Bayes Linear Statistics. Theory and Methods

Transcription

1 Bayes Linear Statistics Theory and Methods Michael Goldstein and David Wooff Durham University, UK BICENTENNI AL BICENTENNIAL

2 Contents r Preface xvii 1 The Bayes linear approach Combining beliefs with data The Bayesian approach Features of the Bayes linear approach Example Expectation, variance, and standardization Prior inputs Adjusted expectations Adjusted versions Adjusted variances Checking data inputs Observed adjusted expectations Diagnostics for adjusted beliefs Further diagnostics for the adjusted versions Summary of basic adjustment Diagnostics for collections Exploring collections of beliefs via canonical structure Modifying the original specifications Repeating the analysis for the revised model Global analysis of collections of observations Partial adjustments Partial diagnostics Summary Overview 30 2 Expectation Expectation as a primitive Discussion: expectation as a primitive Quantifying collections of uncertainties Specifying prior beliefs Example: oral glucose tolerance test 39

3 Vlll CONTENTS 2.5 Qualitative and quantitative prior specification Example: qualitative representation of uncertainty Identifying the quantities of interest Identifying relevant prior Information Sources of Variation Representing population Variation The qualitative representation Graphical modeis Example: quantifying uncertainty Prior expectations Prior variances Prior covariances Summary of belief specifications Discussion: on the various methods for assigning expectations Adjusting beliefs Adjusted expectation Properties of adjusted expectation Adjusted variance Interpretations of belief adjustment Foundational issues concerning belief adjustment Example: one-dimensional problem Collections of adjusted beliefs Examples Algebraic example Oral glucose tolerance test Many oral glucose tolerance tests Canonical analysis for a belief adjustment Canonical directions for the adjustment The resolution transform Partitioning the resolution The reverse adjustment Minimal linear sufficiency The adjusted belief transform matrix The geometric Interpretation of belief adjustment Examples Simple one-dimensional problem Algebraic example ' Oral glucose tolerance test Further reading 93 4 The observed adjustment Discrepancy Discrepancy for a collection 96

4 CONTENTS Evaluating discrepancy over a basis Discrepancy for quantities with variance zero Properties of discrepancy measures Evaluating the discrepancy vector over a basis Examples Simple one-dimensional problem Detecting degeneracy Oral glucose tolerance test The observed adjustment Adjustment discrepancy Adjustment discrepancy for a collection Maximal discrepancy Construction over a basis Partitioning the discrepancy Examples Simple one-dimensional problem Oral glucose tolerance test The size of an adjustment./ The size of an adjustment for a collection The bearing for an adjustment Construction via a basis Representing discrepancy vectors as bearings Joint bearings Size diagnostics Geometrie Interpretation Linear likelihood Examples Algebraic example Oral glucose tolerance test Partial Bayes linear analysis Partial adjustment Partial variance Partial resolution transforms Relative belief adjustment Example: oral glucose tolerance test Performing an initial adjustment Partial resolved variances Partial canonical directions Deducing changes for other linear combinations Relative belief adjustment Withdrawing quantities from the adjustment Partial bearings Partial data size 137 ix

5 CONTENTS 5.8 Bearing and size for a relative adjustment Path correlation Example: oral glucose tolerance test The initial observed adjustment Observed partial expectations The size of the partial adjustment The bearing for the partial adjustment The path correlation for the partial adjustment Sequential adjustment The data trajectory The canonical trajectory Detection of systematic bias Examples Anscombe data sets Regression with correlated responses Bayes linear sufficiency and belief Separation Properties of generalized conditional independence Properties of belief Separation Example: regression with correlated responses Exploiting Separation Heart of the transform Further reading 176 Exchangeable beliefs Exchangeabilityi Coin tossing J Exchangeable belief structures The representation theorem Finite exchangeability Example: oral glucose tolerance test Example: analysing exchangeable regressions Introduction Error structure and specifications Regression coefficient specifications Structural implications Adjusting exchangeable beliefs Predictive sufficiency for exchangeable modeis Bayes linear sufficiency for sample means Belief adjustment for scalar exchangeable quantities Canonical structure for an exchangeable adjustment Standard form for the adjustment Further properties of exchangeable adjustments Algebraic example Representation 203

6 CONTENTS Coherence Bayes linear sufficiency Example: adjusting exchangeable regressions Bayes linear sufficiency Adjustment Resolution transforms Resolution partition for exchangeable cases Data diagnostics Sample size choice Adjustment for an equivalent linear space Data diagnostics for an equivalent linear space Compatibility of data sources Predictive adjustment Example: oral glucose tolerance test Context of exchangeability Mean component adjustment Variance reduction for a predictive adjustment Observed exchangeable adjustments Path diagnostics Example: predictive analysis for exchangeable regressions Choice of canonical directions., Further reading J Co-exchangeable beliefs Respecting exchangeability Adjustments respecting exchangeability Example: simple algebraic problem Coherence Resolution transform Co-exchangeable adjustments Example: analysing further exchangeable regressions The resolution envelope Example: exchangeability in a population dynamics experiment Model Specifications Issues Analysis Learning about population variances Assessing a population variance with known population mean Assessing a population variance with unknown population mean Choice of prior values Example: oral glucose tolerance test 271 xi

