Section: Research Program
Perfect Simulation
Simulation approaches can be used to efficiently estimate the stationary behavior of Markov chains by providing independent samples distributed according to their stationary distribution, even when it is impossible to compute this distribution numerically.
The classical Markov Chain Monte Carlo simulation techniques suffer from two main problems:
To overcome these issues, Propp and Wilson [56] have introduced a perfect sampling algorithm (PSA) that has later been extended and applied in various contexts, including statistical physics [47], stochastic geometry [52], theoretical computer science [33], and communications networks [30], [46] (see also the bibliography at http://dimacs.rutgers.edu/~dbwilson/exact.html/ annotated by David B. Wilson.
Perfect sampling uses coupling arguments to give an unbiased sample
from the stationary distribution of an ergodic Markov chain on a
finite state space
The algorithm is based on a backward coupling scheme: it computes the trajectories from all
Any ergodic Markov chain on a finite state space has a representation of type (1) that couples in finite time with probability 1, so Propp and Wilson's PSA gives a “perfect” algorithm in the sense that it provides an unbiased sample in finite time. Furthermore, the stopping criterion is given by the coupling from the past scheme, and knowing the explicit bounds on the coupling time is not needed for the validity of the algorithm.
However, from the computational side, PSA is efficient only under some monotonicity assumptions that allow reducing the number of trajectories considered in the coupling from the past procedure only to extremal initial conditions. Our goal is to propose new algorithms solving this issue by exploiting semantic and geometric properties of the event space and the state space.