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贝叶斯统计建模 Bayesian_Statistical_Modelling.pdf
Bayesian Statistical Modelling
Contents
Preface
Chapter 1 Introduction: The Bayesian Method, its Benefits and Implementation
1.1 The Bayes Approach and its Potential Advantages
1.2 Expressing Prior Uncertainty About Parameters and Bayesian Updating
1.3 MCMC Sampling and Inferences from Posterior Densities
1.4 The Main MCMC Sampling Algorithms
1.4.1 Gibbs Sampling
1.5 Convergence of MCMC Samples
1.6 Predictions from Sampling: Using the Posterior Predictive Density
1.7 The Present Book
References
Chapter 2 Bayesian Model Choice, Comparison and Checking
2.1 Introduction: The Formal Approach to Bayes Model Choice and Averaging
2.2 Analytic Marginal Likelihood Approximations and the Bayes Information Criterion
2.3 Marginal Likelihood Approximations from the MCMC Output
2.4 Approximating Bayes Factors or Model Probabilities
2.5 Joint Space Search Methods
2.6 Direct Model Averaging by Binary and Continuous Selection Indicators
2.7 Predictive Model Comparison via Cross-Validation
2.8 Predictive Fit Criteria and Posterior Predictive Model Checks
2.9 The DIC Criterion
2.10 Posterior and Iteration-Specific Comparisons of Likelihoods and Penalised Likelihoods
2.11 Monte Carlo Estimates of Model Probabilities
References
Chapter 3 The Major Densities and their Application
3.1 Introduction
3.2 Univariate Normal with Known Variance
3.2.1 Testing Hypotheses on Normal Parameters
3.3 Inference on Univariate Normal Parameters, Mean and Variance Unknown
3.4 Heavy Tailed and Skew Density Alternatives to the Normal
3.5 Categorical Distributions: Binomial and Binary Data
3.5.1 Simulating Controls Through Historical Exposure
3.6 Poisson Distribution for Event Counts
3.7 The Multinomial and Cirichlet Densities for Categorical and Proportional Data
3.8 Multivariate Continuous Data: Multivariate Normal and t Densities
3.8.1 Partitioning Multivariate Priors
3.8.2 The Multivariate t Density
3.9 Applications of Standard Densities: Classification Rules
3.10 Applications of Standard Densities: Multivariate Discrimination
Exercises
References
Chapter 4 Normal Linear Regression, General Linear Models and Log-Linear Models
4.1 The Context for Bayesian Regression Methods
4.2 The Normal Linear Regression Model
4.2.1 Unknown Regression Variance
4.3 Normal Linear Regression: Variable and Model Selection, Outlier Detection and Error form
4.3.1 Other Predictor and Model Search Methods
4.4 Bayesian Ridge Priors for Multicollinearity
4.5 General Linear Models
4.6 Binary and Binomial Regression
4.6.1 Priors on Regression Coefficients
4.6.2 Model Checks
4.7 Latent Data Sampling for Binary Regression
4.8 Poisson Regression
4.8.1 Poisson Regression for Contingency Tables
4.8.2 Log-Linear Model Selection
4.9 Multivariate Responses
Exercises
References
Chapter 5 Hierarchical Priors for Pooling Strength and Overdispersed Regression Modelling
5.1 Hierarchical Priors for Pooling Strength and in General Linear Model Regression
5.2 Hierarchical Priors: Conjugate and Non-Conjugate Mixing
5.3 Hierarchical Priors for Normal Data with Applications in Meta-Analysis
5.3.1 Prior for Second-Stage Variance
5.4 Pooling Strength under Exchangeable Models for Poisson Outcomes
5.4.1 Hierarchical Prior Choices
5.4.2 Parameter Sampling
5.5 Combining Information for Binomial Outcomes
5.6 Random Effects Regression for Overdispersed Count and Binomial Data
5.7 Overdispersed Normal Regression: The Scale-Mixture Student t Model
5.8 The Normal Meta-Analysis Model Allowing for Heterogeneity in Study Design or Patient Risk
5.9 Hierarchical Priors for Multinomial Data
5.9.1 Histogram Smoothing
Exercises
References
Chapter 6 Discrete Mixture Priors
6.1 Introduction: The Relevance and Applicability of Discrete Mixtures
6.2 Discrete Mixtures of Parametric Densities
6.2.1 Model Choice
6.3 Identifiability Constraints
6.4 Hurdle and Zero-Inflated Models for Discrete Data
6.5 Regression Mixtures for Heterogeneous Subpopulations
6.6 Discrete Mixtures Combined with Parametric Random Effects
