Bayesian Networks for Probabilistic Inference and Decision Analysis in Forensic Science

FRANCO TARONI, University of Lausanne, Switzerland ALEX BIEDERMANN, University of Lausanne, Switzerland SILVIA BOZZA, University Ca’ Foscari of Venice, Italy PAOLO GARBOLINO, University IUAV of Venice, Italy COLIN AITKEN, University ofEdinburgh, UK

• Foreword xiiiPreface to the second edition xviiPreface to the first edition xxi1 The logic of decision 11.1 Uncertainty and probability 11.1.1 Probability is not about numbers, it is about coherent reasoning under uncertainty 11.1.2 The first two laws of probability 21.1.3 Relevance and independence 31.1.4 The third law of probability 51.1.5 Extension of the conversation 61.1.6 Bayes’ theorem 61.1.7 Probability trees 71.1.8 Likelihood and probability 91.1.9 The calculus of (probable) truths 101.2 Reasoning under uncertainty 121.2.1 The Hound of the Baskervilles 121.2.2 Combination of background information and evidence 131.2.3 The odds form of Bayes’ theorem 151.2.4 Combination of evidence 161.2.5 Reasoning with total evidence 161.2.6 Reasoning with uncertain evidence 181.3 Population proportions, probabilities and induction 191.3.1 The statistical syllogism 191.3.2 Expectations and population proportions 211.3.3 Probabilistic explanations 221.3.4 Abduction and inference to the best explanation 251.3.5 Induction the Bayesian way 261.4 Decision making under uncertainty 281.4.1 Bookmakers in the Courtrooms? 281.4.2 Utility theory 291.4.3 The rule of maximizing expected utility 331.4.4 The loss function 341.4.5 Decision trees 351.4.6 The expected value of information 381.5 Further readings 422 The logic of Bayesian networks and influence diagrams 452.1 Reasoning with graphical models 452.1.1 Beyond detective stories 452.1.2 Bayesian networks 462.1.3 A graphical model for relevance 482.1.4 Conditional independence 502.1.5 Graphical models for conditional independence: d-separation 512.1.6 A decision rule for conditional independence 532.1.7 Networks for evidential reasoning 532.1.8 The Markov property 562.1.9 Influence diagrams 582.1.10 Conditional independence in influence diagrams 602.1.11 Relevance and causality 612.1.12 The Hound of the Baskervilles revisited 632.2 Reasoning with Bayesian networks and influence diagrams 652.2.1 Divide and conquer 662.2.2 From directed to triangulated graphs 672.2.3 From triangulated graphs to junction trees 692.2.4 Solving influence diagrams 712.2.5 Object-oriented Bayesian networks 742.2.6 Solving object-oriented Bayesian networks 792.3 Further readings 822.3.1 General 822.3.2 Bayesian networks and their predecessors in judicial contexts 833 Evaluation of scientific findings in forensic science 853.1 Introduction 853.2 The value of scientific findings 863.3 Principles of forensic evaluation and relevant propositions 903.3.1 Source level propositions 923.3.2 Activity level propositions 943.3.3 Crime level propositions 973.4 Pre-assessment of the case 1003.5 Evaluation using graphical models 1033.5.1 Introduction 1033.5.2 General aspects of the construction of Bayesian networks 1033.5.3 Eliciting structural relationships 1053.5.4 Level of detail of variables and quantification of influences 1063.5.5 Deriving an alternative network structure 1084 Evaluation given source level propositions 1134.1 General considerations 1134.2 Standard statistical distributions 1154.3 Two stains, no putative source 1174.3.1 Likelihood ratio for source inference when no putative source is available 1174.3.2 Bayesian network for a two-trace case with no putative source 1194.3.3 An alternative network structure for a two trace no putative source case 1214.4 Multiple propositions 1224.4.1 Form of the likelihood ratio 1224.4.2 Bayesian networks for evaluation given multiple propositions 1235 Evaluation given activity level propositions 1295.1 Evaluation of transfer material given activity level propositions assuming a direct source relationship 1305.1.1 Preliminaries 1305.1.2 Derivation of a basic structure for a Bayesian network 1315.1.3 Modifying the basic network 1345.1.4 Further considerations about background presence 1375.1.5 Background from different sources 1395.1.6 An alternative description of the findings 1425.1.7 Bayesian network for an alternative description of findings 1455.1.8 Increasing the level of detail of selected propositions 1475.1.9 Evaluation of the proposed model 1495.2 Cross- or two-way transfer of trace material 1505.3 Evaluation of transfer material given activity level propositions with uncertainty about the true source 1545.3.1 Network structure 1545.3.2 Evaluation of the network 1545.3.3 Effect of varying assumptions about key factors 1576 Evaluation given crime level propositions 1596.1 Material found on a crime scene: A general approach 1596.1.1 Generic network construction for single offender 1596.1.2 Evaluation of the network 1616.1.3 Extending the single-offender scenario 1636.1.4 Multiple offenders 1666.1.5 The role of the relevant population 1686.2 Findings with more than one component: The example of marks 1686.2.1 General considerations 1686.2.2 Adding further propositions 1696.2.3 Derivation of the likelihood ratio 1706.2.4 Consideration of distinct components 1726.2.5 An extension to firearm examinations 1776.2.6 A note on the likelihood ratio 1816.3 Scenarios with more than one trace: ‘Two stain-one offender’ cases 1826.4 Material found on a person of interest 1856.4.1 General form 1856.4.2 Extending the numerator 1876.4.3 Extending the denominator 1896.4.4 Extended form of the likelihood ratio 1906.4.5 Network construction and examples 1907 Evaluation