BAYESIAN STATISTICAL MODELLING
CONTENTS CHAPTER INTRODUCTION: THE BAYESIAN METHOD, ITS BENE?TS AND IMPLEMENTATION THE BAYES APPROACH AND ITS POTENTIAL ADVANTAGES EXPRESSING PRIOR UNCERTAINTY ABOUT PARAMETERS AND BAYESIAN UPDATING MCMC SAMPLING AND INFERENCES FROM POSTERIOR DENSITIES THE MAIN MCMC SAMPLING ALGORITHMS GIBBS SAMPLING CONVERGENCE OF MCMC SAMPLES PREDICTIONS FROM SAMPLING: USING THE POSTERIOR PREDICTIVE DENSITY THE PRESENT BOOK REFERENCES CHAPTER BAYESIAN MODEL CHOICE, COMPARISON AND CHECKING INTRODUCTION: THE FORMAL APPROACH TO BAYES MODEL CHOICE AND AVERAGING ANALYTIC MARGINAL LIKELIHOOD APPROXIMATIONS AND THE BAYES INFORMATION CRITERION MARGINAL LIKELIHOOD APPROXIMATIONS FROM THE MCMC OUTPUT APPROXIMATING BAYES FACTORS OR MODEL PROBABILITIES JOINT SPACE SEARCH METHODS DIRECT MODEL AVERAGING BY BINARY AND CONTINUOUS SELECTION INDICATORS PREDICTIVE MODEL COMPARISON VIA CROSS-VALIDATION PREDICTIVE ?T CRITERIA AND POSTERIOR PREDICTIVE MODEL CHECKS THE DIC CRITERION POSTERIOR AND ITERATION-SPECI?C COMPARISONS OF LIKELIHOODS AND PENALISED LIKELIHOODS MONTE CARLO ESTIMATES OF MODEL PROBABILITIES THE MAJOR DENSITIES AND THEIR APPLICATION INTRODUCTION UNIVARIATE NORMAL WITH KNOWN VARIANCE TESTING HYPOTHESES ON NORMAL PARAMETERS INFERENCE ON UNIVARIATE NORMAL PARAMETERS, MEAN AND VARIANCE UNKNOWN HEAVY TAILED AND SKEW DENSITY ALTERNATIVES TO THE NORMAL CATEGORICAL DISTRIBUTIONS: BINOMIAL AND BINARY DATA SIMULATING CONTROLS THROUGH HISTORICAL EXPOSURE POISSON DISTRIBUTION FOR EVENT COUNTS THE MULTINOMIAL AND DIRICHLET DENSITIES FOR CATEGORICAL AND PROPORTIONAL DATA MULTIVARIATE CONTINUOUS DATA: MULTIVARIATE NORMAL AND T DENSITIES PARTITIONING MULTIVARIATE PRIORS THE MULTIVARIATE T DENSITY APPLICATIONS OF STANDARD DENSITIES: CLASSI?CATION RULES APPLICATIONS OF STANDARD DENSITIES: MULTIVARIATE DISCRIMINATION EXERCISES NORMAL LINEAR REGRESSION, GENERAL LINEAR MODELS AND LOG-LINEAR MODELS THE CONTEXT FOR BAYESIAN REGRESSION METHODS THE NORMAL LINEAR REGRESSION MODEL UNKNOWN REGRESSION VARIANCE NORMAL LINEAR REGRESSION: VARIABLE AND MODEL SELECTION, OUTLIER DETECTION AND ERROR FORM OTHER PREDICTOR AND MODEL SEARCH METHODS BAYESIAN RIDGE PRIORS FOR MULTICOLLINEARITY GENERAL LINEAR MODELS BINARY AND BINOMIAL REGRESSION PRIORS ON REGRESSION COEF?CIENTS MODEL CHECKS LATENT DATA SAMPLING FOR BINARY REGRESSION POISSON REGRESSION POISSON REGRESSION FOR CONTINGENCY TABLES LOG-LINEAR MODEL SELECTION MULTIVARIATE RESPONSES HIERARCHICAL PRIORS FOR POOLING STRENGTH AND OVERDISPERSED REGRESSION MODELLING HIERARCHICAL PRIORS FOR POOLING STRENGTH AND IN GENERAL LINEAR MODEL REGRESSION HIERARCHICAL PRIORS: CONJUGATE AND NON-CONJUGATE MIXING HIERARCHICAL PRIORS FOR NORMAL DATA WITH APPLICATIONS