Abstract
We establish, for various scenarios, whether or not interruptible exact stationary sampling is possible when a finite-state Markov chain can only be viewed passively. In particular, we prove that such sampling is not possible using a single copy of the chain. Such sampling is possible when enough copies of the chain are available, and we provide an algorithm that terminates with probability one.
Similar content being viewed by others
References
D. J. Aldous, “On simulating a Markov chain stationary distribution when transition probabilities are unknown,” In Discrete Probability and Algorithms (Aldous, Diaconis, Spencer and Steele, eds.) volume 72 of IMA Volumes in Mathematics and its Applications, Springer-Verlag, 1995, pp. 1–9.
D. J. Aldous and J. A. Fill, Reversible Markov Chains and Random Walks on Graphs, Book in preparation, 200x, Draft available from http://www.stat.berkeley.edu/users/aldous/.
S. Asmussen, P. W. Glynn, and H. Thorisson, “Stationary detection in the initial transient problem,” ACM Transactions on Modeling and Computer Simulation vol. 2 pp. 130–157, 1992.
J. Barshay, Topics in Ring Theory, W. A. Benjamin: New York, 1969.
A. Broder, “Generating random spanning trees,” In Proceedings of the 30th Annual Symposium on Foundations of Computer Science, 1989, pp. 442–447.
J. A. Fill, “An interruptible algorithm for perfect sampling via Markov chains,” Annals of Applied Probability vol. 8 pp. 131–162, 1998.
J. A. Fill and M. Huber, “The Randomness Recycler: A new technique for perfect sampling,” In Proceedings of the 41st Annual Symposium on Foundations of Computer Science, pp. 503–511, 2000.
J. A. Fill, M. Machida, D. J. Murdoch, and J. S. Rosenthal, “Extension of Fill's perfect rejection sampling algorithm to general chains,” Random Structures & Algorithms vol. 17 pp. 290–316, 2000.
S. Karlin and H. M. Taylor, A First Course in Stochastic Processes, 2nd Edition, Academic Press: New York, 1975.
D. E. Knuth, The Art of Computer Programming. Volume 1 (2nd Edition). Fundamental Algorithms, Addison-Wesley: Reading, MA, 1973.
L. Lovász and P. Winkler, “Exact mixing in an unknown Markov chain,” Electronic Journal of Combinatorics vol. 2 Paper #R15, 1995.
J. G. Propp and D. B. Wilson, “Exact sampling with coupled Markov chains and applications to statistical mechanics,” Random Structures & Algorithms vol. 9 pp. 223–252, 1996.
J. G. Propp and D. B. Wilson, “How to get a perfectly random sample from a generic Markov chain and generate a random spanning tree of a directed graph,” Journal of Algorithms vol. 27 pp. 170–217, 1998.
D. B. Wilson, “Annotated bibliography of perfectly random sampling with Markov chains,” In Microsurveys in discrete probability, Princeton, NJ, 1997, pp. 209–220, Amer. Math. Soc., Providence, RI, 1998. Latest updated version is posted at http://dbwilson.com/exact/.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Crank, K., Fill, J.A. Interruptible Exact Sampling in the Passive Case. Methodology and Computing in Applied Probability 4, 359–376 (2002). https://doi.org/10.1023/A:1023514501208
Issue Date:
DOI: https://doi.org/10.1023/A:1023514501208