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Franco Taroni. Statistics and the Evaluation of Evidence for Forensic Scientists
Table of Contents
List of Tables
List of Illustrations
Guide
Pages
Wiley Series in Statistics in Practice
Statistics and the Evaluation of Evidence for Forensic Scientists
Dedication
Foreword
Preface to Third Edition
Preface to Second Edition
Note
1Uncertainty in Forensic Science
1.1 Introduction
1.2 Statistics and the Law
1.3 Uncertainty in Scientific Evidence
1.3.1 The Frequentist Method
1.3.2 Stains of Body Fluids
1.3.3 Glass Fragments
1.4 Terminology
1.5 Types of Data
1.6 Populations
1.7 Probability
1.7.1 Introduction
1.7.2 A Standard for Uncertainty
1.7.3 Events
1.7.4 Classical and Frequentist Definitions of Probability and Their Limitations
1.7.5 Subjective Definition of Probability
1.7.6 The Quantification of Probability Through a Betting Scheme
1.7.7 Probabilities and Frequencies: The Role of Exchangeability
1.7.8 Laws of Probability
First Law of Probability
Second Law of Probability
1.7.9 Dependent Events and Background Information
First law of probability for dependent events
Second law of probability for dependent events
Third law of probability for dependent events
1.7.9.1 Independence
Third law of probability for independent events
1.7.10 Law of Total Probability
Law of Total Probability
1.7.11 Updating of Probabilities
Example Example 1.4
Notes
2The Evaluation of Evidence
2.1 Odds
2.1.1 Complementary Events
2.1.2 Examples
2.1.3 Definition of Odds
2.2 Bayes' Theorem
2.2.1 Statement of the Theorem
2.2.2 Examples
2.3 The Odds Form of Bayes' Theorem
2.3.1 Likelihood Ratio
2.3.2 Bayes' Factor and Likelihood Ratio
2.3.2.1 Derivation of the Bayes' Factor: Simple Versus Simple Propositions
2.3.2.2 Derivation of the Bayes' Factor: Simple Versus Composite Propositions
2.3.3 Three‐Way Tables
2.3.4 Logarithm of the Likelihood Ratio
2.4 The Value of Evidence
2.4.1 Evaluation of Forensic Evidence
2.4.2 Justification of the Use of the Likelihood Ratio
2.4.3 Single Value for the Likelihood Ratio
2.4.4 Role of Background Information
2.4.5 Summary of Competing Propositions
2.4.6 Qualitative Scale for the Value of the Evidence
2.5 Errors in Interpretation
2.5.1 Fallacy of the Transposed Conditional
2.5.2 Source Probability Error
2.5.3 Ultimate Issue Error
2.5.4 Defence Attorney's Fallacy
2.5.5 Probability (Another Match) Error
2.5.6 Numerical Conversion Error
2.5.7 False Positive Fallacy
2.5.8 Expected Value Fallacy
2.5.9 Uniqueness
2.5.10 Other Difficulties
2.5.10.1 Relative Frequency of Occurrence
2.5.10.2 ‘Could Have’ Approach
2.5.10.3 ‘Cannot Be Excluded’ Approach
2.5.10.4 ‘Consistent with’ Approach
2.5.11 Empirical Evidence of Errors in Interpretation
2.6 Misinterpretations
2.7 Explanation of Transposed Conditional, Defence Attorney's and False Positive Fallacies
2.7.1 Explanation of the Fallacy of the Transposed Conditional
2.7.2 Explanation of the Defence Attorney's Fallacy
2.7.3 Explanation of the False Positive Fallacy
2.8 Making Coherent Decisions
2.8.1 Elements of Statistical Decision Theory
2.8.2 Decision Analysis: An Example
2.9 Graphical Probabilistic Models: Bayesian Networks
2.9.1 Elements of the Bayesian Networks
