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PROBABILITY AND STATISTICS IN THE LAW

PROBABILITY AND STATISTICS IN THE LAW. Philip Dawid University College London. STATISTICS = LAW. Interpretation of evidence Hypothesis testing Decision-making under uncertainty. Prosecution Hypothesis. INGREDIENTS. Defence Hypothesis. Evidence. BAYESIAN APPROACH.

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PROBABILITY AND STATISTICS IN THE LAW

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  1. PROBABILITY AND STATISTICS IN THE LAW Philip Dawid University College London

  2. STATISTICS = LAW • Interpretation of evidence • Hypothesis testing • Decision-making under uncertainty

  3. Prosecution Hypothesis INGREDIENTS • Defence Hypothesis • Evidence

  4. BAYESIAN APPROACH Find posterior probability of guilt: – or posterior odds: • FREQUENTIST APPROACH Look at & effect on decision rules and

  5. SALLY CLARK • Sally and Stephen Clark’s sons Christopher and Harry died suddenly at ages 11 and 8 weeks, in Sally’s care • The Clarks claimed that their children had died from natural causes (SIDS??) • Contested prosecution medical evidence of maltreatment • SALLY CONVICTED OF MURDER

  6. At Trial: • A paediatrician testified that, for a family like the Clarks, the probability of one child dying from SIDS is 1 in 8,543 • He was asked if the report calculated “the risk of two infants dying in that family by chance.” • Answer: Yes, you have to multiply 1 in 8,543 times 1 in 8,543 …. [the CESDI study] points out that it’s approximately a chance of • 1 in 73 million

  7. WHAT TO THINK? • Clear intuitive argument against independence (and thus calculation of “1 in 73 million”) • BUT probability of 2 natural deaths remains very small HOW TO CONSIDER?

  8. Prosecutor’s Fallacy • = 1 in 73 million • Probability of deaths arising from natural causes is 1 in 73 million • = 1 in 73 million • Probability of innocence is 1 in 73 million

  9. Alternatively… • P(2 babies die of SIDS) = 1/73 million • P(2 babies die of murder) = 1/2000 million BOTH figures are equally relevant to the decision between the two possible causes

  10. BAYES: POSTERIOR ODDS LIKELIHOOD RATIO PRIOR ODDS  = 73m ?? If prior odds = 1/2000 million posterior odds = 0.0365

  11. IDENTIFICATION EVIDENCE Individual i Match Criminal Suspect Evidence: Assume “match probability”

  12. PROSECUTOR’S ARGUMENT The probability of a match having arisen by innocent means is 1/10 million. So = 1/10 million – i.e. is overwhelmingly close to 1 –CONVICT

  13. DEFENCE ARGUMENT • Absent other evidence, there are 30 million potential culprits • 1 is GUILTY (and matches) • ~3 are INNOCENT and match • Knowing only that the suspect matches, he could be any one of these 4 individuals • So –ACQUIT

  14. BAYES • POSTERIOR ODDS = (10 MILLION)  “PRIOR” ODDS • PROSECUTOR’S argument OK if • DEFENCE argument OK if Only BAYES allows for explicit incorporation of B

  15. The Island Problem • N+1 on island: N(100) innocent, 1 guilty • Match, probability = P (0.004) • Prosecution: • Defence: (0.996) (0.714)

  16. Other Arguments Let number of individuals i having Ii = x be M So – need distribution of M given Note: Initially

  17. Argument 1 • Evidence tells us • So (0.902)

  18. Argument 2 • Evidence tells us 1 (guilty) individual has x • Our of remaining N innocents, number with x is ; while • So (0.824)

  19. Argument 3 • Evidence E is equivalent to 2 successes on 2 Bernoulli trials with replacement • So • So • Then (0.714 – as for defence)

  20. DENIS ADAMS • Sexual assault • DNA match • Match probability = 1/200 million 1/20 million 1/2 million • Doesn’t fit description • Victim: “not him” • Unshaken alibi • No other evidence to link to crime

  21. BAYES’S THEOREM POSTERIOR ODDS on guilt = LIKELIHOOD RATIOPRIOR ODDS = 2 million  (1 / 200,000) =10(10:1) • Posterior probability of guilt = 10/11 • =91% Reasonable doubt – ACQUIT

  22. WHAT ABOUT OTHER EVIDENCE? } • Didn’t fit description • Victim: “not him” • Unshaken alibi LR = 0.1 / 0.9 = 1/9 LR = 0.25 / 0.5 = 1/2 Apply Bayes’s Theorem again: Final odds on guilt = 10 1/9  1/2 = 5/9 (5:9) (probability of guilt =5/14 = 35%)

