Statistics and the Evaluation of Evidence for Forensic Scientists
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Statistics and the Evaluation of Evidence for Forensic Scientists
Taroni, Franco; Bozza, Silvia; Aitken, Colin
John Wiley & Sons Inc
12/2020
1248
Dura
Inglês
9781119245223
15 a 20 dias
1278
Descrição não disponível.
Foreword xvii
Preface to Third Edition xxi
Preface to Second Edition xxx
Preface to First Edition xxxvii
1 Uncertainty in Forensic Science 1
1.1 Introduction 1
1.2 Statistics and the Law 3
1.3 Uncertainty in Scientific Evidence 11
1.3.1 The Frequentist Method 15
1.3.2 Stains of Body Fluids 17
1.3.3 Glass Fragments 21
1.4 Terminology 29
1.5 Types of Data 34
1.6 Populations 36
1.7 Probability 41
1.7.1 Introduction 41
1.7.2 A Standard for Uncertainty 46
1.7.3 Events 55
1.7.4 Classical and Frequentist Definitions of Probability and Their Limitations 57
1.7.5 Subjective Definition of Probability 60
1.7.6 The Quantification of Probability Through a Betting Scheme 64
1.7.7 Probabilities and Frequencies: The Role of Exchangeability 69
1.7.8 Laws of Probability 78
1.7.9 Dependent Events and Background Information 82
1.7.10 Law of Total Probability 91
1.7.11 Updating of Probabilities 96
2 The Evaluation of Evidence 101
2.1 Odds 101
2.1.1 Complementary Events 101
2.1.2 Examples 104
2.1.3 Definition of Odds 105
2.2 Bayes' Theorem 108
2.2.1 Statement of the Theorem 109
2.2.2 Examples 109
2.3 The Odds Form of Bayes' Theorem 121
2.3.1 Likelihood Ratio 121
2.3.2 Bayes' Factor and Likelihood Ratio 125
2.3.3 Three-Way Tables 130
2.3.4 Logarithm of the Likelihood Ratio 134
2.4 The Value of Evidence 138
2.4.1 Evaluation of Forensic Evidence 138
2.4.2 Justification of the Use of the Likelihood Ratio 154
2.4.3 Single Value for the Likelihood Ratio 158
2.4.4 Role of Background Information 161
2.4.5 Summary of Competing Propositions 163
2.4.6 Qualitative Scale for the Value of the Evidence 168
2.5 Errors in Interpretation 180
2.5.1 Fallacy of the Transposed Conditional 186
2.5.2 Source Probability Error 190
2.5.3 Ultimate Issue Error 194
2.5.4 Defence Attorney's Fallacy 194
2.5.5 Probability (Another Match) Error 196
2.5.6 Numerical Conversion Error 199
2.5.7 False Positive Fallacy 202
2.5.8 Expected Value Fallacy 203
2.5.9 Uniqueness 206
2.5.10 Other Difficulties 209
2.5.11 Empirical Evidence of Errors in Interpretation 220
2.6 Misinterpretations 233
2.7 Explanation of Transposed Conditional, Defence Attorney's and False Positive Fallacies 236
2.7.1 Explanation of the Fallacy of the Transposed Conditional 236
2.7.2 Explanation of the Defence Attorney's Fallacy 239
2.7.3 Explanation of the False Positive Fallacy 241
2.8 Making Coherent Decisions 245
2.8.1 Elements of Statistical Decision Theory 246
2.8.2 Decision Analysis: An Example 249
2.9 Graphical Probabilistic Models: Bayesian Networks 254
2.9.1 Elements of the Bayesian Networks 256
2.9.2 The Construction of Bayesian Networks 261
2.9.3 Bayesian Decision Networks (Influence Diagrams) 272
3 Historical Review 279
3.1 Early History 279
3.2 The Dreyfus Case 286
3.3 Statistical Arguments by Early Twentieth- Century Forensic Scientists 293
3.4 People v.Collins 299
3.5 Discriminating Power 307
3.5.1 Derivation 307
3.5.2 Evaluation of Evidence by Discriminating Power 310
3.5.3 Finite Samples 316
3.5.4 Combination of Independent Systems 319
3.5.5 Correlated Attributes 321
3.6 Significance Probabilities 325
3.6.1 Calculation of Significance Probabilities 326
3.6.2 Relationship to Likelihood Ratio 333
3.6.3 Combination of Significance Probabilities 338
3.7 Coincidence Probabilities 342
3.7.1 Introduction 342
3.7.2 Comparison Stage 346
3.7.3 Significance Stage 347
3.8 Likelihood Ratio 351
4 Bayesian Inference 359
4.1 Introduction 359
4.2 Inference for a Proportion 368
4.2.1 Interval Estimation 374
4.2.2 Estimation with Zero Occurrences in a Sample 381
4.2.3 Uncertainty on Sensitivity and Specificity 387
4.3 Sampling 392
4.3.1 Choice of Sample Size in Large Consignments 398
4.3.2 Choice of Sample Size in Small Consignments 413
4.4 Bayesian Networks for Sampling Inspection 420
4.4.1 Large Consignments 420
4.4.2 Small Consignments 425
4.5 Inference for a Normal Mean 429
4.5.1 Known Variance 431
4.5.2 Unknown Variance 438
4.5.3 Interval Estimation 445
4.6 Quantity Estimation 449
4.6.1 Predictive Approach in Small Consignments 452
4.6.2 Predictive Approach in Large Consignments 461
4.7 Decision Analysis 464
4.7.1 Standard Loss Functions 465
4.7.2 Decision Analysis for Forensic Sampling 471
5 Evidence and Propositions: Theory 483
5.1 The Choice of Propositions and Pre-Assessment 483
5.2 Levels of Propositions and Roles of the Forensic Scientist 485
5.3 The Formal Development of a Likelihood Ratio for Different Propositions and Discrete Characteristics 499
5.3.1 Likelihood Ratio with Source Level Propositions 499
5.3.2 Likelihood Ratio with Activity Level Propositions 519