2.9.2 The Construction of Bayesian Networks
2.9.2.1 ‐Separation Properties
2.9.2.2 Chain Rule for the Bayesian Network
2.9.2.3 Bayesian Network Examples
2.9.3 Bayesian Decision Networks (Influence Diagrams)
Notes
3Historical Review
3.1 Early History
3.2 The Dreyfus Case
3.3 Statistical Arguments by Early Twentieth‐Century Forensic Scientists
3.4 People v. Collins
3.5 Discriminating Power
3.5.1 Derivation
3.5.2 Evaluation of Evidence by Discriminating Power
3.5.3 Finite Samples
3.5.4 Combination of Independent Systems
3.5.5 Correlated Attributes
3.6 Significance Probabilities
3.6.1 Calculation of Significance Probabilities
3.6.2 Relationship to Likelihood Ratio
3.6.3 Combination of Significance Probabilities
3.7 Coincidence Probabilities
3.7.1 Introduction
3.7.2 Comparison Stage
3.7.3 Significance Stage
3.8 Likelihood Ratio
Notes
4Bayesian Inference
4.1 Introduction
4.2 Inference for a Proportion
4.2.1 Interval Estimation
4.2.2 Estimation with Zero Occurrences in a Sample
4.2.3 Uncertainty on Sensitivity and Specificity
4.3 Sampling
4.3.1 Choice of Sample Size in Large Consignments
4.3.2 Choice of Sample Size in Small Consignments
4.4 Bayesian Networks for Sampling Inspection
4.4.1 Large Consignments
4.4.2 Small Consignments
4.5 Inference for a Normal Mean
4.5.1 Known Variance
4.5.2 Unknown Variance
4.5.3 Interval Estimation
4.6 Quantity Estimation
4.6.1 Predictive Approach in Small Consignments
Estimative Approach
4.6.2 Predictive Approach in Large Consignments
4.7 Decision Analysis
4.7.1 Standard Loss Functions
4.7.2 Decision Analysis for Forensic Sampling
Notes
5Evidence and Propositions: Theory. 5.1 The Choice of Propositions and Pre‐Assessment
5.2 Levels of Propositions and Roles of the Forensic Scientist
5.3 The Formal Development of a Likelihood Ratio for Different Propositions and Discrete Characteristics
5.3.1 Likelihood Ratio with Source Level Propositions
5.3.1.1 General Population
5.3.1.2 Particular Population
5.3.1.3 Scene‐ and Suspect‐Anchored Perspectives
5.3.1.4 Some Remarks on Scene‐ and Suspect‐Anchored Perspectives
5.3.1.5 Types of Evidence: Recovered, Control, and Background Data
5.3.2 Likelihood Ratio with Activity Level Propositions
5.3.2.1 Transfer Material Left by an Offender
5.3.2.2 Transfer Material Not Left by the Offender
5.3.2.3 Innocent Transfer of Material
5.3.2.4 Uncertainty About the True Source
5.3.2.5 Cross‐ (or Two‐Way) Transfer of Trace Material
5.3.3 Likelihood Ratio with Offence Level Propositions
5.3.3.1 Probabilities of Innocent Acquisition and Relevance
5.3.3.2 Association Propositions
5.3.3.3 Intermediate Association Propositions
5.3.3.4 A Note on Evidence Assessment Given Offence Level Propositions
5.4 Validation of Bayesian Network Structures: An Example
5.5 Pre‐Assessment
5.5.1 Pre‐assessment of the Case
5.5.2 Pre‐assessment of Evidence
5.5.3 Pre‐assessment: A Practical Example
5.5.3.1 Task‐Relevant Information
5.5.3.2 Formulation of Propositions of Interest and Events
5.5.3.3 Expected Likelihood Ratios
5.5.3.4 The Relevant Population of Fibres
5.5.3.5 The Defence and a Change in the Proposition
5.6 Combination of Items of Evidence
5.6.1 A Difficulty in Combining Evidence: The Problem of Conjunction
5.6.2 Generic Patterns of Inference in Combining Evidence
