Perfect Simulation Discussion

1. Perfect Simulation Discussion David B. Wilson (다비드윌슨) Microsoft 53rd ISI meeting, Seoul, Korea

2. Perfect Simulation Discussion David B. Wilson (다비드 윌슨) Microsoft 53rd ISI meeting, Seoul, Korea

3. How long to run the Markov chain? Convergence diagnostics Workhorse of MCMC Never sure of equilibration Mathematical analysis Sure of equilibration Have to be smart to get good bounds Perfect simulation Sure of equilibration Computer determines on its own how long to run Relies on special structure (Sometimes Markov chain not used)

4. Perfect Simulation Methods (partial list) • Asmussen-Glynn-Thorisson ’92 • Aldous ’95 • Lovász-Winkler ’95 • Coupling from the past (CFTP) Propp-Wilson ’96 (related ideas in Letac ’86, Broder ’89, Aldous ’90, Johnson ’96) • Fill’s algorithm (FMMR) Fill ’98, Fill-Machida-Murdoch-Rosenthal ’00 • Cycle-popping, sink-popping Wilson ’96, Propp-Wilson ’98, Cohn-Propp-Pemantle ’01 • Dominated CFTP Kendall ’98, Kendall-Møller ’99 • Read-once CFTP Wilson ’00 • Clan of ancestors Fernández-Ferrari-Garcia ’00 • Randomness recycler (RR) Fill-Huber ’00

5. Many variables, homogenous and simple interactions Fewer models that get studied intensively (universality) ad hoc methods Focus on special points (phase transitions) where mixing is slow More complicated interactions More different types of models General methods to mechanize study of new models (e.g. BUGS) Focus on generic points (real world data) Statistical Mechanics vs Statistics

6. Perfect Simulation → Mathematics • Cycle popping algorithm used by Benjamini, Lyons, Peres, & Schramm to study uniform spanning forests on Z and other graphs • CFTP used by Van den Berg & Steif to show Ising model on Z² above critical point has finitary codings • CFTP used by Häggström, Jonasson, & Lyons to show that the Potts model on amenable graphs at any temperature exhibits Bernoullicity d

7. Coupling methods (partial list) • Monotone coupling performance guarantee, efficient if the Markov chain is • Antimonotone coupling Kendall ’98, Häggström-Nelander ’98 • Coupling for Markov random fields Häggström-Nelander ’99, Huber ’98 • Coupling for Bayesian inference Murdoch-Green ’98, Green-Murdoch ’99 • Slice sampling (auxillary variables) Mira-Møller-Roberts ’01, Casella-Mengersen-Robert-Titterington ’0x • Simulated tempering (enlarges state space) (in context of perfect simulation) Møller-Nicholls ’0x

8. Random Tiling by Lozenges • Perfect matchings on hexagonal lattice • Diatomic molecules on surface • Product formulas, circular boundary • Monotone Markov chain

9. Coupling from the past (CFTP) • Run Markov chain for very long (infinitely long) time • Final state is random • Figure out final state

10. Square-Ice model (physics) • Boundary between blue & white regions visit every site once • Monotone Markov chain (monotonicity not always apparent)

11. Autonormal model (statistics)Gaussian free field (physics) • Random height at each vertex, Guassian distribution conditional on neighboring heights • Agricultural experiments • Monotone Markov chain • No top or bottom state

12. Ising model • Spins on vertices • Neighboring spins prefer to be aligned • Models magnetism, certain forms of brass • Two different monotone Markov chains (spin & FK representations)

13. Random independent set (CS)Hard-core model (physics) Set of vertices on graph, no two adjacent Monotone on bipartite graphs Even & odd sites shown in different colors

14. Potts model • Generalizes Ising model to multiple spins • Studied extensively in physics • Image restoration • Monotone Markov chain (FK representation)

15. Uniformly Random Spanning Tree • Connected acyclic subgraph • Generated via cycle-popping • Also CFTP algorithm • No monotonicity

16. Example from stochastic geometry • Impenetrable spheres model • Antimonotone coupling (Kendall, Häggström-Nelander) • No top state

17. Fortuin-Kasteleyn (FK) model(random cluster model) 13 edges 11 missing edges 5 connected components Different q’s give • percolation • Ising ferromagnet • Potts model

18. Different embeddings of graph -> different maps Enumerated by Tutte Linear time random generation by Schaeffer Random Planar Maps

19. FK model on random planar maps Annealed Pick planar map G and subgraph σ together Quenched First pick planar map G Then pick subgraph σ

20. Experimental values of quenched exponents

21. Torpid mixing of Swendsen-Wang for large q • Complete graph q≥ 3 Gore-Jerrum • Grid graph q≥ big Borgs-Chayes-Frieze-Kim-Tetali-Vigoda-Vu ≈98% cancelation

22. “It is also noteworthy that the q=10 measurements (and also the q=4 quenched theory predictions) violate a supposedly general bound derived by Chayes et al. [23] for quenched systems, νD>2, since νD~1.72 from the q=10 measurements.” from Janke-Johnston

23. Quenched exponent work still preliminary • Many headaches associated with extracting exponents • Many realizations of disorder, many burn-in’s • Torpid mixing / burn-in is one headache we don’t have

24. “Chance favors the prepared mind.”-Pasteur • Most Markov chains do not have nice special properties useful for perfect simulation • Special Markov chains more interesting than “typical” Markov chains • Look for monotonicity or other features that can be used for perfect simulation, sometimes one gets lucky

25. Further Information • http://dimacs.rutgers.edu/~dbwilson/exact • http://front.math.ucdavis.edu/math.PR • perfect@list.research.microsoft.com