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Rational Proof in Criminal Cases: Approaches and Models

This lecture explores three approaches to rational proof in criminal cases, including Bayesian models, argumentation-based models, and story-based models. It examines the challenges of uncertainty in legal proof and the use of generalizations and expert testimony in constructing rational legal arguments.

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Rational Proof in Criminal Cases: Approaches and Models

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  1. Spring School on Argumentation in AI & LawDay 3 – lecture 2:Three approaches to rational proof in criminal cases Henry Prakken Guangzhou (China) 12 April 2018

  2. Uncertainty in legal proof in criminal cases • Legal proof of facts is never completely certain • Eye witnesses can be unreliable • Expert witnesses sometimes disagree • Documents can be manipulated • Generalisations can have exceptions • DNA tests have an error margin • Confessions might be false • …

  3. Models of proof that leave room for uncertainty • Bayesian models • Determine the prior probability of guilt • Determine how probable the evidence is given guilt and given innocence • Then apply Bayes’ theorem to determine how probable guilt is given the evidence • Argumentation-based models • Construct arguments for and against guilt, starting with the available evidence • Determine which argument survives the competition between the conflicting arguments. • Story-based models • Construct stories about what might have happened • Choose the best • The most coherent, the best explanation of the evidence

  4. Problem • We want a model of rational proof that is: • rationally well-founded • cognitively feasible

  5. Rational legal proof with arguments F.J. Bex, H. Prakken, C. Reed & D. Walton (2003), Towards a formal account of reasoning about evidence: argumentation schemes and generalisations. Artificial Intelligence and Law 11: (2003) 125-165.

  6. Wigmore Charts

  7. Reasoning with generalisations Involved • Critical questions: • Is there an exception? • Are there conflicting generalisations? Defeasible Modus Ponens: P If P then normally Q So (presumably), Q Flees If flees then normally involved People who flee from a crime scene are normally involved in the crime

  8. Implicit generalisation “X entered the office at 4:30 since witness W says that he saw X entering the office at 4:30” X entered the office at 4:30 W says that he saw X entering the office at 4:30

  9. Implicit generalisation made explicit “X entered the office at 4:30 since witness W says that he saw X entering the office at 4:30” Unless W’s memory is flawed Unless W could not observe well X entered the office at 4:30 Unless W has a reason to lie What witnesses say is usually true W says that he saw X entering the office at 4:30

  10. Witness testimony • Critical questions: • Is W sincere? • Does W’s memory function properly? • Did W’s senses function properly? W says P W was in the position to observe P Therefore (presumably), P P is usually of the form “I remember that I observed that ...”

  11. Implicit generalisation (2) “X was in the office at 4:45 since he entered the office at 4:30” X was in the office at 4:45 X entered the office at 4:30

  12. Implicit generalisation made explicit Unless X has meanwhile left the office “X was in the office at 4:45 since he entered the office at 4:30” X was in the office at 4:45 X entered the office at 4:30 People who enter a building are usually still inside a short while later

  13. Temporal persistence(Forward) • Critical questions: • Was P known to be false between T1 and T2? • … P is true at T1 Therefore (presumably), P is still true at later time T2

  14. Temporal persistence(Backward) • Critical questions: • Was P known to be false between T1 and T2? • … P is true at T1 Therefore (presumably), P was already true at earlier time T2

  15. X killed Y Generalisation If V is killed at L at time T, and S was there then, then S killed V X was in the office at 4:45 Y was killed in the office at 4:45 Backward temporal persistence Forward temporal persistence X entered the office at 4:30 X left the office at 5:00 testimony testimony W1: “X entered the office at 4:30” W2: “X entered the office at 4:30” W3: “X left the office at 5:00”

  16. How are generalisations justified? Scientific research (induction) Experts Commonsense Individual opinions Prejudice? Very reliable Very unreliable

  17. Inducing generalisations Critical questions: Is the size of the sample large enough? was the sample selection biased? Almost all observed P’s were Q’s Therefore (presumably), If P then usually Q A ballpoint shot with this type of bow will usually cause this type of eye injury In 16 of 17 tests the ballpoint shot with this bow caused this type of eye injury

  18. Expert testimony • Critical questions: • Is E biased? • Is P consistent with what other experts say? • Is P consistent with known evidence? E is expert on D E says that P P is within D Therefore (presumably), P is the case

  19. V’s injury was caused by a fall This type of eye injury is usually caused by a fall V has this type of injury E says that his type of injury is usually caused by a fall E is an expert on this type of injury Supporting and using generalisations Generalisation scheme Expert testimony scheme