IN META-ANALYSIS PRIOR FOR SECOND-STAGE VARIANCE POOLING STRENGTH UNDER EXCHANGEABLE MODELS FOR POISSON OUTCOMES HIERARCHICAL PRIOR CHOICES PARAMETER SAMPLING COMBINING INFORMATION FOR BINOMIAL OUTCOMES RANDOM EFFECTS REGRESSION FOR OVERDISPERSED COUNT AND BINOMIAL DATA OVERDISPERSED NORMAL REGRESSION: THE SCALE-MIXTURE STUDENT T MODEL THE NORMAL META-ANALYSIS MODEL ALLOWING FOR HETEROGENEITY IN STUDY DESIGN OR PATIENT RISK HIERARCHICAL PRIORS FOR MULTINOMIAL DATA HISTOGRAM SMOOTHING DISCRETE MIXTURE PRIORS INTRODUCTION: THE RELEVANCE AND APPLICABILITY OF DISCRETE MIXTURES DISCRETE MIXTURES OF PARAMETRIC DENSITIES MODEL CHOICE IDENTI?ABILITY CONSTRAINTS HURDLE AND ZERO-IN?ATED MODELS FOR DISCRETE DATA REGRESSION MIXTURES FOR HETEROGENEOUS SUBPOPULATIONS DISCRETE MIXTURES COMBINED WITH PARAMETRIC RANDOM EFFECTS NON-PARAMETRIC MIXTURE MODELLING VIA DIRICHLET PROCESS PRIORS OTHER NON-PARAMETRIC PRIORS MULTINOMIAL AND ORDINAL REGRESSION MODELS INTRODUCTION: APPLICATIONS WITH CATEGORIC AND ORDINAL DATA MULTINOMIAL LOGIT CHOICE MODELS THE MULTINOMIAL PROBIT REPRESENTATION OF INTERDEPENDENT CHOICES MIXED MULTINOMIAL LOGIT MODELS INDIVIDUAL LEVEL ORDINAL REGRESSION SCORES FOR ORDERED FACTORS IN CONTINGENCY TABLES TIME SERIES MODELS INTRODUCTION: ALTERNATIVE APPROACHES TO TIME SERIES MODELS AUTOREGRESSIVE MODELS IN THE OBSERVATIONS PRIORS ON AUTOREGRESSIVE COEF?CIENTS INITIAL CONDITIONS AS LATENT DATA TREND STATIONARITY IN THE AR1 MODEL AUTOREGRESSIVE MOVING AVERAGE MODELS AUTOREGRESSIVE ERRORS MULTIVARIATE SERIES TIME SERIES MODELS FOR DISCRETE OUTCOMES OBSERVATION-DRIVEN AUTODEPENDENCE INAR MODELS ERROR AUTOCORRELATION DYNAMIC LINEAR MODELS AND TIME VARYING COEF?CIENTS SOME COMMON FORMS OF DLM PRIORS FOR TIME-SPECI?C VARIANCES OR INTERVENTIONS NONLINEAR AND NON-GAUSSIAN STATE-SPACE MODELS MODELS FOR VARIANCE EVOLUTION ARCH AND GARCH MODELS STOCHASTIC VOLATILITY MODELS MODELLING STRUCTURAL SHIFTS AND OUTLIERS MARKOV MIXTURES AND TRANSITION FUNCTIONS OTHER NONLINEAR MODELS MODELLING SPATIAL DEPENDENCIES INTRODUCTION: IMPLICATIONS OF SPATIAL DEPENDENCE DISCRETE SPACE REGRESSIONS FOR METRIC DATA DISCRETE SPATIAL REGRESSION WITH STRUCTURED AND UNSTRUCTURED RANDOM EFFECTS PROPER CAR PRIORS MOVING AVERAGE PRIORS MULTIVARIATE SPATIAL PRIORS AND SPATIALLY VARYING REGRESSION EFFECTS ROBUST MODELS FOR DISCONTINUITIES AND NON-STANDARD ERRORS CONTINUOUS SPACE MODELLING IN REGRESSION AND INTERPOLATION NONLINEAR AND NONPARAMETRIC REGRESSION APPROACHES TO MODELLING NONLINEARITY NONLINEAR METRIC DATA MODELS WITH KNOWN FUNCTIONAL FORM BOX–COX TRANSFORMATIONS AND FRACTIONAL POLYNOMIALS NONLINEAR REGRESSION THROUGH SPLINE AND RADIAL BASIS FUNCTIONS SHRINKAGE MODELS FOR SPLINE COEF?CIENTS MODELLING