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Preface xvii
The Bayes linear approach 1
Combining beliefs with data 2
The Bayesian approach 3
Features of the Bayes linear approach 6
Example 7
Expectation, variance, and standardization 8
Prior inputs 8
Adjusted expectations 9
Adjusted versions 10
Adjusted variances 10
Checking data inputs 11
Observed adjusted expectations 12
Diagnostics for adjusted beliefs 12
Further diagnostics for the adjusted versions 12
Summary of basic adjustment 13
Diagnostics for collections 14
Exploring collections of beliefs via canonical structure 17
Modifying the original specifications 19
Repeating the analysis for the revised model 20
Global analysis of collections of observations 22
Partial adjustments 24
Partial diagnostics 27
Summary 29
Overview 30
Expectation 33
Expectation as a primitive 33
Discussion: expectation as a primitive 35
Quantifying collections of uncertainties 37
Specifying prior beliefs 39
Example: oral glucose tolerance test 39
Qualitative and quantitative prior specification 41
Example: qualitative representation of uncertainty 42
Identifying the quantities of interest 42
Identifying relevant prior information 42
Sources of variation 43
Representing population variation 44
The qualitative representation 44
Graphical models 46
Example: quantifying uncertainty 48
Prior expectations 48
Prior variances 49
Prior covariances 51
Summary of belief specifications 52
Discussion: on the various methods for assigning expectations 52
Adjusting beliefs 55
Adjusted expectation 55
Properties of adjusted expectation 56
Adjusted variance 57
Interpretations of belief adjustment 58
Foundational issues concerning belief adjustment 60
Example: one-dimensional problem 63
Collections of adjusted beliefs 64
Examples 65
Algebraic example 65
Oral glucose tolerance test 69
Many oral glucose tolerance tests 73
Canonical analysis for a belief adjustment 75
Canonical directions for the adjustment 75
The resolution transform 77
Partitioning the resolution 79
The reverse adjustment 80
Minimal linear sufficiency 81
The adjusted belief transform matrix 81
The geometric interpretation of belief adjustment 82
Examples 84
Simple one-dimensional problem 84
Algebraic example 84
Oral glucose tolerance test 90
Further reading 93
The observed adjustment 95
Discrepancy 95
Discrepancy for a collection 96
Evaluating discrepancy over a basis 98
Discrepancy for quantities with variance zero 98
Properties of discrepancy measures 98
Evaluating the discrepancy vector over a basis 100
Examples 100
Simple one-dimensional problem 100
Detecting degeneracy 100
Oral glucose tolerance test 102
The observed adjustment 104
Adjustment discrepancy 104
Adjustment discrepancy for a collection 105
Maximal discrepancy 106
Construction over a basis 107
Partitioning the discrepancy 108
Examples 108
Simple one-dimensional problem 108
Oral glucose tolerance test 109
The size of an adjustment 113
The size of an adjustment for a collection 113
The bearing for an adjustment 114
Construction via a basis 115
Representing discrepancy vectors as bearings 115
Joint bearings 116
Size diagnostics 116
Geometric interpretation 117
Linear likelihood 118
Examples 119
Algebraic example 119
Oral glucose tolerance test 120
Partial Bayes linear analysis 125
Partial adjustment 125
Partial variance 127
Partial resolution transforms 128
Relative belief adjustment 129
Example: oral glucose tolerance test 130
Performing an initial adjustment 131
Partial resolved variances 132
Partial canonical directions 132
Deducing changes for other linear combinations 133
Relative belief adjustment 133
Withdrawing quantities from the adjustment 134
Partial bearings 135
Partial data size 137
Bearing and size for a relative adjustment 137
Path correlation 138
Example: oral glucose tolerance test 139
The initial observed adjustment 139
Observed partial expectations 140
The size of the partial adjustment 141
The bearing for the partial adjustment 142
The path correlation for the partial adjustment 143
Sequential adjustment 144
The data trajectory 144
The canonical trajectory 145
Detection of systematic bias 146
Examples 147
Anscombe data sets 147
Regression with correlated responses 153
Bayes linear sufficiency and belief separation 166