  20. Rational legal proof with stories/scenarios

  21. Pennington & Hastie (1993) • Judges and jurors can only understand evidential reasoning in the form of stories about what may have happened. • A good story: • Explains the evidence • Is coherent: • Is internally consistent • Is plausible (conforms to our general knowledge about the world) • Is complete: contains initiating events which cause the main actor to have intentions, which give rise to actions, which have consequences

  22. Argumentation vs. scenario approach • Argumentation: reasons from evidence to hypotheses by applying evidential generalisations / argument schemes • Critical testing by searching for counterarguments based on exceptions • Scenario approach: assumes the hypotheses and then tests whether the evidence (likely) follows by applying causal generalisations • Critical testing by checking under which hypothesis the evidence most likely follows

  23. Evidential vs. causal generalisations • Evidential: P is evidence of Q (smoke means fire) • Causal: P causes Q (fire causes smoke)

  24. Y was killed in the office at 4:45 X entered the office at 4:30 X met Y at 4:45 X left the office at 5:00 X killed Y at 4:45 X wanted to kill Y time 24

  25. W2: “X entered the office at 4:30” W1: “X entered the office at 4:30” W3: “X left the office at 5:00” Y was killed in the office at 4:45 X entered the office at 4:30 X met Y at 4:45 X left the office at 5:00 X killed Y at 4:45 X wanted to kill Y time 25

  26. W2: “X entered the office at 4:30” W1: “X entered the office at 4:30” Y was killed in the office at 4:45 W3: “X left the office at 5:00” X met Z at 4:45 X entered the office at 4:30 X met Y at 4:45 X left the office at 5:00 X killed Y at 4:45 X wanted to kill Y X wanted to visit his friend Z time 26

  27. Example: Adams v Regina A rape took place near London (UK) in 1991. In 1993 Denis John Adams was arrested for another offence and a routine check showed that his DNA matched with that of a sample of semen obtained from the victim of the 1991 rape. The prosecution’s forensic expert estimated the random match probability as 1 in 200 million; the defence thought that 1 in 2 million was a better estimate. A line up took place but the victim did not recognize Adams, and she said he did not resemble her attacker. Adams was 37 and looked older, while the victim claimed the rapist was in his early twenties. Adams’girlfriend testified that he had spent the night of the attack with her.

  28. Arguments and counterarguments John is the rapist John’s DNA matches with DNA found with the victim The forensic scientist’s report says so Regina v. Adams 1995

  29. John is the rapist John is not the rapist John’s DNA matches with DNA found with the victim John does not look like the rapist The forensic scientist’s report says so Victim: “John does not look like the rapist” 29

  30. John is the rapist John is not the rapist The a priori prob that John is the rapist is not too low John does not look like the rapist John was elsewhere Mary has reason to lie The forensic scientist’s report says so Victim: “John does not look like the rapist” Mary: “John was with me” Mary is John’s girlfriend 30

  31. Legal proof with arguments:critical questions • Which evidential generalisations are the glue in the arguments? • Do they hold in general? • If so: are there exceptions in this case? • Critical questions of argument schemes can help • Are there counterarguments on other grounds? • Critical questions of argument schemes can help • Can counterarguments (if any) be refuted?

  32. Scenario construction again John’s DNA matches with DNA found with the victim John is the rapist Mary: “John was with me” Mary is John’s girlfriend Victim: “John does not look like the rapist” 32

  33. Scenario construction John’s DNA matches with DNA found with the victim Mary: “John was with me” Someone else with the same DNA profile raped the victim Victim: “John does not look like the rapist” John does not look like the rapist John was with Mary 33

  34. Scenario construction John’s DNA matches with DNA found with the victim John is the rapist Mary: “John was with me” Mary is John’s girlfriend Someone else with the same DNA profile raped the victim Victim: “John does not look like the rapist” John does not look like the rapist Van Koppen (2011): Under which scenario is the evidence the most likely? John was with Mary 34

  35. Legal proof with scenarios: critical questions • Is the scenario plausible,consistent and complete? • Does it explain the evidence? • Are there alternative scenarios that explain the evidence? • If so: • How plausible are the various scenarios? • how likely is the evidence given the various scenarios?

