# Handbook of Forensic Statistics

Section I: Perspectives on Forensic StatisticsThe History of Forensic Inference and Statistics: A Thematic PerspectiveIntroductionForensic Science and the Evaluation of EvidenceThe Need for an Interpretative ModelSupport of Judicial Disciplines for a Scientific Presentation of the Value of EvidenceProbability of Proposition Given Evidence and of Evidence Given PropositionQuantification of the Value of Evidence Using Alternative Numerical SummariesChange from Two-Stage Approach to Continuous ApproachPresentation of Evidence: New Challenges to SolveThe Island Problem and Results of a Database SelectionProfile Probability vs Conditional Profile ProbabilityEvaluation by Taking Errors into AccountA Minimum Value for the Profile ProbabilityPropositions and Pre-AssessmentThe Choice of PropositionsThe Pre-AssessmentTranslation of a Numerical Value into a Verbal EquivalentAssessment of PerformanceRole for Likelihood Ratio as aMeasure for Investigation as Well as for EvaluationProbabilistic GraphicalModelsBayesian NetworksBayesian Networks to Manage ‘Masses’ of EvidenceBayesian Networks in Judicial ContextsBayesian Networks in Forensic Science: Particular Case ModelingBayesian Networks in Forensic Science: Generic Patterns of InferenceNot Only Inference: The Way to Make a DecisionThe Objectives and Ingredients of Decision TheoryGraphical ModelsThe Existence or Otherwise of a True Value of the EvidenceAcknowledgmentsReferencesSection II: General Concepts andMethodsFrequentist Methods for Statistical InferenceIntroductionDefinitions and NotationData and EvidenceRandom Variables and Probability DistributionsSampling from a Distribution or PopulationEstimationProperties of Point EstimatorsEstimating Allele ProportionsA Point EstimateConstructing a Confidence IntervalChoosing a Confidence CoefficientEstimating a False Positive Probability Through an ExperimentThe Design of Experiments to Test Categorical SourceAn Experiment to Test Categorical Judgments of Latent Print ExaminersConstructing Confidence IntervalsInterpreting Confidence Intervalsp-Valuesp-Values in a Comparison of Glass FragmentsInterpreting p-ValuesHypothesis TestsClassical Hypothesis Tests for Refractive Index MatchingType I Errors and the Size of a TestType II Errors and the Power of a TestHypothesis Testing with p-ValuesHypothesis Testing with Confidence IntervalsIssues in Interpreting the Results of Hypothesis Tests, p-Values, and Confidence CoefficientsTranspositionMultiple Tests: Proof of the Null Hypothesis and Adjusted p-ValuesArbitrary LinesAlternatives and LikelihoodsResampling MethodsBootstrap EstimatesPermutation TestsAcknowledgmentsReferencesBayesian Methods and Forensic InferenceIntroductionThe BasicsA Beta-Binomial Mock ExampleA Gamma-Poisson Mock ExampleMarkov Chain Monte CarloBroad ApplicationsSummaryAcknowledgmentsReferencesComparing Philosophies of Statistical InferenceInferential PhilosophiesFrequentist InferenceBayesian InferenceOther Approaches to InferenceFiducial InferenceLikelihood InferenceConfidence DistributionsComparing the ApproachesPlanning Studies Using Frequentist InferenceChallenges for Frequentist InferenceFlexible Inference with Bayesian MethodsModel Modifications and AdjustmentsThe Prior Distribution and the Definition of ProbabilityRelevance to Forensic StatisticsLikelihood Ratios and Bayes FactorsTwo-Stage Procedures in Forensic ScienceForensic Evidence as Expert Opinion and Error RatesSummaryReferencesDecision TheoryIntroductionConcepts of Statistical Decision TheoryPreliminaries: Basic Elements of Decision ProblemsUtility TheoryImplications of the Expected Utility Maximisation PrincipleThe Loss FunctionParticular Forms of the Expected Utility Maximisation PrincipleLikelihood Ratios in the Decision FrameworkDecision Theory in the Law and Forensic ScienceLegal ApplicationsForensic Science ApplicationsForensic IdentificationUnderstanding Probability Assignment as a Decision: The Use of Proper Scoring RulesOther Forensic Decision Problems: Consignment InspectionDiscussion and ConclusionsFurther ReadingsForensic ScienceGeneralAcknowledgmentsReferencesAssociation Does Not Imply Discrimination: Clarifying When Matches Are (and Are Not) MeaningfulIntroductionAssociation and DiscriminationQuality