Properties of generalized conditional independence 168
Properties of belief separation 169
Example: regression with correlated responses 172
Exploiting separation 172
Heart of the transform 173
Further reading 176
Exchangeable beliefs 177
Exchangeability 177
Coin tossing 180
Exchangeable belief structures 183
The representation theorem 185
Finite exchangeability 188
Example: oral glucose tolerance test 189
Example: analysing exchangeable regressions 191
Introduction 191
Error structure and specifications 192
Regression coefficient specifications 193
Structural implications 194
Adjusting exchangeable beliefs 194
Predictive sufficiency for exchangeable models 195
Bayes linear sufficiency for sample means 196
Belief adjustment for scalar exchangeable quantities 197
Canonical structure for an exchangeable adjustment 198
Standard form for the adjustment 200
Further properties of exchangeable adjustments 201
Algebraic example 202
Representation 203
Coherence 203
Bayes linear sufficiency 204
Example: adjusting exchangeable regressions 205
Bayes linear sufficiency 205
Adjustment 206
Resolution transforms 208
Resolution partition for exchangeable cases 210
Data diagnostics 211
Sample size choice 212
Adjustment for an equivalent linear space 214
Data diagnostics for an equivalent linear space 214
Compatibility of data sources 215
Predictive adjustment 218
Example: oral glucose tolerance test 220
Context of exchangeability 220
Mean component adjustment 220
Variance reduction for a predictive adjustment 221
Observed exchangeable adjustments 223
Path diagnostics 226
Example: predictive analysis for exchangeable regressions 228
Choice of canonical directions 230
Further reading 231
Co-exchangeable beliefs 233
Respecting exchangeability 233
Adjustments respecting exchangeability 234
Example: simple algebraic problem 235
Coherence 236
Resolution transform 236
Co-exchangeable adjustments 238
Example: analysing further exchangeable regressions 240
The resolution envelope 243
Example: exchangeability in a population dynamics experiment 244
Model 244
Specifications 248
Issues 251
Analysis 251
Learning about population variances 265
Assessing a population variance with known population mean 265
Assessing a population variance with unknown population mean 266
Choice of prior values 268
Example: oral glucose tolerance test 271
Adjusting the population residual variance in multiple linear regression: uncorrelated errors 273
Sample information 274
Choice of prior values 276
Example: Anscombe data sets 276
Adjusting the population residual variance in multiple linear regression: correlated errors 277
Example: regression with correlated responses 278
Example: analysing exchangeable regressions 280
Adjusting a collection of population variances and covariances 282
Direct adjustment for a population variance matrix 283
Example: regression with correlated responses 284
Separating direct adjustment for population variances and for correlation structure 285
Assessing the equivalent sample size 286
Example: oral glucose tolerance test 287
Two-stage Bayes linear analysis 288
Example: oral glucose tolerance test 290
Example: analysing exchangeable regressions 290
Further reading 292
Belief comparison 293
Comparing variance specifications 294
Rank-degenerate case 296
Comparison of orthogonal subspaces 298
Example: variance comparison 298
Canonical structure for the comparison 299
Consistency checks 301
Comparisons for further constructed quantities 301
Construction of specifications 302
Comparing many variance specifications 302
Example: comparing some simple nested hypotheses 304
General belief transforms 306
General belief transforms 306
Properties of general belief transforms 307
Adjusted belief transforms as general belief transforms 309
Example: adjustment of exchangeable structures 310
Example: analysing exchangeable regressions 311
Comparing expectations and variances 312
Geometric interpretation 314
Residual forms for mean and variance comparisons 315
Rank-degenerate case 317
The observed comparison 318
Combined directions 319
Example: mean and variance comparison 320
The observed comparison 323
Graphical comparison of specifications 324
Belief comparison diagram 325
The observed comparison 327
Combining information 329
