PERFECT SAMPLING USING BOUNDING CHAINS A Dissertation Presented to the Faculty of the Graduate Scho ol of Cornell University in Partial Fulllmentofthe Requirements for the Degree of Do ctor of Philosophy by Mark Lawrence Hub er May c Mark Lawrence Hub er ALL RIGHTS RESERVED PERFECT SAMPLING USING BOUNDING CHAINS Mark Lawrence Hub er PhD Cornell University In Monte Carlo simulation samples are drawn from a distribution to estimate prop erties of the distribution that are to o dicult to compute analytically This has applications in numerous elds including optimization statistics statistical mechanics genetics and the design of approximation algorithms In the Monte Carlo Markovchain metho d a Markovchain is constructed which has the target distribution as its stationary distribution After running the Markov chain long enough the distribution of the nal state will b e close to the stationary distribution of the chain Unfortunately for most Markov chains the time needed to converge to the stationary distribution the mixing time is completely unknown Here we develop several new techniques for dealing with unknown mixing times First weintro duce the idea of a b ounding chain whichdelivers a wealth of informa tion ab out the chain Once a b ounding chain is created for a particular chain it is p ossible to empirically estimate the mixing time of the chain Using ideas such as coupling from the past and the FillMurdo chRosenthal algorithm b ounding chains can also b ecome the basis of p erfect sampling algorithms Unlike traditional Monte Carlo Markov chain metho ds these algorithms draw samples which are exactly distributed according to the stationary distribution We develop b ounding chains for several Markov chains of practical interest chains from statistical mechanics like the SwendsenWang chain for the Ising mo del the DyerGreenhill chain for the discrete hard core gas mo del and the continuous WidomRowlinson mixture mo del with more than three comp onents in the mix ture We also givetechniques for sampling from weighted p ermutations whichhave applications in database access and nonparametric statistical tests In addition chains for a variety of Markov chains of theoretical in we present here b ounding terest such as the k coloring chain the sink free orientation chain and the anti ferromagnetic Potts mo del with more than three colors Finally we develop new Markov chains and b ounding chains for the continuous hard core gas mo del and the WidomRowlinson mo del which are provably faster in practice Biographical Sketch Mark Lawrence Hub er was b orn in in the quaint town of Austin Minnesota home of the Hormel corp oration Sensing the eventual rise to power of Jesse The Bo dy Ventura Marks family moved out of the state living in Yukon OK Still water OK Ames IA Corvallis OR and Hays KA b efore setting in La Grande OR where he graduated high scho ol in Mark then moved south to sunny Claremont California and Harvey Mudd College where he completed a Bachelor of e land Mark headed Science in Mathematics in Missing the snow of his nativ to Cornell and the Scho ol of Op erations Research and Industrial Engineering Up on completion of his PhD Mark will sp end the next two years at Stanford Univer sity as a National Science Foundation Postdo ctoral Fellow in the Mathematical Sciences iii To my grandmother whose love has always been an inspiration iv Acknowledgements FirstofallI would like to thank my advisor David Shmoys who has lasted stoically through my research ups and downs my eccentric grammar and my complete lack of organizational skills He has b een a go o d friend and a great advisor and I hop e to work with him again in the future Other professors have also had a great impact on my Cornell exp erience The remaining members of my thesis committee Sid Resnick Jon Kleinb erg and Rick Durrett were always willing to lend a hand Id also like to thank Persi Diaconis who rst intro duced me to the b eauty and elegance of rapidly mixing Markovchain hadnt known existed theory aworld of problems and techniques that I This thesis would not be here without the tireless eorts of Nathan Edwards A whose computer exp ertise L T X prowess and willingness to listen to my litanyof E problems is second to none Financially this work was supp orted through numerous sources An Oce of Naval Research Fellowship carried me through the early years with ONR grants N NSF grants CCR CCR DMS and ASSERT grant N nishing the job v I would also like to thank my ro ommates during my stay at Cornell Bas de Blank and Chris Papadop oulos b oth of whom had to suer with my p olicies or lack thereof regarding the prop er storage of all manner of items The friends I have made here are to o many to list however I would like to sp ecically thank my graduate class Greta Pangb orn Nathan Edwards Ed Chan Paulo Zanjacomo Semyon Kruglyak and Fabian Chudak A guy couldnt ask for a better group of p eople to commiserate with Thanks go out to Stephen Gulyas who always had a sp ot on his intermural softball teams for me and who was always willing to try to turn my tennis swing back into a baseball swing come springtime Finally I would like to thank all of my family from the Midwest to the Pacic Northwest whose supp ort has always b een unconditional Some great events have happ ened while Ive b een at Cornell and Im glad to have b een able to b e there for them vi Table of Contents The Need for Markov chains Monte Carlo Markov chain metho ds Markov chains Going the distance The Discrete Mo dels The Ising mo del The antiferromagnetic Ising mo del and MAX CUT The Potts Mo del The hard core gas mo del The WidomRowlinson mixture mo del Q colorings of a graph Sink Free Orientations of a graph Hyp ercub e slices Applying the Monte Carlo metho d Building Markov chains Conditioning chains The Heat Bath chain Metrop olisHastings The acceptance rejection heat bath chain The Swap Move The Ising and Potts mo dels Antiferromagnetic Potts mo del at zero temp erature SwendsenWang Sink Free Orientations WidomRowlinson The Antivoter mo del The List Up date Problem Hyp ercub e slices vii What remains mixing time Bounding Chains Monotonicity Antimonotonicity hain approach The b ounding c Bounding the DyerGreenhill Hard Core chain Martingales Running time of b ounding chain for DyerGreenhill Bounding chains for other mo dels Q coloring chain The Potts mo del SwendsenWang Sink free orientations Hyp ercub e slices WidomRowlinson Nonlo cal conditioning chain The single site heat bath chain The birth death swapping chain The antivoter mo del The list up date problem Application to nonparametric testing Other applications of b ounding chains Perfect sampling using coupling from the past Reversing the chain Coupling from the past CFTP and b ounding chains Coupling from the future Perfect sampling using strong stationary times Upp er b ounds on the strong stationary stopping time Application to lo cal up date chains The Hard Core Gas Mo del Single site WidomRowlinson Application to nonlo cal chains viii Continuous Mo dels Continous state space Markov chains Continuous time Markov chains The Continuous Hard Core Gas Mo del Continuous b ounding chains More on sup ermartingales The Swapping Continuous Hard Core chain WidomRowlinson Continuous swapping chain Final thoughts Bibliography ix List of Tables SwendsenWang approach comparison x List of Figures General Monte Carlo Markov chain metho d The general heat bath Markov chain Single site hard core heat bath chain Nonlo cal hard core heat bath chain The general Metrop olisHastings Markov chain Single site hard core Metrop olisHastings chain The general acceptance rejection heat bath Markov chain Acceptance rejection single site hard core heat bath chain Dyer and Greenhill hard core chain step Single site Potts heat bath chain Single site Q coloring heat bath chain SwendsenWang chain Single edge heat bath sink