Contents
List of Figures xi
List of Tables xiii
Preface xv
Acknowledgments xvii
1 Qualitative Overview 1
1.1 Introduction 1
1.2 Forecasting Mortality 3
1.2.1 The Data 3
1.2.2 The Patterns 5
1.2.3 Scientific versus Optimistic Forecasting Goals 8
1.3 Statistical Modeling 11
1.4 Implications for the Bayesian Modeling Literature 15
1.5 Incorporating Area Studies in Cross-National Comparative Research 16
1.6 Summary 18
Part I Existing Methods for Forecasting Mortality 19
2 Methods without Covariates 21
2.1 Patterns in Mortality Age Profiles 22
2.2 A Unified Statistical Framework 24
2.3 Population Extrapolation Approaches 25
2.4 Parametric Approaches 26
2.5 A Nonparametric Approach: Principal Components 28
2.5.1 Introduction 28
2.5.2 Estimation 32
2.6 The Lee-Carter Approach 34
2.6.1 The Model 34
2.6.2 Estimation 36
2.6.3 Forecasting 36
2.6.4 Properties 38
2.7 Summary 42
3 Methods with Covariates 43
3.1 Equation-by-Equation Maximum Likelihood 43
3.1.1 Poisson Regression 43
3.1.2 Least Squares 44
3.1.3 Computing Forecasts 46
3.1.4 Summary Evaluation 47
3.2 Time-Series, Cross-Sectional Pooling 48
3.2.1 The Model 48
3.2.2 Postestimation Intercept Correction 49
3.2.3 Summary Evaluation 49
3.3 Partially Pooling Cross Sections via Disturbance Correlations 50
3.4 Cause-Specific Methods with Microlevel Information 51
vi • Contents
3.4.1 Direct Decomposition Methods 51
Modeling 51
3.4.2 Microsimulation Methods 52
3.4.3 Interpretation 53
3.5 Summary 53
Part II Statistical Modeling 55
4 The Model 57
4.1 Overview 57
4.2 Priors on Coefficients 59
4.3 Problems with Priors on Coefficients 60
4.3.1 Little Direct Prior Knowledge Exists about Coefficients 61
4.3.2 Normalization Factors Cannot Be Estimated 62
4.3.3 We Know about the Dependent Variable, Not the Coefficients 64
4.3.4 Difficulties with Incomparable Covariates 65
4.4 Priors on the Expected Value of the Dependent Variable 65
4.4.1 Step 1: Specify a Prior for the Dependent Variable 66
4.4.2 Step 2: Translate to a Prior on the Coefficients 67
4.4.3 Interpretation 68
4.5 A Basic Prior for Smoothing over Age Groups 69
4.5.1 Step 1: A Prior for μ 69
4.5.2 Step 2: From the Prior on μ to the Prior on β 71
4.5.3 Interpretation 71
4.6 Concluding Remark 73
5 Priors over Grouped Continuous Variables 74
5.1 Definition and Analysis of Prior Indifference 74
5.1.1 A Simple Special Case 76
5.1.2 General Expressions for Prior Indifference 76
5.1.3 Interpretation 77
5.2 Step 1: A Prior for μ 80
5.2.1 Measuring Smoothness 81
5.2.2 Varying the Degree of Smoothness over Age Groups 83
5.2.3 Null Space and Prior Indifference 83
5.2.4 Nonzero Mean Smoothness Functional 85
5.2.5 Discretizing: From Age to Age Groups 85
5.2.6 Interpretation 86
5.3 Step 2: From the Prior on μ to the Prior on β 92
5.3.1 Analysis 92
5.3.2 Interpretation 92
6 Model Selection 94
6.1 Choosing the Smoothness Functional 94
6.2 Choosing a Prior for the Smoothing Parameter 97
6.2.1 Smoothness Parameter for a Nonparametric Prior 98
6.2.2 Smoothness Parameter for the Prior over the Coefficients 100
6.3 Choosing Where to Smooth 104
Contents • vii
6.4 Choosing Covariates 108
6.4.1 Size of the Null Space 109
6.4.2 Content of the Null Space 110
6.5 Choosing a Likelihood and Variance Function 112
6.5.1 Deriving the Normal Specification 112
6.5.2 Accuracy of the Log-Normal Approximation to the Poisson 114
6.5.3 Variance Specification 120
7 Adding Priors over Time and Space 124
7.1 Smoothing over Time 124
7.1.1 Prior Indifference and the Null Space 125
7.2 Smoothing over Countries 127
7.2.1 Null Space and Prior Indifference 128
7.2.2 Interpretation 130
