1 / 42

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.

adelle
Download Presentation

PROBABILITY AND STATISTICS IN THE LAW

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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 • …

More Related