Data Analysis in Forensic Science: A Bayesian Decision Perspective (Statistics in Practice)

Data Analysis in Forensic Science......Page 6
Contents......Page 10
Foreword......Page 16
Preface......Page 18
I The Foundations of Inference and Decision in Forensic Science......Page 22
1.1 The Inevitability of Uncertainty......Page 24
1.2 Desiderata in Evidential Assessment......Page 26
1.3 The Importance of the Propositional Framework and the Nature of Evidential Assessment......Page 28
1.4 From Desiderata to Applications......Page 29
1.5 The Bayesian Core of Forensic Science......Page 31
1.6 Structure of the Book......Page 33
2.1 Coherent Reasoning Under Uncertainty......Page 36
2.1.1 A rational betting policy......Page 37
2.1.2 A rational policy for combining degrees of belief......Page 40
2.1.3 A rational policy for changing degrees of belief......Page 42
2.2.1 A method for measuring the value of consequences......Page 45
2.2.2 The consequences of rational preferences......Page 48
2.2.3 Intermezzo: some more thoughts about rational preferences......Page 51
2.2.4 The implementation of coherent decision making under uncertainty: Bayesian networks......Page 55
2.2.5 The connection between pragmatic and epistemic standards of reasoning......Page 61
2.3 Scientific Reasoning as Coherent Decision Making......Page 62
2.3.1 Bayes’ theorem......Page 64
2.3.2 The theories’ race......Page 68
2.3.3 Statistical reasoning: the models’ race......Page 72
2.3.4 Probabilistic model building: betting on random quantities......Page 75
2.4.1 Likelihood ratios and the ‘weight of evidence’......Page 80
2.4.2 The expected value of information......Page 84
2.4.3 The hypotheses’ race in the law......Page 91
3.1.1 Univariate random variables......Page 96
3.1.3 Multiple random variables......Page 102
3.2 Statistical Inference and Decision Theory......Page 108
3.2.1 Utility theory......Page 112
3.2.2 Maximizing expected utility......Page 117
3.2.3 The loss function......Page 120
3.3 The Bayesian Paradigm......Page 121
3.3.1 Sequential use of Bayes’ theorem......Page 126
3.3.2 Principles of rational inference in statistics......Page 127
3.3.3 Prior distributions......Page 131
3.3.4 Predictive distributions......Page 141
3.3.5 Markov Chain Monte Carlo methods (MCMC)......Page 143
3.4 Bayesian Decision Theory......Page 146
3.4.1 Optimal decisions......Page 147
3.4.2 Standard loss functions......Page 148
3.5 R Code......Page 153
II Forensic Data Analysis......Page 156
4.1 Introduction......Page 158
4.2 Bayesian Decision for a Proportion......Page 160
4.2.1 Estimation when there are zero occurrences in a sample......Page 161
4.2.2 Prior probabilities......Page 166
4.2.3 Prediction......Page 169
4.2.4 Inference for 0 in the presence of background data on the number of successes......Page 170
4.2.5 Multinomial variables......Page 176
4.3.1 Inference about the Poisson parameter in the absence of background events......Page 179
4.3.2 Inference about the Poisson parameter in the presence of background events......Page 183
4.3.3 Forensic inference using graphical models......Page 184
4.4 Bayesian Decision for Normal Mean......Page 189
4.4.1 Case with known variance......Page 190
4.4.2 Case with unknown variance......Page 194
4.4.3 Estimation of the mean in the presence of background data......Page 196
4.5 R Code......Page 198
5.1 Introduction......Page 206
5.2 Credible Intervals and Lower Bounds......Page 207
5.3 Decision-Theoretic Evaluation of Credible Intervals......Page 215
5.4 R Code......Page 219
6.1 Introduction......Page 222
6.2.1 Posterior odds and Bayes factors......Page 226
6.2.2 Decision-theoretic testing......Page 232
6.3.1 Background......Page 234
6.3.2 Proportion......Page 235
6.3.3 A note on multinomial cases (k categories)......Page 245
6.3.4 Mean......Page 248
6.4.1 Background......Page 253
6.4.2 Proportion......Page 256
6.4.3 Mean......Page 262
6.5 R Code......Page 266
7.1 Introduction......Page 274
7.2.1 Background......Page 275
7.2.2 Large consignments......Page 277
7.2.3 Small consignments......Page 280
7.3.2 Bayesian network for sampling from large consignments......Page 283
7.3.3 Bayesian network for sampling from small consignments......Page 288
7.4.1 Fixed sample size......Page 291
7.4.2 Sequential analysis......Page 295
7.4.3 Sequential probability ratio test......Page 299
7.5 R Code......Page 307
8.1 Introduction......Page 310
8.2 Standards of Coherent Classification......Page 311
8.3.1 Binomial distribution and cocaine on bank notes......Page 314
8.3.2 Poisson distributions and firearms examination......Page 318
8.4.1 Normal distribution and colour dye (case with known variance)......Page 321
8.4.2 A note on the robustness of the likelihood ratio......Page 324
8.4.3 Normal distribution and questioned documents (case with known variance)......Page 326
8.4.4 Normal distribution and sex determination (case with unknown variance)......Page 328
8.5 Non-Normal Distributions and Cocaine on Bank Notes......Page 330
8.6 A Note on Multivariate Continuous Data......Page 335
8.7 R Code......Page 337
9.1 Introduction......Page 344
9.2 What is the Past and Current Position of Statistics in Forensic Science?......Page 345
9.3 Why Should Forensic Scientists Conform to a Bayesian Framework for Inference and Decision Making?......Page 346
9.4 Why Regard Probability as a Personal Degree of Belief?......Page 348
9.5 Why Should Scientists be Aware of Decision Analysis?......Page 353
9.6 How to Implement Bayesian Inference and Decision Analysis?......Page 355
A Discrete Distributions......Page 362
B Continuous Distributions......Page 364
Bibliography......Page 368
Author Index......Page 380
Subject Index......Page 384