5.3.3 Likelihood Ratio with Offence Level Propositions 553
5.4 Validation of Bayesian Network Structures: An Example 562
5.5 Pre-Assessment 568
5.5.1 Pre-assessment of the Case 568
5.5.2 Pre-assessment of Evidence 575
5.5.3 Pre-assessment: A Practical Example 576
5.6 Combination of Items of Evidence 592
5.6.1 A Difficulty in Combining Evidence: The Problem of Conjunction 594
5.6.2 Generic Patterns of Inference in Combining Evidence 598
6 Evidence and Propositions: Practice 615
6.1 Examples for Evaluation given Source Level Propositions 615
6.1.1 General Population 616
6.1.2 Particular Population 617
6.1.3 A Note on The Appropriate Databases for Evaluation Given Source Level Propositions 619
6.1.4 Two Trace Problem 627
6.1.5 Many Samples 633
6.1.6 Multiple Propositions 637
6.1.7 A Note on Biological Traces 654
6.1.8 Additional Considerations on Source Level Propositions 670
6.2 Examples for Evaluation given Activity Level Propositions 699
6.2.1 A Practical Approach to Fibres Evaluation 701
6.2.2 A Practical Approach to Glass Evaluation 704
6.2.3 The Assignment of Probabilities for Transfer Events 713
6.2.4 The Assignment of Probabilities for Background Traces 734
6.2.5 Presence of Material with Non-corresponding Features 739
6.2.6 Absence of Evidence for Activity Level Propositions 741
6.3 Examples for Evaluation given Offence Level Propositions 745
6.3.1 One Stain, k Offenders 745
6.3.2 Two Stains, One Offender 752
6.3.3 Paternity and The Combination of Likelihood Ratios 756
6.3.4 Probability of Paternity 762
6.3.5 Absence of Evidence for Offence Level Propositions 768
6.3.6 A Note on Relevance and Offence Level Propositions 773
6.4 Summary 774
6.4.1 Stain Known to Have Been Left by Offenders: Source-Level Propositions 774
6.4.2 Material Known to Have Been (or Not to Have Been) Left by Offenders: Activity-Level Propositions 777
6.4.3 Stain May Not Have Been Left by Offenders: Offence-Level Propositions 779
7 Data Analysis 783
7.1 Introduction 783
7.2 Theory for Discrete Data 785
7.2.1 Data of Independent Counts with a Poisson Distribution 787
7.2.2 Data of Independent Counts with a Binomial Distribution 791
7.2.3 Data of Independent Counts with a Multinomial Distribution 793
7.3 Theory for Continuous Univariate Data 798
7.3.1 Assessment of Similarity Only 802
7.3.2 Sources of Variation: Two-Level Models 808
7.3.3 Transfer Probability 810
7.4 Normal Between-Source Variation 814
7.4.1 Marginal Distribution of Measurements 814
7.4.2 Approximate Derivation of the Likelihood Ratio 817
7.4.3 Lindley's Approach 820
7.4.4 Interpretation of Result 825
7.4.5 Examples 827
7.5 Non-normal Between-Source Variation 830
7.5.1 Estimation of a Probability Density Function 831
7.5.2 Kernel Density Estimation for Between-Source Data 842
7.5.3 Examples 844
7.6 Multivariate Analysis 849
7.6.1 Introduction 849
7.6.2 Multivariate Two-Level Models 851
7.6.3 A Note on Sensitivity 864
7.6.4 Case Study for Two-Level Data 865
7.6.5 Three-Level Models 876
7.7 Discrimination 882
7.7.1 Discrete Data 884
7.7.2 Continuous Data 889
7.7.3 Autocorrelated Data 893
7.7.4 Multivariate Data 894
7.7.5 Cut-Offs and Legal Thresholds 899
7.8 Score-Based Models 906
7.8.1 Example 910
7.9 Bayes' Factor and Likelihood Ratio (cont.) 913
8 Assessment of the Performance of Methods for the Evaluation of Evidence 919
8.1 Introduction 919
8.2 Properties of Methods for Evaluation 928
8.3 General Topics Relating to Sample Size Estimation and to Assessment 933
8.3.1 Probability of Strong Misleading Evidence: A Sample Size Problem 933
8.3.2 Calibration 948
8.4 Assessment of Performance of a Procedure for the Calculation of the Likelihood Ratio 952
8.4.1 Histograms and Tippett Plots 956
8.4.2 False Positive Rates, False Negative Rates and DET Plots 959
8.4.3 Empirical Cross-Entropy 961
8.5 Case Study: Kinship Analysis 972
8.6 Conclusion 979
Appendix A Probability Distributions 981
A.1 Introduction 981
A.2 Probability Distributions for Counts 988
A.2.1 Probabilities 988
A.2.2 Summary Measures 990
A.2.3 Binomial Distribution 995
A.2.4 Multinomial Distribution 997
A.2.5 Hypergeometric Distribution 998
A.2.6 Poisson Distribution 1000
A.2.7 Beta-Binomial and Dirichlet-Multinomial Distributions 1002
A.3 Measurements 1005
A.3.1 Summary Statistics 1005
A.3.2 Normal Distribution 1007
A.3.3 Jeffreys' Prior Distributions 1021
A.3.4 Student's t-Distribution 1021
A.3.5 Gamma and Chi-Squared Distributions 1025
A.3.6 Inverse Gamma and Inverse Chi-Squared Distributions 1026
A.3.7 Beta Distribution 1028
A.3.8 Dirichlet Distribution 1032
A.3.9 Multivariate Normal Distribution and Correlation 1035
A.3.10 Wishart Distribution 1040
A.3.11 Inverse Wishart Distribution 1041
Appendix B Matrix Properties 1043
B.1 Matrix Terminology 1043
B.1.1 The Trace of a Square Matrix 1044
B.1.2 The Transpose of a Matrix 1044
B.1.3 Addition of Two Matrices 1045
B.1.4 Determinant of a Matrix 1045
B.1.5 Matrix Multiplication 1046
B.1.6 The Inverse of a Matrix 1048