  36. Rational legal proof with Bayesian probability theory

  37. Probability theory • In logic statements are true or false • Uncertainty expressed as defeasibility • In probability theory statements have a probability between 0 and 1 (or between 0% and 100%) • ‘The probability of P is 1 (100%)’means: P is certainly true • ‘The probability of P is 0 (0%)’means: P is certainly false • The probabilities of P and not-P add up to 1 (to 100%). • So the probability of not-P is 1 minus the probability of P (or 100% minus the probability of P) • Notation: ‘Prob’ = probability. • Probabilities can be conditional: the probability of Q given P

  38. The main idea • Determine prior probability of guilt • Determine how probable the evidence is given guilt and given innocence • Then apply Bayes’ theorem to determine how probable guilt is given the evidence

  39. If greater than 1, then E is incriminating evidence, If less than 1, then E is exculpating evidence, otherwise E is irrelevant. Bayes’ theorem (odds version) The prob of G given E The prob of not-G given E The prob of E given G The prob of E given not-G The prob of G The prob of not-G x = Then the computer computes the posterior prob of G given E Determine the prior prob of G Determine or ask an expert to determine the likelihood ratio of E wrt G and not-G Posterior odds Likelihood ratio Prior odds x = G = guilty Not-G = innocent

  40. Example: Adams v Regina • Evidence pro guilt: • The DNA match • Evidence con guilt: • Victim did not recognize Adams during a line up • Alibi of Adam’s girlfiend

  41. ‘Prosecutor fallacy’: inverting conditional probabilities “The probability that Adams’ DNA matches the rapist’s DNA given that Adams is not rapist is very small” Is NOT the same as “The probability that Adams is not the rapist given that Adams’ DNA matches the rapist’s DNA is very small” Compare “The probability that a person is a man given that the person is a rapist is very high” “The probability that a persons is a rapist given that the person is a man is NOT very high”

  42. E: match of Adams DNA with DNA found at the crime scene G: John is the rapist Bayesian Reasoning So the prob that Adams is the donor given the DNA match is 50% The prob of G given E The prob of not-G given E The prob of E given G The prob of E given not-G The prob of G The prob of not-G x = x = 1 2,000,000 1:2,000,000 Random match probability = 1:2,000.000 2,000,000 potential donors Dawid, Philip (2005). Probability and proof. Online appendix to T.J. Anderson, D.A. Schum and W.L. Twining: Analysis of Evidence, http://tinyurl.com/tz85o.

  43. E: match of Adams DNA with DNA found at the crime scene G: John is the rapist Bayesian Reasoning So the prob that Adams is the rapist given the DNA match is 0.91 The prob of G given E The prob of not-G given E The prob of E given G The prob of E given not-G The prob of G The prob of not-G x = x = 10 2,000,000 1:200,000 Random match probability = 1:2,000.000 200,000 potential donors Dawid, Philip (2005). Probability and proof. Online appendix to T.J. Anderson, D.A. Schum and W.L. Twining: Analysis of Evidence, http://tinyurl.com/tz85o.

  44. Bayes with multiple pieces of evidence (1) • Repeat the calculation with the old posterior as the new prior • = successively multiply the prior with the likelihood ratio of every piece of evidence

  45. Bayes with multiple pieces of evidence (2) = Posterior odds Prior odds x Likelihood ratio evidence 1 x Likelihood ratio evidence 2 x Likelihood ratio evidence 3

  46. Bayes with multiple pieces of evidence in Adams (1) Prob of Adams’ guilt given E1&E2&E3 Prob of Adams’ innocence given E1&E2&E3 The prior odds of Adams’ guilt was 1 in 200.000 = x The likelihood ratio of the DNA match for Adams’ guilt was 2.000.000 x The likelihood ratio of the non recognition for Adams’ guilt was ?? E1 = DNA match, E2 = non recognition, E3 = girlfriend’s alibi x The likelihood ratio of the girlfriend’s alibi for Adams’ guilt was ??

  47. Bayes with multiple pieces of evidence in Adams (2) Prob of Adams’ guilt given E1&E2&E3 Prob of Adams’ innocence given E1&E2&E3 The prior odds of Adams’ guilt was 1 in 200.000 = 1/200.000 91% x The likelihood ratio of the DNA match for Adams’ guilt was 2.000.000 10 x 53% The likelihood ratio of the non recognition for Adams’ guilt was 1/9 10/9 E1 = DNA match, E2 = non recognition, E3 = girlfriend’s alibi 36% x The likelihood ratio of the girlfriend’s alibi for Adams’ guilt was 1/2 5/9

  48. Limitations of ‘naive’ Bayes • Repeated updating (or multiplying likelihood ratios) is only justified if the evidence is statistically independent. • Otherwise Bayesian networks are needed • Graphically display statistical dependencies • Can reduce the number of probabilities to be estimated • Their graphical structure can perhaps capture scenarios

  49. BN met klikken https://stats.stackexchange.com/questions/249392/how-to-calculate-causal-inference-in-bayesian-networks

  50. Legal proof with Bayes: critical questions • To which extent is the evidence statistically independent? • Can the prior probabilities be reasonably estimated? • Are the conditional probabilities well-founded? • Are the considered alternatives jointly exhaustive? • …

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