THEORETICAL AND ALGORITHMIC DEVELOPMENTS IN MARKOV CHAIN MONTE CARLO DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Rajib Paul, M.Sc. ***** The Ohio State University 2008 Dissertation Committee: Approved by Professor L. Mark Berliner, Adviser Professor Noel Cressie Adviser Professor Steven N. MacEachern Graduate Program in Statistics c Copyright by Rajib Paul 2008 ABSTRACT This PhD thesis is concerned with three problems: (1) “Bayesian Change Point Analysis for Mechanistic Modeling,” (3) “Assessing Convergence and Mixing of Markov Chain Monte Carlo via Stratification,” and (2) “Bayesian Analysis via Diffusion Monte Carlo.” In Bayesian parametric change point modeling, the primary challenges are de- tection of the number and locations of change points and obtaining the posterior distribution of other unknown parameters involved in the model. Chapter 2 suggests approaches for dealing with the computational burden arising in change point mod- els that involving mechanistic reasoning as well as random explanatory variables. We present an example invovling flow velocities of ice sheets in the Lambert Glacial Basin of east Antarctica. We pay special attention to the development of reasonable priors that can reflect a variety of prior information. In Chapter 3 we apply the notion of post-stratification to develop assessment tools for MCMC analysis. These tools are based on comparison of variances of two natural estimators. Based on the estimates of these variances we propose a test statistic which helps in checking convergence and mixing of MCMC. Our method is illustrated using a Bayesian change-point model for ice flow velocity in East Antarctica, a logistic regres- sion model, a latent variable model for Arsenic concentration in public water systems in Arizona, and a regime-switching model for Pacific sea surface temperatures. ii Markov chain algorithms using langevin-type diffusion sometimes offer potentially useful methods in cases for which other MCMC methods are challenging. The key idea is to develop a stochastic process whose stationary distribution is the target posterior using diffusions represented as solutions to stochastic differential equations (SDE). The main challenge is to solve the required SDE’s. Naive discretizations like the Euler method can lead to lack of ergodicity. Hence, more sophisticated discretizations or Metropolis-Hastings correction steps are prescribed. Chapter 4 of my dissertation research deals with development of diffusion based algorithms for posterior distributions arising from non-conjugate Gaussian models with non-linear mean and variance functions. iii Dedicated to my family, friends, and teachers iv ACKNOWLEDGMENTS Words are not enough to thank those people who mean a lot to me. I express my profuse thanks and deepest appreciation to my friend, philosopher, and guide, Dr. L. Mark Berliner, for his continued support, patience, encouragement, and invaluable suggestions during my PhD program. Having him as an advisor is the best gift from the Almighty. His probability classes are cherishable memories for me. I am highly indebted to Dr. Noel Cressie for motivating and enlightening me in various ways and also for his parent-like affection. It was a great opportunity and wonderful experience for me to work with him in the Spatial Statistics and Environmental Sciences (SSES) program at The Ohio State University. I would like to convey my heartfelt thanks to Dr. Steven N. MacEachern for his very precious intellectual guidance and for being my co-author in one of my dissertation problems. I have enjoyed a lot working with him. I am very much grateful to Dr. Catherine A. Calder, Dr. Peter F. Craigmile, Dr. Thomas J. Santner, Dr. Radu Herbei, Terry England, and Emily Kang for their help during my PhD program. Finally, I would like to acknowledge my loving parents for their support and affection. v VITA 1978 ........................................Born - Calcutta, INDIA 2000 ........................................B.Sc. Statistics, Calcutta University 2002 ........................................M.Sc. Statistics, Indian Institute of Technology, Kanpur 2002-2004 ..................................Graduate Teaching Associate, Department of Statistics, The Ohio State University. 2004-present ................................Graduate Research Associate, Department of Statistics, The Ohio State University. PUBLICATIONS Research Publications Santner, Thomas J., Craigmile, Peter F., Calder, Catherine A., and Paul, Rajib, “Demographic and Behavioral Modifiers of Arsenic Exposure Pathways: A Bayesian Hierarchical Analysis of NHEXAS Data .” Journal of Environmental Science and Technology, in press FIELDS OF STUDY Major Field: Statistics vi TABLE OF CONTENTS Page Abstract....................................... ii Dedication...................................... iv Acknowledgments.................................. v Vita ......................................... vi ListofTables.................................... ix ListofFigures ................................... x Chapters: 1. MarkovChainMonteCarlo ......................... 