  23. Dependence on Match Probability – number of noughts does matter!

  24. DATABASE SEARCH • Crime trace, frequency (match probability) 1 in 1 million • Search Police DNA database (D) of size 10,000 • Find unique match: “John Smith” (S) • No other evidence

  25. Defence Case • Probability of finding a match in database if innocent ~ 10,000  (1/1,000,000) = 1/100 • Match probability of 1/100 is not convincing evidence • Evidence against John Smith is (significantly)weakened by virtue of database search –ACQUIT

  26. Prosecution Case • We have examined 10,000 individuals • Of these, 9,999 found not to match • This has reduced the pool of potential alternative culprits • Evidence against John Smith is (marginally)strengthened by virtue of database search –CONVICT

  27. Which likelihood ratio? • Hypothesis HS: “John Smith did it” is data-dependent • Replace by hypothesis HD: “Someone in database D did it” • equivalent after search identifies S (but not before) • LR = 1/(match probability) is now only 100 • weak evidence? • But HD is a priori 10,000 times more probable than HS • posterior odds the same! • agrees with prosecution argument

  28. Multiple Stains • 2 DNA stains • 1 on sheet, 1 on pillow • assume 2 perpetrators, 1 stain from each • John Smith (S) matches pillow stain • associated “match probability” P • What are appropriate hypotheses, likelihoods, inferences?

  29. S left one of 2 stains S left pillow stain S left pillow stain S left neither stain S left neither stain S didn’t leave pillow stain Hypotheses ( = prior probability S is guilty)

  30. What to present in Court? • Hypotheses equivalent (only) after data • Different prior odds • Identical posterior odds

  31. Mixed Stains • Crime trace containing DNA from more than 1 contributor • Rape • Scuffle etc

  32. O. J. SIMPSON Marker DQ-a “MATCH” to OJS Crime OJS RG Allele Frequency 13% A    20% B   28% C  

  33. MATCH PROBABILITY? • PROSECUTION: Frequency of OJS type AB:5% • DEFENCE: Combined frequency of all matching types AA, AB, AC, BB, BC, CC:39% • LR approach assuming Goldman (AC) in mixture: AB, BB, BC: 21% • LR approach not assuming Goldman in mixture: (more complex calculation) ~ 21%

  34. MISSING DNA DATA • What if we can not obtain DNA from the suspect ? (or other relevant individual?) • Sometimes we can obtain indirect information by DNA profiling of relatives • But analysis is complex and subtle…

  35. HANRATTY (“A6” murder and rape, 1961) • James Hanratty convicted and executed in 1962 • DNA profile from crime items analysed in 1998 • Population frequency less than 1 in 2.5 million • DNA profiles from mother and brother– “consistent with” crime DNA being from Hanratty

  36. PRESS REPORTS • “There is a 1 in 2.5 million chance that Hanratty was not the A6 killer” • “The DNA is 2.5 million times more likely to belong to Hanratty than anyone else” • even though no direct match to Hanratty! Likelihood Ratio based on profiles of mother and brother (complex calculation): 440

  37. DISPUTED PATERNITY brothers undisputed child disputed child • DNA profiles from MOTHER and CHILD • No profile fromPUTATIVE FATHER • MOTHER (m1) of CHILD (c1) claims that PUTATIVE FATHER (pf) is its TRUE FATHER (tf) But DO have DNA profiles from: • Two full BROTHERS (b1, b2) of PUTATIVE FATHER • HisUNDISPUTED CHILD (c2) and its MOTHER (m2)

  38. DECISION AID “PROBABILISTIC EXPERT SYSTEM” – embodies probabilistic relationships (between inherited genes)

  39. ANALYSIS • Measurements for 12 DNA markers on all 6 individuals • Enter data, “propagate” through system • Overall Likelihood Ratio in favour of paternity: ~1300

  40. FURTHER COMPLEX DNA CASES • Contamination • Laboratory errors, mix-up, fraud • Relatives • …

  41. Statistics Law Crime Science Psychology Economics Philosophy of Science Geography Medicine Ancient History Computer Science Education … EVIDENCE, INFERENCE AND ENQUIRY www.evidencescience.org

  42. EVIDENCE SCIENCE • Subject- and substance-blind approach • Inference, explanation, causality • Recurrent patterns of evidence • Narrative, argumentation, analysis, synthesis • Cognitive biases • Formal rules • Decision aids • Interdisciplinary studies • …

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