B.1.7 Completion of the Square 1049
References 1051
Notation 1143
Cases 1157
Author Index 1163
Subject Index 1187
Preface to Third Edition xxi
Preface to Second Edition xxx
Preface to First Edition xxxvii
1 Uncertainty in Forensic Science 1
1.1 Introduction 1
1.2 Statistics and the Law 3
1.3 Uncertainty in Scientific Evidence 11
1.3.1 The Frequentist Method 15
1.3.2 Stains of Body Fluids 17
1.3.3 Glass Fragments 21
1.4 Terminology 29
1.5 Types of Data 34
1.6 Populations 36
1.7 Probability 41
1.7.1 Introduction 41
1.7.2 A Standard for Uncertainty 46
1.7.3 Events 55
1.7.4 Classical and Frequentist Definitions of Probability and Their Limitations 57
1.7.5 Subjective Definition of Probability 60
1.7.6 The Quantification of Probability Through a Betting Scheme 64
1.7.7 Probabilities and Frequencies: The Role of Exchangeability 69
1.7.8 Laws of Probability 78
1.7.9 Dependent Events and Background Information 82
1.7.10 Law of Total Probability 91
1.7.11 Updating of Probabilities 96
2 The Evaluation of Evidence 101
2.1 Odds 101
2.1.1 Complementary Events 101
2.1.2 Examples 104
2.1.3 Definition of Odds 105
2.2 Bayes' Theorem 108
2.2.1 Statement of the Theorem 109
2.2.2 Examples 109
2.3 The Odds Form of Bayes' Theorem 121
2.3.1 Likelihood Ratio 121
2.3.2 Bayes' Factor and Likelihood Ratio 125
2.3.3 Three-Way Tables 130
2.3.4 Logarithm of the Likelihood Ratio 134
2.4 The Value of Evidence 138
2.4.1 Evaluation of Forensic Evidence 138
2.4.2 Justification of the Use of the Likelihood Ratio 154
2.4.3 Single Value for the Likelihood Ratio 158
2.4.4 Role of Background Information 161
2.4.5 Summary of Competing Propositions 163
2.4.6 Qualitative Scale for the Value of the Evidence 168
2.5 Errors in Interpretation 180
2.5.1 Fallacy of the Transposed Conditional 186
2.5.2 Source Probability Error 190
2.5.3 Ultimate Issue Error 194
2.5.4 Defence Attorney's Fallacy 194
2.5.5 Probability (Another Match) Error 196
2.5.6 Numerical Conversion Error 199
2.5.7 False Positive Fallacy 202
2.5.8 Expected Value Fallacy 203
2.5.9 Uniqueness 206
2.5.10 Other Difficulties 209
2.5.11 Empirical Evidence of Errors in Interpretation 220
2.6 Misinterpretations 233
2.7 Explanation of Transposed Conditional, Defence Attorney's and False Positive Fallacies 236
2.7.1 Explanation of the Fallacy of the Transposed Conditional 236
2.7.2 Explanation of the Defence Attorney's Fallacy 239
2.7.3 Explanation of the False Positive Fallacy 241
2.8 Making Coherent Decisions 245
2.8.1 Elements of Statistical Decision Theory 246
2.8.2 Decision Analysis: An Example 249
2.9 Graphical Probabilistic Models: Bayesian Networks 254
2.9.1 Elements of the Bayesian Networks 256
2.9.2 The Construction of Bayesian Networks 261
2.9.3 Bayesian Decision Networks (Influence Diagrams) 272
3 Historical Review 279
3.1 Early History 279
3.2 The Dreyfus Case 286
3.3 Statistical Arguments by Early Twentieth- Century Forensic Scientists 293
3.4 People v.Collins 299
3.5 Discriminating Power 307
3.5.1 Derivation 307
3.5.2 Evaluation of Evidence by Discriminating Power 310
3.5.3 Finite Samples 316
3.5.4 Combination of Independent Systems 319
3.5.5 Correlated Attributes 321
3.6 Significance Probabilities 325
3.6.1 Calculation of Significance Probabilities 326
3.6.2 Relationship to Likelihood Ratio 333
3.6.3 Combination of Significance Probabilities 338
3.7 Coincidence Probabilities 342
3.7.1 Introduction 342
3.7.2 Comparison Stage 346
3.7.3 Significance Stage 347
3.8 Likelihood Ratio 351
4 Bayesian Inference 359
4.1 Introduction 359
4.2 Inference for a Proportion 368
4.2.1 Interval Estimation 374
4.2.2 Estimation with Zero Occurrences in a Sample 381
4.2.3 Uncertainty on Sensitivity and Specificity 387
4.3 Sampling 392
4.3.1 Choice of Sample Size in Large Consignments 398
4.3.2 Choice of Sample Size in Small Consignments 413
4.4 Bayesian Networks for Sampling Inspection 420
4.4.1 Large Consignments 420
4.4.2 Small Consignments 425
4.5 Inference for a Normal Mean 429
4.5.1 Known Variance 431
4.5.2 Unknown Variance 438
4.5.3 Interval Estimation 445
4.6 Quantity Estimation 449
4.6.1 Predictive Approach in Small Consignments 452
4.6.2 Predictive Approach in Large Consignments 461
4.7 Decision Analysis 464
4.7.1 Standard Loss Functions 465
4.7.2 Decision Analysis for Forensic Sampling 471
5 Evidence and Propositions: Theory 483
5.1 The Choice of Propositions and Pre-Assessment 483
5.2 Levels of Propositions and Roles of the Forensic Scientist 485
5.3 The Formal Development of a Likelihood Ratio for Different Propositions and Discrete Characteristics 499
5.3.1 Likelihood Ratio with Source Level Propositions 499
5.3.2 Likelihood Ratio with Activity Level Propositions 519
5.3.3 Likelihood Ratio with Offence Level Propositions 553
5.4 Validation of Bayesian Network Structures: An Example 562
5.5 Pre-Assessment 568
5.5.1 Pre-assessment of the Case 568
5.5.2 Pre-assessment of Evidence 575
5.5.3 Pre-assessment: A Practical Example 576
