perfect simulation discussion david b. wilson ( ) microsoft 53 rd isi meeting, seoul, korea
TRANSCRIPT
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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
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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) 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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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-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
Title:
Creator:random and prettyPreview:This EPS picture was not savedwith a preview included in it.Comment:This EPS picture will print to aPostScript printer, but not toother types of printers.
Title:Matchings, Tilings, and PathsCreator:vaxmacs & dbwilsonPreview:This EPS picture was not savedwith a preview included in it.Comment:This EPS picture will print to aPostScript printer, but not toother types of printers.
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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 11 5weight( ) (1 )p p q
13 edges
11 missing edges
5 connected components
weight( )G
Z
weight( )Pr( )
Z
Different q’s give • percolation• Ising ferromagnet• Potts model
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Random Planar Maps
Different embeddings of graph -> different maps
Enumerated by Tutte Linear time random
generation by Schaeffer
4
vertices
diameter
n
n
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FK model on random planar maps
AnnealedPick planar map G and
subgraph σ together
maps
weight( )Pr[ , ]
weight( )H H
G
QuenchedFirst pick planar map G
Then pick subgraph σ1 weight( )
Pr[ , ]#maps weight( )
G
G
KPZannealed exponents exponents on square lattice
quenched exponents exponents for ``dirty'' systems
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Experimental values of quenched exponents
1/(νd) or 1/(2-α) β/(νd) or β/(2-α)
q=2 q=4 q=10 q=2 q=4 q=10
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
0 0
1 1[ #edges] [ ] [#edges]
1
dE fE f E f E
dp p p
≈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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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
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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