Perfect Simulation Discussion

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Perfect Simulation Discussion

• David B. Wilson (?????)
• Microsoft
• 53rd ISI meeting, Seoul, Korea

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Perfect Simulation Discussion

• David B. Wilson (??? ??)
• Microsoft
• 53rd ISI meeting, Seoul, Korea

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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)
• Fills 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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Statistical Mechanics vs Statistics

• 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)

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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-Ro
  bert-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 qs give

• percolation
• Ising ferromagnet
• Potts model

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Random Planar Maps

• Different embeddings of graph -gt different maps
• Enumerated by Tutte
• Linear time random generation by Schaeffer

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FK model on random planar maps

• Annealed
• Pick planar map G and subgraph s together

Quenched First pick planar map G Then pick
subgraph s
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Experimental values of quenched exponents
1/(?d) or 1/(2-a) 1/(?d) or 1/(2-a) 1/(?d) or 1/(2-a) ß/(?d) or ß/(2-a) ß/(?d) or ß/(2-a) ß/(?d) or ß/(2-a)
q2 q4 q10 q2 q4 q10
conjecture .3486 .5886 none .1452 .1452 none
Janke-Johnston .34 .42 .58 .10 .11 .12
Schaeffer-W (preliminary) .34 .38 .43 .135 .15 .17
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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 q10 measurements
  (and also the q4 quenched theory predictions)
  violate a supposedly general bound derived by
  Chayes et al. 23 for quenched systems, ?Dgt2,
  since ?D1.72 from the q10 measurements.
• from Janke-Johnston

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• Quenched exponent work still preliminary
• Many headaches associated with extracting
  exponents
• Many realizations of disorder, many
  burn-ins
• Torpid mixing / burn-in is one headache we dont
  have

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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
• perfect_at_list.research.microsoft.com