Statistics and the Evaluation of Evidence for Forensic Scientists

Statistics and the Evaluation of Evidence for Forensic Scientists......Page 4
Contents......Page 10
List of tables......Page 16
List of figures......Page 22
Foreword......Page 25
Preface to the first edition......Page 28
Preface to the second edition......Page 31
1.1 Introduction......Page 34
1.2 Statistics and the law......Page 35
1.3 Uncertainty in scientific evidence......Page 38
1.3.1 The frequentist method......Page 39
1.3.2 Stains of body fluids......Page 40
1.3.3 Glass fragments......Page 42
1.4 Terminology......Page 45
1.5 Types of data......Page 48
1.6.1 Introduction......Page 49
1.6.2 A standard for uncertainty......Page 51
1.6.3 Events......Page 53
1.6.4 Subjective probability......Page 54
1.6.5 Laws of probability......Page 56
1.6.6 Dependent events and background information......Page 58
1.6.7 Law of total probability......Page 62
1.6.8 Updating of probabilities......Page 65
2.1 Populations......Page 68
2.2 Samples and estimates......Page 70
2.3.1 Probabilities......Page 73
2.3.2 Summary measures......Page 74
2.3.3 Binomial distribution......Page 76
2.3.4 Multinomial distribution......Page 77
2.3.5 Hypergeometric distribution......Page 78
2.3.6 Poisson distribution......Page 81
2.3.7 Beta-binomial distribution......Page 84
2.4.1 Summary statistics......Page 85
2.4.2 Normal distribution......Page 86
2.4.3 Student’s t-distribution......Page 93
2.4.4 Beta distribution......Page 95
2.4.5 Dirichlet distribution......Page 96
2.4.6 Multivariate Normal and correlation......Page 97
3.1.1 Complementary events......Page 102
3.1.3 Definition......Page 103
3.2.1 Statement of the theorem......Page 105
3.2.2 Examples......Page 106
3.3 Errors in interpretation......Page 111
3.3.1 Fallacy of the transposed conditional......Page 112
3.3.2 Source probability error......Page 114
3.3.4 Defender’s fallacy......Page 115
3.3.5 Probability (another match) error......Page 116
3.3.6 Numerical conversion error......Page 117
3.3.7 False positive fallacy......Page 118
3.3.8 Uniqueness......Page 119
3.3.9 Other difficulties......Page 120
3.3.10 Empirical evidence of errors in interpretation......Page 122
3.4.1 Likelihood ratio......Page 128
3.4.2 Logarithm of the likelihood ratio......Page 132
3.5.1 Evaluation of forensic evidence......Page 134
3.5.2 Summary of competing propositions......Page 138
3.5.3 Qualitative scale for the value of the evidence......Page 140
3.5.4 Misinterpretations......Page 144
3.5.5 Explanation of transposed conditional and defence fallacies......Page 145
3.5.6 The probability of guilt......Page 149
3.6 Summary......Page 151
4.1 Early history......Page 152
4.2 The Dreyfus case......Page 155
4.3 Statistical arguments by early twentieth-century forensic scientists......Page 158
4.4 People v. Collins......Page 159
4.5.1 Derivation......Page 162
4.5.2 Evaluation of evidence by discriminating power......Page 163
4.5.3 Finite samples......Page 166
4.5.4 Combination of independent systems......Page 168
4.5.5 Correlated attributes......Page 169
4.6.1 Calculation of significance probabilities......Page 174
4.6.2 Relationship to likelihood ratio......Page 177
4.6.3 Combination of significance probabilities......Page 180
4.7.1 Introduction......Page 182
4.7.3 Significance stage......Page 184
4.8 Likelihood ratio......Page 186
5.1 Introduction......Page 190
5.2 Bayesian inference for a Bernoulli probability......Page 193
5.3 Estimation with zero occurrences in a sample......Page 195
5.4 Estimation of products in forensic identification......Page 198
5.5 Bayesian inference for a Normal mean......Page 199
5.6.1 Confidence intervals......Page 203
5.6.3 Bootstrap intervals......Page 205
5.6.4 Likelihood intervals......Page 206
5.7 Odds ratios......Page 208
6.1 Introduction......Page 212
6.2.1 Large consignments......Page 215
6.2.2 Small consignments......Page 219
6.3.1 Frequentist approach......Page 223
6.3.2 Bayesian approach......Page 224
6.4 Misleading evidence......Page 230
7.1.1 Relevant population......Page 238
7.1.2 Consideration of odds......Page 239
7.1.4 Specific cases......Page 241
7.2.1 Levels of proposition......Page 247
7.2.2 Pre-assessment of the case......Page 250
7.2.3 Pre-assessment of the evidence......Page 253
7.3.1 Earprints......Page 254
7.3.2 Firearms and toolmarks......Page 256
7.3.3 Fingerprints......Page 259
7.3.4 Speaker recognition......Page 261
7.3.5 Hair......Page 262
7.3.6 Documents......Page 264
7.3.7 Envelopes......Page 266
7.3.8 Handwriting......Page 268
7.4 Pre-data and post-data questions......Page 272