5.6.2.1 Dissonant Evidence: Contradiction and Conflict
5.6.2.2 Harmonious Evidence: Corroboration and Convergence
Notes
6Evidence and Propositions: Practice
6.1 Examples for Evaluation given Source Level Propositions
6.1.1 General Population
6.1.2 Particular Population
6.1.3 A Note on The Appropriate Databases for Evaluation Given Source Level Propositions
6.1.4 Two Trace Problem
6.1.5 Many Samples. 6.1.5.1 Many Different Profiles
6.1.5.2 General Cases
6.1.6 Multiple Propositions
6.1.6.1 Multiple Propositions: Posterior Probabilities
6.1.6.2 Multiple Propositions: Bayes' Factor
6.1.7 A Note on Biological Traces
6.1.8 Additional Considerations on Source Level Propositions
6.1.8.1 A Probabilistic Approach to Database Searching
6.1.8.2 Search and Selection Effect (Double Counting Error)
6.1.8.3 The Island Problem
6.1.8.4 A Probabilistic Approach to Laboratory Error
6.2 Examples for Evaluation given Activity Level Propositions
6.2.1 A Practical Approach to Fibres Evaluation
6.2.2 A Practical Approach to Glass Evaluation
6.2.2.1 Case 1
6.2.2.2 Case 2
6.2.2.3 Case 3
6.2.2.4 Case 4
6.2.3 The Assignment of Probabilities for Transfer Events
6.2.3.1 Probabilities of Transfer in The Context of DNA
6.2.3.2 Transfer Probabilities and Micro‐traces
6.2.3.3 Additional Considerations on Activity Level Propositions and Transfer Probabilities
6.2.4 The Assignment of Probabilities for Background Traces
6.2.5 Presence of Material with Non‐corresponding Features
6.2.6 Absence of Evidence for Activity Level Propositions. 6.2.6.1 A Question of Terminology
6.2.6.2 Some Practical Examples
6.3 Examples for Evaluation given Offence Level Propositions. 6.3.1 One Stain, Offenders
6.3.2 Two Stains, One Offender
6.3.3 Paternity and The Combination of Likelihood Ratios
6.3.4 Probability of Paternity
6.3.5 Absence of Evidence for Offence Level Propositions
6.3.6 A Note on Relevance and Offence Level Propositions
6.4 Summary
6.4.1 Stain Known to Have Been Left by Offenders: Source‐Level Propositions. 6.4.1.1 One Stain Known to Have Come from One Offender
6.4.1.2 Two Stains, One from Each of Two Offenders
6.4.1.3 Multiple () Stains, One from Each of Offenders
6.4.1.4 Multiple () Stains, Different Profiles, Different Offenders
6.4.1.5 Multiple () Stains, Different Profiles, Offenders
6.4.2 Material Known to Have Been (or Not to Have Been) Left by Offenders: Activity‐Level Propositions. 6.4.2.1 Material Left by The Offender
6.4.2.2 Material Not Left by The Offender
6.4.2.3 Innocent Transfer of Material
6.4.2.4 Material Left by The Offender (Uncertainty About The True Source)
6.4.3 Stain May Not Have Been Left by Offenders: Offence‐Level Propositions. 6.4.3.1 One Stain, Offenders
6.4.3.2 Two Stains, One of Which is Relevant, One Offender
6.4.3.3 Two Stains, Neither of Which May Be Relevant, One Offender
Notes
7Data Analysis. 7.1 Introduction
7.2 Theory for Discrete Data
7.2.1 Data of Independent Counts with a Poisson Distribution
7.2.2 Data of Independent Counts with a Binomial Distribution
7.2.3 Data of Independent Counts with a Multinomial Distribution
7.3 Theory for Continuous Univariate Data
7.3.1 Assessment of Similarity Only
7.3.2 Sources of Variation: Two‐Level Models