6.7 Non-Parametric Mixture Modelling via Dirichlet Process Priors
6.8 Other Non-Parametric Priors
Exercises
References
Chapter 7 Multinomial and Ordinal Regression Models
7.1 Introduction: Applications with Categoric and Ordinal Data
7.2 Multinomial Logit Choice Models
7.3 The Multinomial Probit Representation of Interdependent Choices
7.4 Mixed Multinomial Logit Models
7.5 Individual Level Ordinal Regression
7.6 Scores for Ordered Factors in Contingency Tables
Exercises
References
Chapter 8 Time Series Models
8.1 Introduction: Alternative Approaches to Time Series Models
8.2 Autoregressive Models in the Observations
8.2.1 Priors on Autoregressive Coefficients
8.2.2 Initial Conditions as Latent Data
8.3 Trend Stationarity in the AR1 Model
8.4 Autoregressive Moving Average Models
8.5 Autoregressive Errors
8.6 Multivariate Series
8.7 Time Series Models for Discrete Outcomes
8.7.1 Observation-Driven Autodependence
8.7.2 INAR Models
8.7.3 Error Autocorrelation
8.8 Dynamic Linear Models and Time Varying Coefficients
8.8.1 Some Common Forms of DLM
8.8.2 Priors for TIme-Specific Variances or Interventions
8.8.3 Nonlinear and Non-Gaussian State-Space Models
8.9 Models for Variance Evolution
8.9.1 ARCH and GARCH Models
8.9.2 Stochastic Volatility Models
8.10 Modelling Structural Shifts and Outliers
8.10.1 Markov Mixtures and Transition Functions
8.11 Other Nonlinear Models
Exercises
References
Chapter 9 Modelling Spatial Dependencies
9.1 Introduction: Implications of Spatial Dependence
9.2 Discrete Space Regressions for Metric Data
9.3 Discrete Spatial Regression with Structured and Unstructured Random Effects
9.3.1 Proper CAR Priors
9.4 Moving Average Priors
9.5 Multivariate Spatial Priors and Spatially Varying Regression Effects
9.6 Robust Models for Discontinuities and Non-Standard Errors
9.7 Continuous Space Modelling in Regression and Interpolation
Exercises
References
Chapter 10 Nonlinear and Nonparametric Regression
10.1 Approaches to Modelling Nonlinearity
10.2 Nonlinear Metric Data Models with Known Functional Form
10.3 Box–Cox Transformations and Fractional Polynomials
10.4 Nonlinear Regression Through Spline and Radial Basis Functions
10.4.1 Shrinkage Models for Spline Coefficients
10.4.2 Modelling Interaction Effects
10.5 Application of State-Space Priors in General Additive Nonparametric Regression
10.5.1 Continuous Predictor Space Prior
10.5.2 Discrete Predictor Space Priors
Exercises
References
Chapter 11 Multilevel and Panel Data Models
11.1 Introduction: Nested Data Structures
11.2 Multilevel Structures
11.2.1 The Multilevel Normal Linear Model
11.2.2 General Linear Mixed Models for Discrete Outcomes
11.2.3 Multinomial and Ordinal Multilevel Models
11.2.4 Robustness Regarding Cluster Effects
11.2.5 Conjugate Approaches for Discrete Data
11.3 Heteroscedasticity in Multilevel Models
11.4 Random Effects for Crossed Factors
11.5 Panel Data Models: The Normal Mixed Model and Extensions
11.5.1 Autocorrelated Errors
11.5.2 Autoregression in y
11.6 Models for Panel Discrete (Binary, Count and Categorical) Observations
11.6.1 Binary Panel Data
11.6.2 Repeated Counts
11.6.3 Panel Categorical Data
11.7 Growth Curve Models
11.8 Dynamic Models for Longitudinal Data: Pooling Strength Over Units and Times
11.9 Area APC and Spatiotemporal Models
11.9.1 Age–Period Data
11.9.2 Area–Time Data
11.9.3 Age–Area–Period Data
11.9.4 Interaction Priors
Exercises
References
Chapter 12 Latent Variable and Structural Equation Models for Multivariate Data
12.1 Introduction: Latent Traits and Latent Classes
12.2 Factor Analysis and SEMS for Continuous Data
12.2.1 Identifiability Constraints in Latent Trait (Factor Analysis) Models
12.3 Latent Class Models
12.3.1 Local Dependence
12.4 Factor Analysis and SEMS for Multivariate Discrete Data
12.5 Nonlinear Factor Models
Exercises
References
Chapter 13 Survival and Event History Analysis
13.1 Introduction
13.2 Parametric Survival Analysis in Continuous Time
13.2.1 Censored Observations
13.2.2 Forms of Parametric Hazard and Survival Curves