of Test: Sensitivity and SpecificitySources of ErrorWeight of Evidence: The Likelihood RatioUseful Databases for Ascertaining Discriminatory PowerConflating Conditional Statements: The Prosecutor’s FallacyExamples: The Discriminatory Power of Forensic EvidenceArson InvestigationOther Types of Forensic Evidence: DNA, Fingerprints, and Shoe PrintsAbusive Head TraumaConclusionReferencesValidation of Forensic Automatic Likelihood Ratio MethodsIntroductionScopeAimStructureValidation ProcessStandardizationValidation of Theoretical and Empirical AspectsPerformance Characteristics for Automatic LR MethodsEmpirical ValidationValidation ProtocolPrimary Performance CharacteristicsPerformance of Probabilities by Proper Scoring RulesDiscrimination and Calibration of ProbabilitiesPerformance of Likelihood RatiosProperties of Well-Calibrated Likelihood RatiosExamples with Primary Performance CharacteristicsSecondary Performance CharacteristicsRobustnessMonotonicityGeneralizationConclusionReferencesBayesian Networks in Forensic ScienceIntroductionProbability LogicSimple Bayesian Networks for Forensic ProblemsObject-Oriented Bayesian NetworksForensic GeneticsBayesian Networks for Simple Criminal IdentificationSimple Disputed PaternityBayesian Networks for Complex Criminal Cases Involving Family RelationshipsMutationBayesian Networks for Analysing Mixed DNA ProfilesDiscrete FeaturesContinuous FeaturesAnalysis of Sensitivity to Assumptions on Founder GenesUncertainty in Allele FrequenciesHeterogeneous Reference PopulationConclusionsAppendix 8A: Bayesian Network BasicsA.1 Qualitative StructureA.2 Independence PropertiesA.3 Quantitative StructureA.4 ComputationReferencesSection III: Legal and Psychological DimensionsHow Well Do Lay People Comprehend Statistical Statements from Forensic Scientists?Methodological OverviewConsistencyFramingFormatSensitivity(In)CoherenceProsecutor’s FallacyDefense Attorney’s FallacyDirectional ErrorsAggregation ErrorsAbilityOrthodoxyDiscussionConclusionReferencesForensic Statistics in the CourtroomThe Purpose, Form, and Prerequisites of Expert TestimonyLay and Expert TestimonyQualifications for Statistical Experts (and Experts Who Use Statistics)Forms of Statistical Expert TestimonyReasonable Scientific or Statistical CertaintySpecial Rules for Scientific Expert Testimony*The General-Acceptance StandardThe Scientific-Validity StandardSelected Evidentiary Issues in Forensic StatisticsTwo Uses of Statistical Analysis as EvidenceTheory and ApplicationError Rates in Determining AdmissibilityError Rates, Likelihood Ratios, and Bayes Factors for Quantifying Probative ValueConclusionReferencesCases and RulesSection IV: Applications of Statistics to Particular Fields in Forensic ScienceDNA Frequencies and ProbabilitiesIntroductionLikelihood RatiosPopulation GeneticsSingle LociMultiple LociPopulation StructureLineage MarkersMixturesSemi-Continuous ModelContinuous ModelDNA Sequence DataFuture DirectionsConclusionsReferencesKinshipIntroductionGenetic Models for Allele and Genotype FrequenciesPopulation StructureRelatednessRelatedness and Population StructureParentage Calculations for Structured PopulationsIdentifying RemainsSNP DataGenealogy DataConclusionsReferencesStatistical Support for Conclusions in Fingerprint ExaminationsIntroductionFingerprint Examination Framework (ACE-V)Analysis StageComparison StageEvaluation StageVerification StageDecision Making in the ACE-V FrameworkSummaryCommon Source vs. Specific Source ScenariosCommon Source ScenarioSpecific Source ScenarioSimulationsSummaryWeight of Fingerprint EvidenceSimilarity Metrics and Kernel FunctionsScore-Based Likelihood RatiosCommon Source Score-based ModelsSuspect-Centred Score-based ModelsTrace-Centred Score-based ModelsApproximate Bayes Factor for Fingerprint EvidenceSummaryFactors Affecting Examiners’ Decision Making and Error RatesDecision Making During ACEPCAST ControversyStatistical Analysis of Experiments Designed to Study the ACE-V FrameworkSummaryU.S. Defense Forensic Science Center’sSummaryConclusionsReferencesProbabilistic Considerations When Interpreting Database Search and Selection EffectsIntroductionProbabilistic AnalysisScientific Debate: The Database Search ControversyThe Number of Adventitious MatchesReporting