of DNA profiling results 1967.1 DNA likelihood ratio 1967.2 Network approaches to the DNA likelihood ratio 1987.2.1 The ‘match’ approach 1987.2.2 Representation of individual alleles 1987.2.3 Alternative representation of a genotype 2027.3 Missing suspect 2037.4 Analysis when the alternative proposition is that a brother of the suspect left the crime stain 2067.4.1 Revision of probabilities and networks 2067.4.2 Further considerations on conditional genotype probabilities 2127.5 Interpretation with more than two propositions 2147.6 Evaluation with more than two propositions 2177.7 Partially corresponding profiles 2207.8 Mixtures 2237.8.1 Considering multiple crime stain contributors 2237.8.2 Bayesian network for a three-allele mixture scenario 2257.9 Kinship analyses 2277.9.1 A disputed paternity 2277.9.2 An extended paternity scenario 2307.9.3 A case of questioned maternity 2327.10 Database search 2347.10.1 Likelihood ratio after database searching 2347.10.2 An analysis focussing on posterior probabilities 2377.11 Probabilistic approaches to laboratory error 2417.11.1 Implicit approach to typing error 2417.11.2 Explicit approach to typing error 2437.12 Further reading 2467.12.1 A note on object-oriented Bayesian networks 2467.12.2 Additional topics 2468 Aspects of combining evidence 2498.1 Introduction 2498.2 A difficulty in combining evidence: The ‘problem of conjunction’ 2508.3 Generic patterns of inference in combining evidence 2528.3.1 Preliminaries 2528.3.2 Dissonant evidence: Contradiction and conflict 2528.3.3 Harmonious evidence: Corroboration and convergence 2568.3.4 Drag coefficient 2618.4 Examples of the combination of distinct items of evidence 2628.4.1 Handwriting and fingermarks 2628.4.2 Issues in DNA analyses 2668.4.3 One offender and two corresponding traces 2678.4.4 Firearms and gunshot residues 2718.4.5 Comments 2799 Networks for continuous models 2819.1 Random variables and distribution functions 2819.1.1 Normal distribution 2839.1.2 Bivariate Normal distribution 2879.1.3 Conditional expectation and variance 2889.2 Samples and estimates 2899.2.1 Summary statistics 2899.2.2 The Bayesian paradigm 2919.3 Continuous Bayesian networks 2929.3.1 Propagation in a continuous Bayesian network 2959.3.2 Background data 3009.3.3 Intervals for a continuous entity 3029.4 Mixed networks 3069.4.1 Bayesian network for a continuous variable with a discrete parent 3089.4.2 Bayesian network for a continuous variable with a continuous parent and a binary parent, unmarried 31010 Pre-assessment 31410.1 Introduction 31410.2 General elements of pre-assessment 31510.3 Pre-assessment in a fibre case: A worked through example 31610.3.1 Preliminaries 31610.3.2 Propositions and relevant events 31710.3.3 Expected likelihood ratios 31910.3.4 Construction of a Bayesian network 32110.4 Pre-assessment in a cross-transfer scenario 32110.4.1 Bidirectional transfer 32110.4.2 A Bayesian network for a pre-assessment of a cross-transfer scenario 32410.4.3 The value of the findings 32510.5 Pre-assessment for consignment inspection 32810.5.1 Inspecting small consignments 32810.5.2 Bayesian network for inference about small consignments 33010.5.3 Pre-assessment for inspection of small consignments 33310.6 Pre-assessment for gunshot residue particles 33510.6.1 Formation and deposition of gunshot residue particles 33510.6.2 Bayesian network for grouped expected findings (GSR counts) 33610.6.3 Examples for GSR count pre-assessment using a Bayesian network 33911 Bayesian decision networks 34311.1 Decision making in forensic science 34311.2 Examples of forensic decision analyses 34411.2.1 Deciding about whether or not to perform a DNA analysis 34411.2.2 Probability assignment as a question of decision making 35211.2.3 Decision analysis for consignment inspection 35711.2.4 Decision after database searching 36611.3 Further readings 36812 Object-oriented networks 37012.1 Object orientation 37012.2 General elements of object-oriented networks 37112.2.1 Static versus dynamic networks 37112.2.2 Dynamic Bayesian networks as object-oriented networks 37312.2.3 Refining internal class descriptions 37412.3 Object-oriented networks for evaluating DNA profiling results 37812.3.1 Basic disputed paternity case 37812.3.2 Useful class networks for modelling kinship analyses 37912.3.3 Object-oriented networks for kinship analyses 38112.3.4 Object-oriented networks for inference of source 38312.3.5 Refining internal class descriptions and further considerations 38513 Qualitative, sensitivity and conflict analyses 38813.1 Qualitative probability models 38913.1.1 Qualitative influence 38913.1.2 Additive synergy 39213.1.3 Product synergy 39413.1.4 Properties of qualitative relationships 39613.1.5 Implications of qualitative graphical models 40113.2 Sensitivity analyses 40213.2.1 Preliminaries 40213.2.2 Sensitivity to a single probability assignment 40313.2.3 Sensitivity to two probability assignments 40513.2.4 Sensitivity to prior distribution 40813.3 Conflict analysis 41013.3.1 Conflict detection 41113.3.2 Tracing a conflict 41413.3.3 Conflict resolution 415References 419Author index 433Subject index 438

“The clear and accessible style of this second edition makes this book ideal for all forensic scientists, applied statisticians and graduate students wishing to evaluate forensic  findings from the perspective of probability and decision analysis. It will also appeal to lawyers and other scientists and professionals interested in the evaluation and interpretation of forensic findings, including decision making based on scientific information.”  (Zentralblatt MATH, 1 October 2014)