APPLIED BAYESIAN MODELLING

By W. BEVIK

CHAPTER 1 THE BASIS FOR, AND ADVANTAGES OF, BAYESIAN MODEL ESTIMATION VIA REPEATED SAMPLING 11 INTRODUCTION 5 12 GIBBS SAMPLING 12 13 SIMULATING RANDOM VARIABLES FROM STANDARD DENSITIES 18 14 MONITORING MCMC CHAINS AND ASSESSING CONVERGENCE 20 15 MODEL ASSESSMENT AND SENSITIVITY 27 16 REVIEW REFERENCES 28 CHAPTER 2 HIERARCHICAL MIXTURE MODELS 31 21 INTRODUCTION: SMOOTHING TO THE POPULATION 31 22 GENERAL ISSUES OF MODEL ASSESSMENT: MARGINAL LIKELIHOOD AND OTHER APPROACHES 32 221 BAYES MODEL SELECTION USING MARGINAL LIKELIHOODS 33 222 OBTAINING MARGINAL LIKELIHOODS IN PRACTICE 35 223 APPROXIMATING THE POSTERIOR 37 224 PREDICTIVE CRITERIA FOR MODEL CHECKING AND SELECTION 39 225 REPLICATE SAMPLING 40 23 ENSEMBLE ESTIMATES: POOLING OVER SIMILAR UNITS 41 231 MIXTURES FOR POISSON AND BINOMIAL DATA 43 232 SMOOTHING METHODS FOR CONTINUOUS DATA 51 24 DISCRETE MIXTURES AND DIRICHLET PROCESSES 58 241 DISCRETE PARAMETRIC MIXTURES 58 242 DPP PRIORS 60 25 GENERAL ADDITIVE AND HISTOGRAM SMOOTHING PRIORS 67 251 SMOOTHNESS PRIORS 68 252 HISTOGRAM SMOOTHING 69 26 REVIEW 74 REFERENCES 75 EXERCISES 1 1 78 CHAPTER 3 REGRESSION MODELS 79 31 INTRODUCTION: BAYESIAN REGRESSION 79 311 SPECIFYING PRIORS: CONSTRAINTS ON PARAMETERS 80 312 PRIOR SPECIFICATION: ADOPTING ROBUST OR INFORMATIVE PRIORS 81 313 REGRESSION MODELS FOR OVERDISPERSED DISCRETE OUTCOMES 82 32 CHOICE BETWEEN REGRESSION MODELS AND SETS OF PREDICTORS IN REGRESSION 84 321 PREDICTOR SELECTION 85 322 CROSS-VALIDATION REGRESSION MODEL ASSESSMENT 86 33 POLYTOMOUS AND ORDINAL REGRESSION 98 331 MULTINOMIAL LOGISTIC CHOICE MODELS 99 332 NESTED LOGIT SPECIFICATION 100 101 333 ORDINAL OUTCOMES 102 334 LINK FUNCTIONS 110 34 REGRESSIONS WITH LATENT MIXTURES 35 GENERAL ADDITIVE MODELS FOR NONLINEAR REGRESSION EFFECTS 115 36 ROBUST REGRESSION METHODS 118 361 BINARY SELECTION MODELS FOR ROBUSTNESS 119 362 DIAGNOSTICS FOR DISCORDANT OBSERVATIONS 120 37 REVIEW 126 129 REFERENCES EXERCISES 132 CHAPTER 4 ANALYSIS OF MULTI-LEVEL DATA 135 41 INTRODUCTION 135 42 MULTI-LEVEL MODELS: UNIVARIATE CONTINUOUS AND DISCRETE OUTCOMES 137 421 DISCRETE OUTCOMES 139 43 MODELLING HETEROSCEDASTICITY 145 44 ROBUSTNESS IN MULTI-LEVEL MODELLING 151 45 MULTI-LEVEL DATA ON MULTIVARIATE INDICES 156 46 SMALL DOMAIN ESTIMATION 163 47 REVIEW 167 168 REFERENCES 169 EXERCISES CHAPTER 5 MODELS FOR TIME SERIES 51 INTRODUCTION 171 52 AUTOREGRESSIVE AND MOVING AVERAGE MODELS UNDER STATIONARITY AND NON-STATIONARITY 171 521 SPECIFYING PRIORS 172 174 522 FURTHER TYPES OF TIME DEPENDENCE 179 523 FORMAL TESTS OF STATIONARITY IN THE AR(1) MODEL 180 524 MODEL ASSESSMENT 182 53 DISCRETE OUTCOMES 191 531 AUTO REGRESSION ON TRANSFORMED OUTCOME 193 532 INAR MODELS FOR COUNTS 193 533 CONTINUITY PARAMETER MODELS 195 534 MULTIPLE DISCRETE OUTCOMES 195 54 ERROR CORRECTION MODELS 200 55 DYNAMIC LINEAR MODELS AND TIME VARYING COEFFICIENTS 203 205 551 STATE SPACE SMOOTHING 210 56 STOCHASTIC VARIANCES AND STOCHASTIC VOLATILITY 210 561 ARCH AND GARCH MODELS 211 562 STOCHASTIC VOLATILITY MODELS 215 57 MODELLING STRUCTURAL SHIFTS 571 BINARY INDICATORS FOR MEAN AND VARIANCE SHIFTS 215 572 MARKOV MIXTURES 216 573 SWITCHING REGRESSIONS 216 58 REVIEW 221 REFERENCES 222 225 CHAPTER 6 ANALYSIS OF PANEL DATA EXERCISES 227 61 INTRODUCTION 227 611 TWO STAGE MODELS 228 612 FIXED VS RANDOM EFFECTS 230 613 TIME DEPENDENT EFFECTS 231 62 NORMAL LINEAR PANEL MODELS AND GROWTH CURVES FOR METRIC OUTCOMES 231 621 GROWTH CURVE VARIABILITY 232 622 THE LINEAR MIXED MODEL 234 623 VARIABLE AUTOREGRESSIVE PARAMETERS 235 63 LONGITUDINAL DISCRETE DATA: BINARY, ORDINAL AND MULTINOMIAL AND POISSON PANEL DATA 243 631 BETA-BINOMIAL MIXTURE FOR PANEL DATA 244 64 PANELS FOR FORECASTING 257 641 DEMOGRAPHIC DATA BY AGE AND TIME PERIOD 261 65 MISSING DATA IN LONGITUDINAL STUDIES 264 66 REVIEW 268 269 REFERENCES 271 EXERCISES CHAPTER 7 MODELS FOR SPATIAL OUTCOMES AND GEOGRAPHICAL ASSOCIATION 273 71 INTRODUCTION 273 72 SPATIAL REGRESSIONS FOR CONTINUOUS DATA WITH FIXED INTERACTION SCHEMES 275 721 JOINT VS CONDITIONAL PRIORS 276 73 SPATIAL EFFECTS FOR DISCRETE OUTCOMES: ECOLOGICAL ANALYSIS INVOLVING COUNT DATA 278 731 ALTERNATIVE SPATIAL PRIORS IN DISEASE MODELS 279 732 MODELS RECOGNISING DISCONTINUITIES 281 733 BINARY OUTCOMES 282