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Perfect Simulation Discussion. David B. Wilson ( 다비드윌슨 ) Microsoft 53 rd ISI meeting, Seoul, Korea. Perfect Simulation Discussion. David B. Wilson ( 다비드 윌슨 ) Microsoft 53 rd ISI meeting, Seoul, Korea. How long to run the Markov chain?. Convergence diagnostics

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Perfect simulation discussion l.jpg

Perfect Simulation Discussion

David B. Wilson (다비드윌슨)

Microsoft

53rd ISI meeting, Seoul, Korea


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Perfect Simulation Discussion

David B. Wilson (다비드 윌슨)

Microsoft

53rd ISI meeting, Seoul, Korea


How long to run the markov chain l.jpg
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)


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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


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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


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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


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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


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Random Tiling by Lozenges

  • Perfect matchings on hexagonal lattice

  • Diatomic molecules on surface

  • Product formulas, circular boundary

  • Monotone Markov chain


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Coupling from the past (CFTP)

  • Run Markov chain for very long (infinitely long) time

  • Final state is random

  • Figure out final state


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Square-Ice model (physics)

  • Boundary between blue & white regions visit every site once

  • Monotone Markov chain (monotonicity not always apparent)


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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


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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)


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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


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Potts model

  • Generalizes Ising model to multiple spins

  • Studied extensively in physics

  • Image restoration

  • Monotone Markov chain (FK representation)


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Uniformly Random Spanning Tree

  • Connected acyclic subgraph

  • Generated via cycle-popping

  • Also CFTP algorithm

  • No monotonicity


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Example from stochastic geometry

  • Impenetrable spheres model

  • Antimonotone coupling (Kendall, Häggström-Nelander)

  • No top state


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Fortuin-Kasteleyn (FK) model(random cluster model)

13 edges

11 missing edges

5 connected components

Different q’s give

  • percolation

  • Ising ferromagnet

  • Potts model


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Different embeddings of graph -> different maps

Enumerated by Tutte

Linear time random generation by Schaeffer

Random Planar Maps


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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



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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


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“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


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“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


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Further Information

  • http://dimacs.rutgers.edu/~dbwilson/exact

  • http://front.math.ucdavis.edu/math.PR

  • [email protected]


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