# Statistics and the Evaluation of Evidence for Forensic Scientists, 3rd Edition PDF by Colin Aitken, Franco Taroni and Silvia Bozza

## Statistics and the Evaluation of Evidence for Forensic Scientists, Third Edition

By Colin Aitken, Franco Taroni and Silvia Bozza

Contents

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

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

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 InverseWishart 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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