7.3.3 Transfer Probability
7.4 Normal Between‐Source Variation
7.4.1 Marginal Distribution of Measurements
7.4.2 Approximate Derivation of the Likelihood Ratio
7.4.3 Lindley's Approach
7.4.4 Interpretation of Result
7.4.5 Examples
7.5 Non‐normal Between‐Source Variation
7.5.1 Estimation of a Probability Density Function
7.5.2 Kernel Density Estimation for Between‐Source Data
7.5.3 Examples. 7.5.3.1 Medullary Widths of Cat Hairs
7.5.3.2 Refractive Index
7.6 Multivariate Analysis. 7.6.1 Introduction
7.6.2 Multivariate Two‐Level Models
7.6.2.1 Parameter Estimation
7.6.2.2 Likelihood Ratio Using a Multivariate Random Effects Model and Assumptions of Normality
7.6.2.3 Likelihood Ratio Using a Multivariate Random Effects Model and Non‐normal Between‐Source Variation
7.6.2.4 Non‐constant Within‐Group Covariance Matrix
7.6.3 A Note on Sensitivity
7.6.4 Case Study for Two‐Level Data
7.6.4.1 Normal Density for Between‐Group Distribution
7.6.4.2 Kernel Density for Between‐Group Distribution
7.6.5 Three‐Level Models
7.6.5.1 Parameter Estimation
7.6.5.2 Likelihood Ratio Using a Multivariate Random Effects Model and Assumptions of Normality
7.7 Discrimination
7.7.1 Discrete Data
7.7.2 Continuous Data
7.7.3 Autocorrelated Data
7.7.4 Multivariate Data
7.7.5 Cut‐Offs and Legal Thresholds
7.8 Score‐Based Models
7.8.1 Example
7.9 Bayes' Factor and Likelihood Ratio (cont.)
Notes
8Assessment of the Performance of Methods for the Evaluation of Evidence
8.1 Introduction
8.2 Properties of Methods for Evaluation
8.3 General Topics Relating to Sample Size Estimation and to Assessment
8.3.1 Probability of Strong Misleading Evidence: A Sample Size Problem
8.3.2 Calibration
8.4 Assessment of Performance of a Procedure for the Calculation of the Likelihood Ratio
8.4.1 Histograms and Tippett Plots
8.4.2 False Positive Rates, False Negative Rates and DET Plots
8.4.3 Empirical Cross‐Entropy
8.4.3.1 Pool Adjacent Violators Algorithm
8.4.3.2 Implementation of the PAV Algorithm
8.4.3.3 Transformation of ECE
8.5 Case Study: Kinship Analysis
8.6 Conclusion
Notes
A Probability Distributions
A.1 Introduction
A.2 Probability Distributions for Counts
A.2.1 Probabilities
A.2.2 Summary Measures
A.2.3 Binomial Distribution
A.2.4 Multinomial Distribution
A.2.5 Hypergeometric Distribution
A.2.6 Poisson Distribution
A.2.7 Beta‐Binomial and Dirichlet‐Multinomial Distributions
A.3 Measurements
A.3.1 Summary Statistics
A.3.2 Normal Distribution
A.3.2.1 Normal Approximations to the Binomial and Poisson Distributions
A.3.3 Jeffreys' Prior Distributions
A.3.4 Student's ‐Distribution
A.3.5 Gamma and Chi‐Squared Distributions
A.3.6 Inverse Gamma and Inverse Chi‐Squared Distributions
A.3.7 Beta Distribution
A.3.8 Dirichlet Distribution
A.3.9 Multivariate Normal Distribution and Correlation
A.3.10 Wishart Distribution
A.3.11 Inverse Wishart Distribution
Notes
B Matrix Properties. B.1 Matrix Terminology
B.1.1 The Trace of a Square Matrix
B.1.2 The Transpose of a Matrix
B.1.3 Addition of Two Matrices
B.1.4 Determinant of a Matrix
B.1.5 Matrix Multiplication
B.1.6 The Inverse of a Matrix
B.1.7 Completion of the Square
References
1 Notation
Cases
Author Index
Subject Index
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