5.6 Combination of Items of Evidence 592
5.6.1 A Difficulty in Combining Evidence: The Problem of Conjunction 594
5.6.2 Generic Patterns of Inference in Combining Evidence 598
6 Evidence and Propositions: Practice 615
6.1 Examples for Evaluation given Source Level Propositions 615
6.1.1 General Population 616
6.1.2 Particular Population 617
6.1.3 A Note on The Appropriate Databases for Evaluation Given Source Level Propositions 619
6.1.4 Two Trace Problem 627
6.1.5 Many Samples 633
6.1.6 Multiple Propositions 637
6.1.7 A Note on Biological Traces 654
6.1.8 Additional Considerations on Source Level Propositions 670
6.2 Examples for Evaluation given Activity Level Propositions 699
6.2.1 A Practical Approach to Fibres Evaluation 701
6.2.2 A Practical Approach to Glass Evaluation 704
6.2.3 The Assignment of Probabilities for Transfer Events 713
6.2.4 The Assignment of Probabilities for Background Traces 734
6.2.5 Presence of Material with Non-corresponding Features 739
6.2.6 Absence of Evidence for Activity Level Propositions 741
6.3 Examples for Evaluation given Offence Level Propositions 745
6.3.1 One Stain, k Offenders 745
6.3.2 Two Stains, One Offender 752
6.3.3 Paternity and The Combination of Likelihood Ratios 756
6.3.4 Probability of Paternity 762
6.3.5 Absence of Evidence for Offence Level Propositions 768
6.3.6 A Note on Relevance and Offence Level Propositions 773
6.4 Summary 774
6.4.1 Stain Known to Have Been Left by Offenders: Source-Level Propositions 774
6.4.2 Material Known to Have Been (or Not to Have Been) Left by Offenders: Activity-Level Propositions 777
6.4.3 Stain May Not Have Been Left by Offenders: Offence-Level Propositions 779
7 Data Analysis 783
7.1 Introduction 783
7.2 Theory for Discrete Data 785
7.2.1 Data of Independent Counts with a Poisson Distribution 787
7.2.2 Data of Independent Counts with a Binomial Distribution 791
7.2.3 Data of Independent Counts with a Multinomial Distribution 793
7.3 Theory for Continuous Univariate Data 798
7.3.1 Assessment of Similarity Only 802
7.3.2 Sources of Variation: Two-Level Models 808
7.3.3 Transfer Probability 810
7.4 Normal Between-Source Variation 814
7.4.1 Marginal Distribution of Measurements 814
7.4.2 Approximate Derivation of the Likelihood Ratio 817
7.4.3 Lindley's Approach 820
7.4.4 Interpretation of Result 825
7.4.5 Examples 827
7.5 Non-normal Between-Source Variation 830
7.5.1 Estimation of a Probability Density Function 831
7.5.2 Kernel Density Estimation for Between-Source Data 842
7.5.3 Examples 844
7.6 Multivariate Analysis 849
7.6.1 Introduction 849
7.6.2 Multivariate Two-Level Models 851
7.6.3 A Note on Sensitivity 864
7.6.4 Case Study for Two-Level Data 865
7.6.5 Three-Level Models 876
7.7 Discrimination 882
7.7.1 Discrete Data 884
7.7.2 Continuous Data 889
7.7.3 Autocorrelated Data 893
7.7.4 Multivariate Data 894
7.7.5 Cut-Offs and Legal Thresholds 899
7.8 Score-Based Models 906
7.8.1 Example 910
7.9 Bayes' Factor and Likelihood Ratio (cont.) 913
8 Assessment of the Performance of Methods for the Evaluation of Evidence 919
8.1 Introduction 919
8.2 Properties of Methods for Evaluation 928
8.3 General Topics Relating to Sample Size Estimation and to Assessment 933
8.3.1 Probability of Strong Misleading Evidence: A Sample Size Problem 933
8.3.2 Calibration 948
8.4 Assessment of Performance of a Procedure for the Calculation of the Likelihood Ratio 952
8.4.1 Histograms and Tippett Plots 956
8.4.2 False Positive Rates, False Negative Rates and DET Plots 959
8.4.3 Empirical Cross-Entropy 961
8.5 Case Study: Kinship Analysis 972
8.6 Conclusion 979
Appendix A Probability Distributions 981
A.1 Introduction 981
A.2 Probability Distributions for Counts 988
A.2.1 Probabilities 988
A.2.2 Summary Measures 990
A.2.3 Binomial Distribution 995
A.2.4 Multinomial Distribution 997
A.2.5 Hypergeometric Distribution 998
A.2.6 Poisson Distribution 1000
A.2.7 Beta-Binomial and Dirichlet-Multinomial Distributions 1002
A.3 Measurements 1005
A.3.1 Summary Statistics 1005
A.3.2 Normal Distribution 1007
A.3.3 Jeffreys' Prior Distributions 1021
A.3.4 Student's t-Distribution 1021
A.3.5 Gamma and Chi-Squared Distributions 1025
A.3.6 Inverse Gamma and Inverse Chi-Squared Distributions 1026
A.3.7 Beta Distribution 1028
A.3.8 Dirichlet Distribution 1032
A.3.9 Multivariate Normal Distribution and Correlation 1035
A.3.10 Wishart Distribution 1040
A.3.11 Inverse Wishart Distribution 1041
Appendix B Matrix Properties 1043
B.1 Matrix Terminology 1043
B.1.1 The Trace of a Square Matrix 1044
B.1.2 The Transpose of a Matrix 1044
B.1.3 Addition of Two Matrices 1045
B.1.4 Determinant of a Matrix 1045
B.1.5 Matrix Multiplication 1046
B.1.6 The Inverse of a Matrix 1048
B.1.7 Completion of the Square 1049
References 1051
Notation 1143
Cases 1157
Author Index 1163
Subject Index 1187
Este título pertence ao(s) assunto(s) indicados(s). Para ver outros títulos clique no assunto desejado.
Bayesian networks; decision-making; Bayesian reasoning; belief updates; chain of reasoning; evidence assessment; evidence evaluation; evidence interpretation; scientific evidence; statistical evaluation and interpretation of scientific evidence; trace evidence
Foreword xvii
Preface to Third Edition xxi
Preface to Second Edition xxx
Preface to First Edition xxxvii
1 Uncertainty in Forensic Science 1
1.1 Introduction 1
1.2 Statistics and the Law 3
1.3 Uncertainty in Scientific Evidence 11
1.3.1 The Frequentist Method 15
1.3.2 Stains of Body Fluids 17
1.3.3 Glass Fragments 21
1.4 Terminology 29
1.5 Types of Data 34
1.6 Populations 36
1.7 Probability 41
1.7.1 Introduction 41
1.7.2 A Standard for Uncertainty 46
1.7.3 Events 55
1.7.4 Classical and Frequentist Definitions of Probability and Their Limitations 57
1.7.5 Subjective Definition of Probability 60
1.7.6 The Quantification of Probability Through a Betting Scheme 64
1.7.7 Probabilities and Frequencies: The Role of Exchangeability 69
1.7.8 Laws of Probability 78
1.7.9 Dependent Events and Background Information 82
1.7.10 Law of Total Probability 91
1.7.11 Updating of Probabilities 96
2 The Evaluation of Evidence 101
2.1 Odds 101
2.1.1 Complementary Events 101
2.1.2 Examples 104
2.1.3 Definition of Odds 105
2.2 Bayes' Theorem 108
2.2.1 Statement of the Theorem 109
2.2.2 Examples 109
2.3 The Odds Form of Bayes' Theorem 121
2.3.1 Likelihood Ratio 121
2.3.2 Bayes' Factor and Likelihood Ratio 125
2.3.3 Three-Way Tables 130
2.3.4 Logarithm of the Likelihood Ratio 134
2.4 The Value of Evidence 138
2.4.1 Evaluation of Forensic Evidence 138
2.4.2 Justification of the Use of the Likelihood Ratio 154
2.4.3 Single Value for the Likelihood Ratio 158
2.4.4 Role of Background Information 161
2.4.5 Summary of Competing Propositions 163
2.4.6 Qualitative Scale for the Value of the Evidence 168
2.5 Errors in Interpretation 180
2.5.1 Fallacy of the Transposed Conditional 186
2.5.2 Source Probability Error 190
2.5.3 Ultimate Issue Error 194
2.5.4 Defence Attorney's Fallacy 194
2.5.5 Probability (Another Match) Error 196
2.5.6 Numerical Conversion Error 199
2.5.7 False Positive Fallacy 202
2.5.8 Expected Value Fallacy 203
2.5.9 Uniqueness 206
2.5.10 Other Difficulties 209
2.5.11 Empirical Evidence of Errors in Interpretation 220
2.6 Misinterpretations 233
2.7 Explanation of Transposed Conditional, Defence Attorney's and False Positive Fallacies 236
2.7.1 Explanation of the Fallacy of the Transposed Conditional 236
2.7.2 Explanation of the Defence Attorney's Fallacy 239
2.7.3 Explanation of the False Positive Fallacy 241
2.8 Making Coherent Decisions 245
2.8.1 Elements of Statistical Decision Theory 246
2.8.2 Decision Analysis: An Example 249
2.9 Graphical Probabilistic Models: Bayesian Networks 254
2.9.1 Elements of the Bayesian Networks 256
2.9.2 The Construction of Bayesian Networks 261
2.9.3 Bayesian Decision Networks (Influence Diagrams) 272
3 Historical Review 279
3.1 Early History 279
3.2 The Dreyfus Case 286
3.3 Statistical Arguments by Early Twentieth- Century Forensic Scientists 293
3.4 People v.Collins 299
3.5 Discriminating Power 307
3.5.1 Derivation 307
3.5.2 Evaluation of Evidence by Discriminating Power 310
3.5.3 Finite Samples 316
3.5.4 Combination of Independent Systems 319
3.5.5 Correlated Attributes 321
3.6 Significance Probabilities 325
3.6.1 Calculation of Significance Probabilities 326
3.6.2 Relationship to Likelihood Ratio 333
3.6.3 Combination of Significance Probabilities 338
3.7 Coincidence Probabilities 342
3.7.1 Introduction 342
3.7.2 Comparison Stage 346
3.7.3 Significance Stage 347
3.8 Likelihood Ratio 351
4 Bayesian Inference 359
4.1 Introduction 359
4.2 Inference for a Proportion 368
4.2.1 Interval Estimation 374
4.2.2 Estimation with Zero Occurrences in a Sample 381
4.2.3 Uncertainty on Sensitivity and Specificity 387
4.3 Sampling 392
4.3.1 Choice of Sample Size in Large Consignments 398
4.3.2 Choice of Sample Size in Small Consignments 413
4.4 Bayesian Networks for Sampling Inspection 420
4.4.1 Large Consignments 420
4.4.2 Small Consignments 425
4.5 Inference for a Normal Mean 429
4.5.1 Known Variance 431
4.5.2 Unknown Variance 438
4.5.3 Interval Estimation 445
4.6 Quantity Estimation 449
4.6.1 Predictive Approach in Small Consignments 452
4.6.2 Predictive Approach in Large Consignments 461
4.7 Decision Analysis 464
4.7.1 Standard Loss Functions 465
4.7.2 Decision Analysis for Forensic Sampling 471
5 Evidence and Propositions: Theory 483
5.1 The Choice of Propositions and Pre-Assessment 483
5.2 Levels of Propositions and Roles of the Forensic Scientist 485
5.3 The Formal Development of a Likelihood Ratio for Different Propositions and Discrete Characteristics 499
5.3.1 Likelihood Ratio with Source Level Propositions 499
5.3.2 Likelihood Ratio with Activity Level Propositions 519
5.3.3 Likelihood Ratio with Offence Level Propositions 553
5.4 Validation of Bayesian Network Structures: An Example 562
5.5 Pre-Assessment 568
5.5.1 Pre-assessment of the Case 568
5.5.2 Pre-assessment of Evidence 575
5.5.3 Pre-assessment: A Practical Example 576
5.6 Combination of Items of Evidence 592
5.6.1 A Difficulty in Combining Evidence: The Problem of Conjunction 594
5.6.2 Generic Patterns of Inference in Combining Evidence 598
6 Evidence and Propositions: Practice 615
6.1 Examples for Evaluation given Source Level Propositions 615
6.1.1 General Population 616
6.1.2 Particular Population 617
6.1.3 A Note on The Appropriate Databases for Evaluation Given Source Level Propositions 619
6.1.4 Two Trace Problem 627
6.1.5 Many Samples 633
6.1.6 Multiple Propositions 637
6.1.7 A Note on Biological Traces 654
6.1.8 Additional Considerations on Source Level Propositions 670
6.2 Examples for Evaluation given Activity Level Propositions 699
6.2.1 A Practical Approach to Fibres Evaluation 701
6.2.2 A Practical Approach to Glass Evaluation 704
6.2.3 The Assignment of Probabilities for Transfer Events 713
6.2.4 The Assignment of Probabilities for Background Traces 734
6.2.5 Presence of Material with Non-corresponding Features 739
6.2.6 Absence of Evidence for Activity Level Propositions 741
6.3 Examples for Evaluation given Offence Level Propositions 745
6.3.1 One Stain, k Offenders 745
6.3.2 Two Stains, One Offender 752
6.3.3 Paternity and The Combination of Likelihood Ratios 756
6.3.4 Probability of Paternity 762
6.3.5 Absence of Evidence for Offence Level Propositions 768
6.3.6 A Note on Relevance and Offence Level Propositions 773
6.4 Summary 774
6.4.1 Stain Known to Have Been Left by Offenders: Source-Level Propositions 774
6.4.2 Material Known to Have Been (or Not to Have Been) Left by Offenders: Activity-Level Propositions 777
6.4.3 Stain May Not Have Been Left by Offenders: Offence-Level Propositions 779
7 Data Analysis 783
7.1 Introduction 783
7.2 Theory for Discrete Data 785
7.2.1 Data of Independent Counts with a Poisson Distribution 787
7.2.2 Data of Independent Counts with a Binomial Distribution 791
7.2.3 Data of Independent Counts with a Multinomial Distribution 793
7.3 Theory for Continuous Univariate Data 798
7.3.1 Assessment of Similarity Only 802
7.3.2 Sources of Variation: Two-Level Models 808
7.3.3 Transfer Probability 810
7.4 Normal Between-Source Variation 814
7.4.1 Marginal Distribution of Measurements 814
7.4.2 Approximate Derivation of the Likelihood Ratio 817
7.4.3 Lindley's Approach 820
7.4.4 Interpretation of Result 825
7.4.5 Examples 827
7.5 Non-normal Between-Source Variation 830
7.5.1 Estimation of a Probability Density Function 831
7.5.2 Kernel Density Estimation for Between-Source Data 842
7.5.3 Examples 844
7.6 Multivariate Analysis 849
7.6.1 Introduction 849
7.6.2 Multivariate Two-Level Models 851
7.6.3 A Note on Sensitivity 864
7.6.4 Case Study for Two-Level Data 865
7.6.5 Three-Level Models 876
7.7 Discrimination 882
7.7.1 Discrete Data 884
7.7.2 Continuous Data 889
7.7.3 Autocorrelated Data 893
7.7.4 Multivariate Data 894
7.7.5 Cut-Offs and Legal Thresholds 899
7.8 Score-Based Models 906
7.8.1 Example 910
7.9 Bayes' Factor and Likelihood Ratio (cont.) 913
8 Assessment of the Performance of Methods for the Evaluation of Evidence 919
8.1 Introduction 919
8.2 Properties of Methods for Evaluation 928
8.3 General Topics Relating to Sample Size Estimation and to Assessment 933
8.3.1 Probability of Strong Misleading Evidence: A Sample Size Problem 933
8.3.2 Calibration 948
8.4 Assessment of Performance of a Procedure for the Calculation of the Likelihood Ratio 952
8.4.1 Histograms and Tippett Plots 956
8.4.2 False Positive Rates, False Negative Rates and DET Plots 959
8.4.3 Empirical Cross-Entropy 961
8.5 Case Study: Kinship Analysis 972
8.6 Conclusion 979
Appendix A Probability Distributions 981
A.1 Introduction 981
A.2 Probability Distributions for Counts 988
A.2.1 Probabilities 988
A.2.2 Summary Measures 990
A.2.3 Binomial Distribution 995
A.2.4 Multinomial Distribution 997
A.2.5 Hypergeometric Distribution 998
A.2.6 Poisson Distribution 1000
A.2.7 Beta-Binomial and Dirichlet-Multinomial Distributions 1002
A.3 Measurements 1005
A.3.1 Summary Statistics 1005
A.3.2 Normal Distribution 1007
A.3.3 Jeffreys' Prior Distributions 1021
A.3.4 Student's t-Distribution 1021
A.3.5 Gamma and Chi-Squared Distributions 1025
A.3.6 Inverse Gamma and Inverse Chi-Squared Distributions 1026
A.3.7 Beta Distribution 1028
A.3.8 Dirichlet Distribution 1032
A.3.9 Multivariate Normal Distribution and Correlation 1035
A.3.10 Wishart Distribution 1040
A.3.11 Inverse Wishart Distribution 1041
Appendix B Matrix Properties 1043
B.1 Matrix Terminology 1043
B.1.1 The Trace of a Square Matrix 1044
B.1.2 The Transpose of a Matrix 1044
B.1.3 Addition of Two Matrices 1045
B.1.4 Determinant of a Matrix 1045
B.1.5 Matrix Multiplication 1046
B.1.6 The Inverse of a Matrix 1048