1 2. Bayesian Change Point Analysis for Mechanistic Modeling ........ 4 2.1 Introduction .............................. 4 2.2 Bayesian Hierarchical Modeling of Ice Flow Velocities . ....... 7 2.2.1 PhysicalTheory ........................ 7 2.2.2 Data............................... 8 2.2.3 BayesianModeling ....................... 9 2.2.4 Detecting the Number of Change Points . 13 2.3 ImplementingMCMC ......................... 15 2.4 Results ................................. 17 2.5 Discussion................................ 20 2.6 Appendix: Full Conditional Distributions . ... 24 vii 3. Assessing Convergence and Mixing of MCMC via Stratification ..... 28 3.1 Introduction .............................. 28 3.2 Motivation ............................... 31 3.3 ANewMethod............................. 33 3.3.1 Choosing n, K, and J ..................... 38 3.3.2 Assessingburn-in. 38 3.3.3 Parametricbootstrapsamplesize . 41 3.4 Example 1: Bayesian Change Point Analysis: Velocity of Antarctic Glaciers................................. 44 3.5 Example 2: Latent Variable Models: Arsenic Concentrations in Pub- licWaterSystem(PWS). .. .. 45 3.6 Example 3: Logistic Regression (WinBUGS manual) . .. 48 3.7 Example 4: Regime-switching model: Pacific Sea Surface Tempera- ture(SST) ............................... 50 3.8 PoweroftheTest............................ 51 3.9 Discussion................................ 54 4. Bayesian Analysis via Diffusion Monte Carlo . .... 58 4.1 Introduction .............................. 58 4.2 Background............................... 61 4.3 UnivariateCase............................. 64 4.4 MultivariateSettings. 67 4.4.1 Multivariate Bayesian Models . 67 4.5 Discretization.............................. 69 4.6 UnivariateExamples . .. .. 70 4.6.1 Normal-CauchyModel . 70 4.6.2 Gaussian Model with Nonlinear Mean Function . 71 4.6.3 Computational Example: Nonlinear Model . 71 4.6.4 MixtureModels......................... 75 4.6.5 Conditional Autoregressive (CAR) Model . 75 4.7 MultivariateExamples . 77 4.7.1 HierarchicalModels . 77 4.7.2 Normal-GammaModel. 79 4.7.3 Conditional Autoregressive Model with Multivariate Param- eters............................... 83 4.7.4 AnExchangeableModel . 88 4.8 Discussion................................ 90 Bibliography .................................... 92 viii LIST OF TABLES Table Page 2.1 Summary of wavelet coefficients of Hτ 3 model in terms of posterior mean (Post. Mean) in (1018m(KP a)3) and posterior standard devia- tion (Post. SDev) in (1018m(KP a)3) .................. 17 2.2 DIC values for m =2,..., 5........................ 20 2.3 Posterior summaries of the slopes of the regression lines in(a−1(kP a)−3) 23 2.4 Posterior mean of the change point locations . .... 23 3.1 Beetles dataset for logistic regression . ..... 48 3.2 Acceptance rates for Geweke’s test, Gelman-Rubin test, and our new test..................................... 52 4.1 Comparing posterior summaries from diffusion MCMC and Gibbs sam- pler for normal-gamma model: E ( Y) = posterior mean based on DM ·| diffusion MCMC; SD ( Y) = posterior standard deviation based on DM ·| diffusion MCMC; E ( Y) = posterior mean based on Gibbs sampler; GS ·| SD ( Y) = posterior standard deviation based on Gibbs sampler . 80 GS ·| 4.2 Posterior summaries of the parameters involved in the CAR model (4.67).E ( Y) = posterior mean based on diffusion MCMC; SD ( Y) DM ·| DM ·| = posterior standard deviation based on diffusion MCMC; E ( Y)= GG ·| posterior mean based on griddy Gibbs sampler; SD ( Y) = posterior GG ·| standard deviation based on griddy Gibbs sampler . .. 88 4.3 Hellinger Distances for the parameters involved in the CAR model (4.67). This compares the distance between two estimated densities based on the samples generated using diffusion MCMC and griddy Gibbssampler. .............................. 88 ix LIST OF FIGURES Figure Page 2.1 IllustrationofUsingthePrior(2.15). .... 12 2.2 Observed and posterior means of the wavelet model for Hτ 3. Blue lines indicate observed values and red dots indicate several realizations from posterior distribution. (a) m = 2, (b) m = 3, (c) m = 4, and (d) m =5. 18 2.3 Observed and posterior means of the velocity model. Cyan circles indicate observed velocities and blue dots indicate posterior means. (a) m = 2, (b) m = 3, (c) m = 4, and (d) m =5. ........... 21 2.4 Residual: