Math on Trial
BackgroundWe will begin with the Collins opinion and the Finkelstein & Fairley article which started the law & probability debate in the United States back in the 70ies. This will allow us to learn about basic probability theory, Bayes' theorem and its applications to the legal fact-finding process.
- People v. Collins (1968) (opinion)
- Finkelstein & Fairley, A Bayesian Approach etc. (article)
Basic questions about DNA evidenceWe will explore how DNA evidence is used and how DNA database searches work.
- Wasserman, Forensic DNA typing (chapter)
- People v. Rush (1995) (opinion) and People v. Nelson (2008) (opinion)
- Roth, Safety in numbers etc. (article)
- Balding, Evaluating DNA etc. (article)
More on DNA evidenceTime and interest permitting, we will explore more advanced questions about the uses and limitations of DNA evidence, such as:
- Are the numbers trustworthy?
- Buckleton, DNA models etc. (chapter)
- How do we interpret, assesss and present DNA evidence to the fact-finders?
- Koehler, Error and Exaggeration etc. (article)
- Buckleton, Framework etc. (chapter)
- Are genetic profiles unique?
- Kaye, Beyond Uniqueness etc. (article)
CoincidencesWe will explore cases in which the putative impossibility of a coincidence was used as a basis to draw an inference of culpability.
- Lucia de Berk
- Derksen, The Fabrication of Fact (book summary)
- Meester et al., On the abuse etc. (article)
- Fienberg & Kaye, Clusters etc. (article)
- Sally Clark
- R v. Clark (2005) (opinion)
Summary - Thursday
- Legal examples of how statistics and probabilities are used (slides).
- Interpretations of the concept of probability (classical, subjective, frequency-based) and Bertrand's paradox (here). See sec. 3.1, 3.3 and 3.4 of the SEP article (here). Which interpretation is best suited for the trial context? Slight preference for the subjective interpretation.
- Axioms; conditional probability; negation lemma. See handout on probability, sec. 2.
- Collins decision, in particular, (1) the facts of the case; (2) the question the court set out to address; (3) the law applicable to the case; (4) the answer of the court; (5) the supporting arguments. See handout on Collins, sec. 1 (the five objections).
Summary - Friday
- Collins mathematical appendix; uniqueness objection; the binomial formula; how it was used to calculate the conditional probability Pr(M>1 | M>=1), i.e. the probability that more than one couple matches the description given that at least one couple does. See Collins handout, sec. 2. The "Island problem" as an illustration.
- The inversion fallacy, i.e. confusing P(A| B) and P(B| A), in Collins. See Colllins handout, section 1 (the four step argument).
- Bayes' theorem and its proof; how it avoids the inversion fallacy. See probability handout, sec. 3. Application of Bayes' theorem to the taxi scenario. See my note (here).
- Using Bayes' theorem for calculating the probability that the Collins were guilty. See Collins handout, sec. 3.
- Comparison between the uniqueness approach (i.e. Collins, math appendix) and Bayes' theorem. Emphasis on the role assumptions. See Fiklestein & Fairley article, in particular, footnote 12.
- DNA evidence (i.e. genetic match & estimated frequency of DNA profile). See slides (here). See DNA evidence handout, sec. 1, 2 and 3. Similarities and differences between DNA evidence and the statistical evidence in Collins.
Summary - Saturday
- Genetic models and underlying assumptions in calculating genetic frequencies (e.g. random mating). See/skim Buckleton article "Population Genetic Models" (here), sec. 3.2.1, 3.2.2, 188.8.131.52 184.108.40.206 and 3.3. See, also, DNA evidence handout, section 7.
- The Rush decision, in particular the claims that (1) DNA evidebe alone (given certain restrictions) is enough for a conviction in a rape case; (2) analogies between fingerprints and DNA evidence; and (3) DNA evidence is less prone to error than eyewitness evidence.
- Difference between the probability of G (=guilt) and the probability of S(=source). See DNA evidence handout, sec. 5. Application of Bayes' theorem to the Rush decision. Odds formulation of Bayes' theorem, easier for calculations. See DNA evidence handout, sec. 9 and 10.
- Standard DNA evidence cases vs. cold-hit cases. The Jenkins case. See Devlin's article (here), only pp. 1-4. See DNA evidence handout, sec. 4.
- The cold-hit controversy. Is a DNA match in a cold-hit case "less significant" Sandy Zabell's affidavit (here), in particular, the three approaches: (1) NRC; (2) Likelihood ratio (see Balding & Donnelly article); and (3) Bayes' theorem. My own take on this (here).
- Can we do away with numbers? What is the best way to present DNA evidence to jurors? See DNA evidence handout, sec. 13 and 14.
Summary - Sunday
- The Lucia the Berk case; pediatricians' testimonies; the prosecutor's case; causation; the court's decision. See "The Fabrication of Fact" (pp. 1-15).
- The data; the calculations; the chance/null hypothesis; hypergeometric distribution. See Meester article, sec. 2.
- Problems with the calculations; using data twice; multiplying p-values; alternative hypotheses. See Meester article, sec. 3.
- A way to fix the calculations and get a less extreme p-value. See Meester article, sec. 5.
- Inversion Fallacy. See Meester article, sec. 4.
- "Hypothesis testing" versus "Bayes' Thm". See Meester article, sec. 6. Also, see "The Fabrication of Fact" (pp. 20-22).
- Concluding remarks. Three ways to use "numbers" in the courtroom: (a) as raw data and frequencies; (b) as probabilities assigned to a proposition on the basis of assumptions embedded in a chosen model; (c) by means of drawing inferences bearing on the defendant's guilt and innocence (using, for example, Bayes' Theorem, Hypothesis Testing, the inversion fallacy). Illustration of (a), (b) and (c) for Collins, DNA evidence cases, and Lucia de Berk.