Perfect Simulation Discussion

David B. Wilson ( 다비드윌슨 ) Microsoft

53rd ISI meeting, Seoul, Korea

Perfect Simulation Discussion

David B. Wilson ( 다비드 윌슨 ) Microsoft

53rd ISI meeting, Seoul, Korea

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)

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

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)

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

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

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

Square-Ice model (physics)

Boundary between blue & white regions visit every site once

Monotone Markov chain (monotonicity not always apparent)

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

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)

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

Potts model

Generalizes Ising model to multiple spins

Studied extensively in physics

Image restoration Monotone Markov chain

(FK representation)

Uniformly Random Spanning Tree

Connected acyclic subgraph

Generated via cycle-popping

Also CFTP algorithm No monotonicity

Example from stochastic geometry

Impenetrable spheres model

Antimonotone coupling (Kendall, Häggström-Nelander)

No top state

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

Random Planar Maps

Different embeddings of graph -> different maps

Enumerated by Tutte Linear time random

generation by Schaeffer

4

vertices

diameter

n

n

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

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

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

“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

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

“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

Further Information

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

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

perfect@list.research.microsoft.com