13.2.3 Modelling Covariate Impacts and Time Dependence in the Hazard Rate
13.3 Accelerated Hazard Parametric Models
13.4 Counting Process Models
13.5 Semiparametric Hazard Models
13.5.1 Priors for the Baseline Hazard
13.5.2 Gamma Process Prior on Cumulative Hazard
13.6 Competing Risk-Continuous Time Models
13.7 Variations in Proneness: Models for Frailty
13.8 Discrete Time Survival Models
Exercises
References
Chapter 14 Missing Data Models
14.1 Introduction: Types of Missingness
14.2 Selection and Pattern Mixture Models for the Joint Data-Missingness Density
14.3 Shared Random Effect and Common Factor Models
14.4 Missing Predictor Data
14.5 Multiple Imputation
14.6 Categorical Response Data with Possible Non-Random Missingness: Hierarchical and Regression Models
14.6.1 Hierarchical Models for Response and Non-Response by Strata
14.6.2 Regression Frameworks
14.7 Missingness with Mixtures of Continuous and Categorical Data
14.8 Missing Cells in Contingency Tables
14.8.1 Ecological Inference
Exercises
References
Chapter 15 Measurement Error, Seemingly Unrelated Regressions, and Simultaneous Equations
15.1 Introduction
15.2 Measurement Error in Both Predictors and Response in Normal Linear Regression
15.2.1 Prior Information on X or its Density
15.2.2 Measurement Error in General Linear Models
15.3 Misclassification of Categorical Variables
15.4 Simultaneous Equations and Instruments for Endogenous Variables
15.5 Endogenous Regression Involving Discrete Variables
Exercises
References
Appendix 1 A Brief Guide to Using WINBUGS
A1.1 Procedure for Compiling and Running Programs
A1.2 Generating Simulated Data
A1.3 Other Advice
Index
Wiley Series in Probability and Statistics
Bayesian Statistical Modelling
Second Edition
PETER CONGDON
Queen Mary, University of London, UK
Bayesian Statistical Modelling
WILEY SERIES IN PROBABILITY AND STATISTICS
established by Walter A. Shewhart and Samuel S. Wilks
Editors
David J. Balding, Peter Bloomfield, Noel A. C. Cressie, Nicholas I. Fisher,
Iain M. Johnstone, J. B. Kadane, Geert Molenberghs, Louise M. Ryan,
David W. Scott, Adrian F. M. Smith, Jozef L. Teugels
Editors Emeriti
Vic Barnett, J. Stuart Hunter, David G. Kendall
A complete list of the titles in this series appears at the end of this volume.
Bayesian Statistical Modelling
Second Edition
PETER CONGDON
Queen Mary, University of London, UK
Copyright C 2006
John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester,
West Sussex PO19 8SQ, England
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ISBN-10 0-470-01875-5 (HB)
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Contents
Preface
Chapter 1
xiii
Introduction: The Bayesian Method, its Benefits and Implementation
1.1
1.2
1.3 MCMC sampling and inferences from posterior densities
1.4
1
1
The Bayes approach and its potential advantages
Expressing prior uncertainty about parameters and Bayesian updating 2
5
9
12
14
18
18
19
The main MCMC sampling algorithms
1.4.1 Gibbs sampling
Convergence of MCMC samples
Predictions from sampling: using the posterior predictive density
The present book
1.5
1.6
1.7
References
Chapter 2 Bayesian Model Choice, Comparison and Checking
25
25
2.1
2.2
Introduction: the formal approach to Bayes model choice and
averaging
Analytic marginal likelihood approximations and the Bayes
information criterion
28
30
36
Approximating Bayes factors or model probabilities
38
Joint space search methods
Direct model averaging by binary and continuous selection indicators 41
43
Predictive model comparison via cross-validation
46
Predictive fit criteria and posterior predictive model checks
The DIC criterion
48
2.3 Marginal likelihood approximations from the MCMC output
2.4
2.5
2.6
2.7
2.8
2.9
2.10 Posterior and iteration-specific comparisons of likelihoods and
penalised likelihoods
2.11 Monte carlo estimates of model probabilities
References
Chapter 3 The Major Densities and their Application
3.1
3.2
Introduction
Univariate normal with known variance
3.2.1 Testing hypotheses on normal parameters
50
52
57
63
63
64
66
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