B.1.7 Completion of the Square 1049
References 1051
Notation 1143
Cases 1157
Author Index 1163
Subject Index 1187
Preface to Third Edition xxi
Preface to Second Edition xxx
Preface to First Edition xxxvii
1 Uncertainty in Forensic Science 1
1.1 Introduction 1
1.2 Statistics and the Law 3
1.3 Uncertainty in Scientific Evidence 11
1.3.1 The Frequentist Method 15
1.3.2 Stains of Body Fluids 17
1.3.3 Glass Fragments 21
1.4 Terminology 29
1.5 Types of Data 34
1.6 Populations 36
1.7 Probability 41
1.7.1 Introduction 41
1.7.2 A Standard for Uncertainty 46
1.7.3 Events 55
1.7.4 Classical and Frequentist Definitions of Probability and Their Limitations 57
1.7.5 Subjective Definition of Probability 60
1.7.6 The Quantification of Probability Through a Betting Scheme 64
1.7.7 Probabilities and Frequencies: The Role of Exchangeability 69
1.7.8 Laws of Probability 78
1.7.9 Dependent Events and Background Information 82
1.7.10 Law of Total Probability 91
1.7.11 Updating of Probabilities 96
2 The Evaluation of Evidence 101
2.1 Odds 101
2.1.1 Complementary Events 101
2.1.2 Examples 104
2.1.3 Definition of Odds 105
2.2 Bayes' Theorem 108
2.2.1 Statement of the Theorem 109
2.2.2 Examples 109
2.3 The Odds Form of Bayes' Theorem 121
2.3.1 Likelihood Ratio 121
2.3.2 Bayes' Factor and Likelihood Ratio 125
2.3.3 Three-Way Tables 130
2.3.4 Logarithm of the Likelihood Ratio 134
2.4 The Value of Evidence 138
2.4.1 Evaluation of Forensic Evidence 138
2.4.2 Justification of the Use of the Likelihood Ratio 154
2.4.3 Single Value for the Likelihood Ratio 158
2.4.4 Role of Background Information 161
2.4.5 Summary of Competing Propositions 163
2.4.6 Qualitative Scale for the Value of the Evidence 168
2.5 Errors in Interpretation 180
2.5.1 Fallacy of the Transposed Conditional 186
2.5.2 Source Probability Error 190
2.5.3 Ultimate Issue Error 194
2.5.4 Defence Attorney's Fallacy 194
2.5.5 Probability (Another Match) Error 196
2.5.6 Numerical Conversion Error 199
2.5.7 False Positive Fallacy 202
2.5.8 Expected Value Fallacy 203
2.5.9 Uniqueness 206
2.5.10 Other Difficulties 209
2.5.11 Empirical Evidence of Errors in Interpretation 220
2.6 Misinterpretations 233
2.7 Explanation of Transposed Conditional, Defence Attorney's and False Positive Fallacies 236
2.7.1 Explanation of the Fallacy of the Transposed Conditional 236
2.7.2 Explanation of the Defence Attorney's Fallacy 239
2.7.3 Explanation of the False Positive Fallacy 241
2.8 Making Coherent Decisions 245
2.8.1 Elements of Statistical Decision Theory 246
2.8.2 Decision Analysis: An Example 249
2.9 Graphical Probabilistic Models: Bayesian Networks 254
2.9.1 Elements of the Bayesian Networks 256
2.9.2 The Construction of Bayesian Networks 261
2.9.3 Bayesian Decision Networks (Influence Diagrams) 272
3 Historical Review 279
3.1 Early History 279
3.2 The Dreyfus Case 286
3.3 Statistical Arguments by Early Twentieth- Century Forensic Scientists 293
3.4 People v.Collins 299
3.5 Discriminating Power 307
3.5.1 Derivation 307
3.5.2 Evaluation of Evidence by Discriminating Power 310
3.5.3 Finite Samples 316
3.5.4 Combination of Independent Systems 319
3.5.5 Correlated Attributes 321
3.6 Significance Probabilities 325
3.6.1 Calculation of Significance Probabilities 326
3.6.2 Relationship to Likelihood Ratio 333
3.6.3 Combination of Significance Probabilities 338
3.7 Coincidence Probabilities 342
3.7.1 Introduction 342
3.7.2 Comparison Stage 346
3.7.3 Significance Stage 347
3.8 Likelihood Ratio 351
4 Bayesian Inference 359
4.1 Introduction 359
4.2 Inference for a Proportion 368
4.2.1 Interval Estimation 374
4.2.2 Estimation with Zero Occurrences in a Sample 381
4.2.3 Uncertainty on Sensitivity and Specificity 387
4.3 Sampling 392
4.3.1 Choice of Sample Size in Large Consignments 398
4.3.2 Choice of Sample Size in Small Consignments 413
4.4 Bayesian Networks for Sampling Inspection 420
4.4.1 Large Consignments 420
4.4.2 Small Consignments 425
4.5 Inference for a Normal Mean 429
4.5.1 Known Variance 431
4.5.2 Unknown Variance 438
4.5.3 Interval Estimation 445
4.6 Quantity Estimation 449
4.6.1 Predictive Approach in Small Consignments 452
4.6.2 Predictive Approach in Large Consignments 461
4.7 Decision Analysis 464
4.7.1 Standard Loss Functions 465
4.7.2 Decision Analysis for Forensic Sampling 471
5 Evidence and Propositions: Theory 483
5.1 The Choice of Propositions and Pre-Assessment 483
5.2 Levels of Propositions and Roles of the Forensic Scientist 485
5.3 The Formal Development of a Likelihood Ratio for Different Propositions and Discrete Characteristics 499
5.3.1 Likelihood Ratio with Source Level Propositions 499
5.3.2 Likelihood Ratio with Activity Level Propositions 519
5.3.3 Likelihood Ratio with Offence Level Propositions 553
5.4 Validation of Bayesian Network Structures: An Example 562
5.5 Pre-Assessment 568
5.5.1 Pre-assessment of the Case 568
5.5.2 Pre-assessment of Evidence 575
