Bayes Linear Statistics: Theory & Methods Book

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