Residual belief comparison diagrams 329
Example: exchangeable regressions 331
Basic canonical analysis 332
Mean and residual comparisons 333
Comparisons for exchangeable structures 337
The observed comparison 338
Example: exchangeable regressions 340
Example: fly population dynamics 342
Differences for the mean part of the average 343
Differences for the residual part of the average 343
Differences for the residual part of the average 344
Assessing robustness of specifications 346
Sensitivity analyses for expectations 347
Example: robustness analysis for exchangeable regressions 349
Sensitivity analyses for variances 350
Example: robustness analysis for variance specifications 351
Further reading 353
Bayes linear graphical models 355
Directed graphical models 356
Construction via statistical models 358
Operations on directed graphs 358
Quantifying a directed graphical model 361
Undirected graphs 362
Node removal via the moral graph 364
Example 364
Plates for duplicated structures 367
Reading properties from the diagram 367
Alternative diagrams 368
Diagrams for inference and prediction 370
Displaying the flow of information 372
Node shading 373
Arc labelling 374
Tracking information as it is received 376
Example 377
Displaying diagnostic information 383
Node diagnostics 385
Arc diagnostics 387
Showing implications across all nodes 388
Interpreting diagnostic warnings 388
Example: inference and prediction 389
Local computation: directed trees 395
Propagation 397
Example 398
Junction trees 399
Sequential local computation on the junction tree 400
Example: correlated regressions 402
Example: problems of prediction in a large brewery 402
Problem summary 402
Identifying the quantities of interest 403
Modelling 404
Initialization values and specifications 406
Examining the generated model 412
Basic adjustment 414
Exploration via graphical models 416
Local computation for global adjustment of the junction tree 424
Merging separate adjustments 425
The global adjustment algorithm 427
Absorption of evidence 427
Further reading 429
Matrix algebra for implementing the theory 431
Basic definitions 431
Covariance matrices and quadratic forms 431
Generalized inverses 432
Basic properties 432
Computing the Moore-Penrose inverse 432
Other properties of generalized inverses 433
Multiplication laws 434
Range and null space of a matrix 435
Rank conditions 436
Partitioned matrices 436
Definiteness for a partitioned real symmetric matrix 436
Generalized inverses for partitioned non-negative definite matrices 437
Solving linear equations 438
Eigensolutions to related matrices 439
Maximizing a ratio of quadratic forms 440
The generalized eigenvalue problem 441
Introduction 441
The QZ algorithm 442
An alternative algorithm 442
An algorithm for B - A non-negative definite 444
Direct products of matrices 447
The Helmert matrix 447
Direct products 448
Implementing Bayes linear statistics 451
Introduction 451
Coherence of belief specifications 452
Coherence for a single collection 452
Coherence for two collections 452
Coherence for three collections 453
Consistency of data with beliefs 455
Consistency for a single collection 455
Consistency for a partitioned collection 456
Adjusted expectation 457
Adjusted and resolved variance 458
The resolved variance matrix 459
Matrix representations of the resolution transform 460
The symmetrized resolution transform matrix 461
The transform for the reverse adjustment 463
Inverses for the resolved variance matrix 464
Canonical quantities 465
Coherence via the resolution transform matrix 466
Assessing discrepant data 467
Consistency of observed adjustments 468
Partitioning the discrepancy 469
The bearing and size of adjustment 472
Partial adjustments 473
Partial and relative adjustment transforms 475
Calculating the partial bearing 475
Exchangeable adjustments 476
Notation 476
Coherence requirements for exchangeable adjustments 477
Data consistency 477
Pure exchangeable adjustments 477
General exchangeable adjustments 481
Implementing comparisons of belief 483
Expectation comparisons 483
Comparison of exchangeable beliefs 483
Notation 487
Index of examples 491
Software for Bayes linear computation 495
[B/D] 495
Bayes-Lin 495
References 497
Index 503
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