7.3 Smoothing Simultaneously over Age, Country, and Time 131
7.4 Smoothing Time Trend Interactions 132
7.4.1 Smoothing Trends over Age Groups 133
7.4.2 Smoothing Trends over Countries 133
7.5 Smoothing with General Interactions 134
7.6 Choosing a Prior for Multiple Smoothing Parameters 136
7.6.1 Example 139
7.6.2 Estimating the Expected Value of the Summary Measures 141
7.7 Summary 144
8 Comparisons and Extensions 145
8.1 Priors on Coefficients versus Dependent Variables 145
8.1.1 Defining Distances 145
8.1.2 Conditional Densities 147
8.1.3 Connections to “Virtual Examples” in Pattern Recognition 147
8.2 Extensions to Hierarchical Models and Empirical Bayes 148
8.2.1 The Advantages of Empirical Bayes without Empirical Bayes 149
8.2.2 Hierarchical Models as Special Cases of Spatial Models 151
8.3 Smoothing Data without Forecasting 151
8.4 Priors When the Dependent Variable Changes Meaning 153
Part III Estimation 159
9 Markov Chain Monte Carlo Estimation 161
9.1 Complete Model Summary 161
9.1.1 Likelihood 162
9.1.2 Prior for β 162
9.1.3 Prior for σi 162
9.1.4 Prior for θ 163
9.1.5 The Posterior Density 164
9.2 The Gibbs Sampling Algorithm 164
9.2.1 Sampling σ 165
The Conditional Density 165
Interpretation 165
viii • Contents
9.2.2 Sampling θ 166
The Conditional Density 166
Interpretation 166
9.2.3 Sampling β 167
The Conditional Density 167
Interpretation 168
9.2.4 Uncertainty Estimates 169
9.3 Summary 169
10 Fast Estimation without Markov Chains 170
10.1 Maximum A Posteriori Estimator 170
10.2 Marginal Maximum A Posteriori Estimator 171
10.3 Conditional Maximum A Posteriori Estimator 172
10.4 Summary 173
Part IV Empirical Evidence 175
11 Illustrative Analyses 177
11.1 Forecasts without Covariates: Linear Trends 178
11.1.1 Smoothing over Age Groups Only 178
11.1.2 Smoothing over Age and Time 181
11.2 Forecasts without Covariates: Nonlinear Trends 182
11.3 Forecasts with Covariates: Smoothing over Age and Time 187
11.4 Smoothing over Countries 189
12 Comparative Analyses 196
12.1 All Causes in Males 197
12.2 Lung Disease in Males 200
12.2.1 Comparison with Least Squares 202
12.2.2 Country-by-Country Analysis 203
12.3 Breast Cancer in Females 205
12.3.1 Comparison with Least Squares 205
12.3.2 Country-by-country Analysis 205
12.4 Comparison on OECD Countries 206
12.4.1 Transportation Accidents in Males 208
12.4.2 Cardiovascular Disease in Males 210
13 Concluding Remarks 211
Appendixes 213
A Notation 215
A.1 Principles 215
A.2 Glossary 216
B Mathematical Refresher 219
B.1 Real Analysis 219
B.1.1 Vector Space 219
Contents • ix
B.1.2 Metric Space 220
B.1.3 Normed Space 221
B.1.4 Scalar Product Space 222
B.1.5 Functions, Mappings, and Operators 223
B.1.6 Functional 224
B.1.7 Span 224
B.1.8 Basis and Dimension 224
B.1.9 Orthonormality 225
B.1.10 Subspace 225
B.1.11 Orthogonal Complement 226
B.1.12 Direct Sum 226
B.1.13 Projection Operators 227
B.2 Linear Algebra 229
B.2.1 Range, Null Space, Rank, and Nullity 229
B.2.2 Eigenvalues and Eigenvectors for Symmetric Matrices 232
B.2.3 Definiteness 234
B.2.4 Singular Values Decomposition 234
Definition 234
For Approximation 235
B.2.5 Generalized Inverse 236
B.2.6 Quadratic Form Identity 238
B.3 Probability Densities 239
B.3.1 The Normal Distribution 239
B.3.2 The Gamma Distribution 239
B.3.3 The Log-Normal Distribution 240
C Improper Normal Priors 241
C.1 Definitions 241
C.2 An Intuitive Special Case 242
C.3 The General Case 243
C.4 Drawing Random Samples 246
D Discretization of the Derivative Operator 247
E Smoothness over Graphs 249
Bibliography 251
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