8.1.1 Probability of guilt......Page 278
8.1.2 Justification......Page 279
8.1.3 Combination of evidence and comparison of more than two propositions......Page 281
8.2 Correspondence probabilities......Page 287
8.3.1 Transfer of evidence from the criminal to the scene......Page 288
8.3.2 Transfer of evidence from the scene to the criminal......Page 293
8.3.3 Transfer probabilities......Page 294
8.3.4 Two-way transfer......Page 303
8.4 Grouping......Page 304
8.5 Relevant populations......Page 307
9.2.1 Introduction......Page 316
9.2.4 Examples......Page 319
9.3.1 Two stains, two offenders......Page 321
9.3.2 DNA profiling......Page 324
9.4.1 Many different profiles......Page 325
9.4.2 General cases......Page 326
9.5.1 Introduction......Page 328
9.5.3 Association propositions......Page 329
9.5.4 Intermediate association propositions......Page 330
9.5.5 Examples......Page 331
9.5.6 Two stains, one offender......Page 335
9.6.1 Stain known to have been left by offenders......Page 337
9.6.2 Relevance: stain may not have been left by offenders......Page 338
9.6.3 Relevance and the crime level......Page 340
9.7 Missing persons......Page 341
9.7.2 Case 2 (Ogino and Gregonis, 1981)......Page 342
9.7.3 Calculation of the likelihood ratio......Page 343
9.8 Paternity: combination of likelihood ratios......Page 345
9.8.1 Likelihood of paternity......Page 347
9.8.2 Probability of exclusion in paternity......Page 350
10.1 The likelihood ratio......Page 352
10.2 Normal distribution for between-source data......Page 354
10.2.2 Derivation of the marginal distribution......Page 355
10.2.3 Approximate derivation of the likelihood ratio......Page 357
10.2.4 Lindley’s approach......Page 359
10.2.5 Interpretation of result......Page 360
10.2.6 Examples......Page 361
10.3 Estimation of a probability density function......Page 363
10.4 Kernel density estimation for between-source data......Page 370
10.4.2 Refractive index of glass......Page 372
10.5.2 Single fragment......Page 375
10.5.3 Two fragments......Page 378
10.5.4 A practical approach to glass evaluation......Page 382
10.5.5 Graphical models for the assessment of transfer probabilities......Page 385
10.6 Approach based on t-distribution......Page 386
10.7 Appendix Derivation of V when the between-source measurements are assumed normally distributed......Page 390
11.1 Introduction......Page 392
11.2 Description of example......Page 393
11.3 Univariate t-tests......Page 395
11.4 Hotelling’s T(2)......Page 396
11.5 Univariate Normality, two sources of variation......Page 398
11.6 Multivariate Normality, two sources of variation......Page 399
11.7 Caveat lector......Page 404
11.8 Summary......Page 405
11.9.1 Matrix terminology......Page 406
11.9.2 Determination of a likelihood ratio with an assumption of Normality......Page 410
12.2 Likelihood ratios in scenarios involving fibres......Page 414
12.2.1 Fibres evidence left by an offender......Page 415
12.2.2 Comments on the fibres scenario......Page 420
12.2.3 Fibres evidence not left by the offender......Page 421
12.2.4 Cross-transfer......Page 422
12.3.2 Formulation of the pairs of propositions and events......Page 425
12.3.3 Assessment of the expected value of the likelihood ratio......Page 427
12.4 The relevant population of fibres......Page 429
13.1 Introduction......Page 432
13.2 Hardy–Weinberg equilibrium......Page 434
13.3 DNA likelihood ratio......Page 437
13.5 Variation in sub-population allele frequencies......Page 438
13.6 Related individuals......Page 442
13.7 More than two propositions......Page 445
13.8 Database searching......Page 447
13.8.1 Search and selection effect (double counting error)......Page 451
13.9 Island problem......Page 452
13.10 Mixtures......Page 454
13.11 Error rate......Page 457
14.1 Introduction......Page 462
14.2 Bayesian networks......Page 463
14.2.1 The construction of Bayesian networks......Page 464
14.3.2 Description of probabilities required......Page 470
14.4.1 Preliminaries......Page 472
14.4.2 Determination of a structure for a Bayesian network......Page 473
14.5.1 Preliminaries......Page 475
14.5.2 Determination of a structure for a Bayesian network......Page 476
14.6.2 Determination of a structure for a Bayesian network......Page 477
14.6.3 Comment on the transfer node......Page 479
14.7 Combination of evidence......Page 480
14.8 Cross-transfer evidence......Page 482
14.8.1 Description of nodes......Page 484
14.9 Factors to consider......Page 485
14.10 Summary......Page 486
References......Page 488
Notation......Page 518
Cases......Page 522
Author index......Page 524
Subject index......Page 532