7 xii CONTENTS 8.5 Adjusting the population residual variance in multiple linear regression: uncorrelated errors Sample Information Choice of prior values Example: Anscombe data sets Adjusting the population residual variance in multiple linear regression: correlated errors Example: regression with correlated responses Example: analysing exchangeable regressions Adjusting a collection of population variances and covariances Direct adjustment for a population variance matrix Example: regression with correlated responses Separating direct adjustment for population variances and for correlation structure Assessing the equivalent sample size Example: oral glucose tolerance test Two-stage Bayes linear analysis Example: oral glucose tolerance test Example: analysing exchangeable regressions Further reading Belief comparison Comparing variance specifications Rank-degenerate case Comparison of orthogonal subspaces Example: variance comparison Canonical structure for the comparison Consistency checks Comparisons for further constructed quantities Construction of specifications Comparing many variance specifications Example: comparing some simple nested hypotheses General belief transforms General belief transforms Properties of general belief transforms Adjusted belief transforms as general belief transforms Example: adjustment of exchangeable structures Example: analysing exchangeable regressions Comparing expectations and variances Geometrie Interpretation Residual forms for mean and variance comparisons Rank-degenerate case 317

8 CONTENTS 9.9 The observed comparison Combined directions Example: mean and variance comparison The observed comparison Graphical comparison of specifications Belief comparison diagram The observed comparison Combining Information Residual belief comparison diagrams Example: exchangeable regressions Basic canonical analysis Mean and residual comparisons Comparisons for exchangeable structures The observed comparison Example: exchangeable regressions Example: fly population dynamics Differences for the mean part of the average Differences for the residual part of the average Differences for the residual part of the average Assessing robustness of specifications Sensitivity analyses for expectations Example: robustness analysis for exchangeable regressions Sensitivity analyses for variances Example: robustness analysis for variance specifications Further reading Bayes linear graphical modeis Directed graphical modeis f Construction via Statistical modeis Operations on directed graphs Quantifying a directed graphical model Undirected graphs Node removal via the moral graph Example Plates for duplicated structures Reading properties from the diagram Alternative diagrams Diagrams for inference and prediction Displaying the flow of information Node shading Arclabelling Tracking information as it is received Example 377 xiii

9 I xiv CONTENTS 10.7 Displaying diagnostic information Node diagnostics Are diagnostics Showing implications across all nodes Interpreting diagnostic warnings Example: inference and prediction Local computation: directed trees Propagation Example Junction trees Sequential local computation on the junction tree Example: correlated regressions Example: problems of prediction in a large brewery Problem summary Identifying the quantities of interest Modelling Initialization values and speeifications Examining the generated model Basic adjustment Exploration via graphical modeis Local computation for global adjustment of the junction tree Merging separate adjustments The global adjustment algorithm Absorption of evidence Further reading Matrix algebra for implementing the theory Basic definitions Covariance matrices and quadratic fonus Generalized inverses Basic properties Computing the Moore-Penrose inverse Other properties of generalized inverses Multiplication laws Range and null space of a matrix Rank conditions Partitioned matrices Definiteness for a partitioned real Symmetrie matrix Generalized inverses for partitioned non-negative definite matrices Solving linear equations Eigensolutions to related matrices 439

10 CONTENTS Maximizing a ratio of quadratic forms The generalized eigenvalue problem Introduction The QZ algorithm An alternative algorithm An algorithm for B A non-negative definite Direct products of matrices The Helmert matrix Direct products Implementing Bayes linear statistics Introduction Coherence of belief specifications Coherence for a Single collection Coherence for two collections Coherence for three collections Consistency of data with beliefs Consistency for a Single collection Consistency for a partitioned collection Adjusted expectation Adjusted and resolved variance The resolved variance matrix Matrix representations of the resolution transform The symmetrized resolution transform matrix The transform for the reverse adjustment Inverses for the resolved variance matrix Canonical quantities Coherence via the resolution transform matrix Assessing discrepant data Consistency of observed adjustments Partitioning the discrepancy, The bearing and size of adjustment Partial adjustments Partial and relative adjustment transforms Calculating the partial bearing Exchangeable adjustments Notation Coherence requirements for exchangeable adjustments Data consistency Pure exchangeable adjustments General exchangeable adjustments 481 xv

11 xvi CONTENTS Implementing comparisons of belief Expectation comparisons Comparison of exchangeable beliefs 483 A Notation 487 B Index of examples 491 C Software for Bayes linear computation 495 C.l [B/D] 495 C.2 BAYES-LIN 495 References 497 Index 503