5.5.3 Pre-assessment: A Practical Example 576
5.6 Combination of Items of Evidence 592
5.6.1 A Difficulty in Combining Evidence: The Problem of Conjunction 594
5.6.2 Generic Patterns of Inference in Combining Evidence 598
6 Evidence and Propositions: Practice 615
6.1 Examples for Evaluation given Source Level Propositions 615
6.1.1 General Population 616
6.1.2 Particular Population 617
6.1.3 A Note on The Appropriate Databases for Evaluation Given Source Level Propositions 619
6.1.4 Two Trace Problem 627
6.1.5 Many Samples 633
6.1.6 Multiple Propositions 637
6.1.7 A Note on Biological Traces 654
6.1.8 Additional Considerations on Source Level Propositions 670
6.2 Examples for Evaluation given Activity Level Propositions 699
6.2.1 A Practical Approach to Fibres Evaluation 701
6.2.2 A Practical Approach to Glass Evaluation 704
6.2.3 The Assignment of Probabilities for Transfer Events 713
6.2.4 The Assignment of Probabilities for Background Traces 734
6.2.5 Presence of Material with Non-corresponding Features 739
6.2.6 Absence of Evidence for Activity Level Propositions 741
6.3 Examples for Evaluation given Offence Level Propositions 745
6.3.1 One Stain, k Offenders 745
6.3.2 Two Stains, One Offender 752
6.3.3 Paternity and The Combination of Likelihood Ratios 756
6.3.4 Probability of Paternity 762
6.3.5 Absence of Evidence for Offence Level Propositions 768
6.3.6 A Note on Relevance and Offence Level Propositions 773
6.4 Summary 774
6.4.1 Stain Known to Have Been Left by Offenders: Source-Level Propositions 774
6.4.2 Material Known to Have Been (or Not to Have Been) Left by Offenders: Activity-Level Propositions 777
6.4.3 Stain May Not Have Been Left by Offenders: Offence-Level Propositions 779
7 Data Analysis 783
7.1 Introduction 783
7.2 Theory for Discrete Data 785
7.2.1 Data of Independent Counts with a Poisson Distribution 787
7.2.2 Data of Independent Counts with a Binomial Distribution 791
7.2.3 Data of Independent Counts with a Multinomial Distribution 793
7.3 Theory for Continuous Univariate Data 798
7.3.1 Assessment of Similarity Only 802
7.3.2 Sources of Variation: Two-Level Models 808
7.3.3 Transfer Probability 810
7.4 Normal Between-Source Variation 814
7.4.1 Marginal Distribution of Measurements 814
7.4.2 Approximate Derivation of the Likelihood Ratio 817
7.4.3 Lindley's Approach 820
7.4.4 Interpretation of Result 825
7.4.5 Examples 827
7.5 Non-normal Between-Source Variation 830
7.5.1 Estimation of a Probability Density Function 831
7.5.2 Kernel Density Estimation for Between-Source Data 842
7.5.3 Examples 844
7.6 Multivariate Analysis 849
7.6.1 Introduction 849
7.6.2 Multivariate Two-Level Models 851
7.6.3 A Note on Sensitivity 864
7.6.4 Case Study for Two-Level Data 865
7.6.5 Three-Level Models 876
7.7 Discrimination 882
7.7.1 Discrete Data 884
7.7.2 Continuous Data 889
7.7.3 Autocorrelated Data 893
7.7.4 Multivariate Data 894
7.7.5 Cut-Offs and Legal Thresholds 899
7.8 Score-Based Models 906
7.8.1 Example 910
7.9 Bayes' Factor and Likelihood Ratio (cont.) 913
8 Assessment of the Performance of Methods for the Evaluation of Evidence 919
8.1 Introduction 919
8.2 Properties of Methods for Evaluation 928
8.3 General Topics Relating to Sample Size Estimation and to Assessment 933
8.3.1 Probability of Strong Misleading Evidence: A Sample Size Problem 933
8.3.2 Calibration 948
8.4 Assessment of Performance of a Procedure for the Calculation of the Likelihood Ratio 952
8.4.1 Histograms and Tippett Plots 956
8.4.2 False Positive Rates, False Negative Rates and DET Plots 959
8.4.3 Empirical Cross-Entropy 961
8.5 Case Study: Kinship Analysis 972
8.6 Conclusion 979
Appendix A Probability Distributions 981
A.1 Introduction 981
A.2 Probability Distributions for Counts 988
A.2.1 Probabilities 988
A.2.2 Summary Measures 990
A.2.3 Binomial Distribution 995
A.2.4 Multinomial Distribution 997
A.2.5 Hypergeometric Distribution 998
A.2.6 Poisson Distribution 1000
A.2.7 Beta-Binomial and Dirichlet-Multinomial Distributions 1002
A.3 Measurements 1005
A.3.1 Summary Statistics 1005
A.3.2 Normal Distribution 1007
A.3.3 Jeffreys' Prior Distributions 1021
A.3.4 Student's t-Distribution 1021
A.3.5 Gamma and Chi-Squared Distributions 1025
A.3.6 Inverse Gamma and Inverse Chi-Squared Distributions 1026
A.3.7 Beta Distribution 1028
A.3.8 Dirichlet Distribution 1032
A.3.9 Multivariate Normal Distribution and Correlation 1035
A.3.10 Wishart Distribution 1040
A.3.11 Inverse Wishart Distribution 1041
Appendix B Matrix Properties 1043
B.1 Matrix Terminology 1043
B.1.1 The Trace of a Square Matrix 1044
B.1.2 The Transpose of a Matrix 1044
B.1.3 Addition of Two Matrices 1045
B.1.4 Determinant of a Matrix 1045
B.1.5 Matrix Multiplication 1046
B.1.6 The Inverse of a Matrix 1048
B.1.7 Completion of the Square 1049
References 1051
Notation 1143
Cases 1157
Author Index 1163
Subject Index 1187
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