DatabaseMatchesComparing a “Probable Cause” and a “Database Search” MatchThe Other Evidence in the CaseGuidance on Interpreting Database MatchesDealing with “Selection Effects”More GenerallyInteresting CasesConclusionReferencesComparing Handwriting in Questioned DocumentsIntroductionOutline of This ChapterHistorical BackgroundRoman LawEnglish LawUnited States LawChallenges in Studying Questioned DocumentsThe Art of Conducting a Handwriting ExaminationPrinciples of Handwriting AnalysisThe ACE-V Process of Comparing HandwritingA Probability Scale for Expressing ConclusionsStatistical Approaches to Handwriting AnalysisComparing Howland Will SignaturesA Bayesian Approach to Comparing SignaturesA Dynamic Model for HandwritingProficiency Studies and Error RatesDiscussionAcknowledgmentsReferencesCases and StatutesAn Introduction to Firearms Examination for Researchers in StatisticsIntroductionThe Anatomy of Guns and of AmmunitionA Brief History of Firearms ExaminationMicroscopic ImperfectionsComparison of Cartridge Case MarksComparison ofMarks on Land Engraved Areas of BulletsPairwise ComparisonsFrom Land-to-Land Comparisons to Bullet-to-Bullet ComparisonsResults for the Hamby Set of BulletsRevisiting the Question of SourceSome Final ThoughtsReferencesShoeprints: The Path from Practice to ScienceIntroductionThe Process of Evaluating ShoeprintsThe Challenges that Arise from Analyzing the Working ProcedureRequirements for Good PracticeHow to Strengthen the Scientific Foundation for Footwear Analysis?Wishlist for and Design of the Semi-Automated ElementAccidentalSensorAn Early Stage Implementation of the Semi-Automated ElementAccidentalSensorPreprocessing and BinarizationElement DetectionDetecting Accidentals and WearDiscussion and ConclusionsAppendix 17AA.1 Shoeprint Comparison StagesA.2 The Conclusion ScaleReferencesForensic Glass EvidenceIntroductionEarly Cases of Forensic Glass EvidenceTypes of GlassGlass Breakage and RecoveryPhysical and Optical PropertiesChemical CompositionMethods forMeasuring Glass Elemental ConcentrationsICP-MS and LA-ICP-MSMicro X-Ray Fluorescence (XRF)Multivariate MethodsGlass DataData SetsVariabilityValidating ProceduresSimulation StrategySimulated Match RatesAnother Interpretation of “Difference”ConclusionsReferencesEstimation of Insect Age for Assessing Minimum Post-Mortem Interval in Forensic Entomology CaseworkBackgroundIsomorphen and Isomegalen DiagramsThermal Summation ModelsCurvilinear ModelsSpectralMeasurementsGrowth Curve ReconstructionConclusionReferencesStatistical Models in Forensic Voice ComparisonIntroductionFeature ExtractionMel-Frequency Cepstral Coefficients (MFCCs)Deltas and Double DeltasVoice-Activity Detection (VAD) and DiarizationMismatch Compensation in the Feature DomainCepstral-Mean Subtraction (CMS) and Cepstral-Mean-and-Variance Normalization (CMVN)Feature WarpingGMM-UBMTraining the Relevant-Population Model (the UBM): Expectation Maximization (EM) AlgorithmTraining the Known-Speaker Model: Maximum a Posteriori (MAP) AdaptationCalculating a ScoreRemarks Regarding UBM Training Datai-Vector PLDAi-Vectorsi-Vector Domain Mismatch Compensation (LDA)PLDADNN-Based SystemsDNN Senone Posterior i-Vector SystemsBottleneck-Feature Based SystemsDNN Speaker Embedding Systems (x-Vector Systems)Score-to-Likelihood-Ratio Conversion (Calibration)ValidationList of Published Validation StudiesConclusionAcknowledgmentsAppendix 20A: Mathematical Details of T Matrix Training and i-Vector ExtractionReferencesLegal ReferencesBringing New Statistical Approaches to Eyewitness EvidenceIntroductionThe Eyewitness TaskSystem and Estimator VariablesCurrent Statistical MethodologiesDiagnosticity Ratio, Discriminability Index, and ROC CurvesCalibration CurveROC CurvesLogistic RegressionExpected UtilityStatistical Models From Diagnostic MedicineLogitnormal Bivariate Random-Effects ModelNonparametric Meta-Analysis for Diagnostic Accuracy StudiesSupervised Learning Classification MethodsMachine Learning Classification ModelsGraphical ModelsTools Based on ROC MethodsMethodology DevelopmentPredictive Receiver Operating Characteristic CurveMultivariate ROC CurvesAUC EstimationConfidence Intervals for ROCEstimating Probability of Eyewitness AccuracyProbability of AccuracyExampleDiscussionAcknowledgementsReferences