INTERACTION EFFECTS APPLICATION OF STATE-SPACE PRIORS IN GENERAL ADDITIVE NONPARAMETRIC REGRESSION CONTINUOUS PREDICTOR SPACE PRIOR DISCRETE PREDICTOR SPACE PRIORS MULTILEVEL AND PANEL DATA MODELS INTRODUCTION: NESTED DATA STRUCTURES MULTILEVEL STRUCTURES THE MULTILEVEL NORMAL LINEAR MODEL GENERAL LINEAR MIXED MODELS FOR DISCRETE OUTCOMES MULTINOMIAL AND ORDINAL MULTILEVEL MODELS ROBUSTNESS REGARDING CLUSTER EFFECTS CONJUGATE APPROACHES FOR DISCRETE DATA HETEROSCEDASTICITY IN MULTILEVEL MODELS RANDOM EFFECTS FOR CROSSED FACTORS PANEL DATA MODELS: THE NORMAL MIXED MODEL AND EXTENSIONS AUTOCORRELATED ERRORS AUTOREGRESSION IN Y MODELS FOR PANEL DISCRETE (BINARY, COUNT AND CATEGORICAL) OBSERVATIONS BINARY PANEL DATA REPEATED COUNTS PANEL CATEGORICAL DATA GROWTH CURVE MODELS DYNAMIC MODELS FOR LONGITUDINAL DATA: POOLING STRENGTH OVER UNITS AND TIMES AREA APC AND SPATIOTEMPORAL MODELS AGE–PERIOD DATA AREA–TIME DATA AGE–AREA–PERIOD DATA INTERACTION PRIORS LATENT VARIABLE AND STRUCTURAL EQUATION MODELS FOR MULTIVARIATE DATA INTRODUCTION: LATENT TRAITS AND LATENT CLASSES FACTOR ANALYSIS AND SEMS FOR CONTINUOUS DATA IDENTI?ABILITY CONSTRAINTS IN LATENT TRAIT (FACTOR ANALYSIS) MODELS LATENT CLASS MODELS LOCAL DEPENDENCE FACTOR ANALYSIS AND SEMS FOR MULTIVARIATE DISCRETE DATA NONLINEAR FACTOR MODELS SURVIVAL AND EVENT HISTORY ANALYSIS PARAMETRIC SURVIVAL ANALYSIS IN CONTINUOUS TIME CENSORED OBSERVATIONS FORMS OF PARAMETRIC HAZARD AND SURVIVAL CURVES MODELLING COVARIATE IMPACTS AND TIME DEPENDENCE IN THE HAZARD RATE ACCELERATED HAZARD PARAMETRIC MODELS COUNTING PROCESS MODELS SEMIPARAMETRIC HAZARD MODELS PRIORS FOR THE BASELINE HAZARD GAMMA PROCESS PRIOR ON CUMULATIVE HAZARD COMPETING RISK-CONTINUOUS TIME MODELS VARIATIONS IN PRONENESS: MODELS FOR FRAILTY DISCRETE TIME SURVIVAL MODELS MISSING DATA MODELS INTRODUCTION: TYPES OF MISSINGNESS SELECTION AND PATTERN MIXTURE MODELS FOR THE JOINT DATA-MISSINGNESS DENSITY SHARED RANDOM EFFECT AND COMMON FACTOR MODELS MISSING PREDICTOR DATA MULTIPLE IMPUTATION CATEGORICAL RESPONSE DATA WITH POSSIBLE NON-RANDOM MISSINGNESS: HIERARCHICAL AND REGRESSION MODELS HIERARCHICAL MODELS FOR RESPONSE AND NON-RESPONSE BY STRATA REGRESSION FRAMEWORKS MISSINGNESS WITH MIXTURES OF CONTINUOUS AND CATEGORICAL DATA MISSING CELLS IN CONTINGENCY TABLES ECOLOGICAL INFERENCE MEASUREMENT ERROR, SEEMINGLY UNRELATED REGRESSIONS, AND SIMULTANEOUS EQUATIONS MEASUREMENT ERROR IN BOTH PREDICTORS AND RESPONSE IN NORMAL LINEAR REGRESSION PRIOR INFORMATION ON X OR ITS DENSITY MEASUREMENT ERROR IN GENERAL LINEAR MODELS MISCLASSI?CATION OF CATEGORICAL VARIABLES SIMULTANEOUS EQUATIONS AND INSTRUMENTS FOR ENDOGENOUS VARIABLES ENDOGENOUS REGRESSION INVOLVING DISCRETE VARIABLES