# PROBABILITY AND STATISTICS IN THE LAW Philip Dawid University College Londo

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## PROBABILITY AND STATISTICS IN THE LAW Philip Dawid University College Londo

Connecticut College, US has reference to this Academic Journal, PROBABILITY AND STATISTICS IN THE LAW Philip Dawid University College London STATISTICS = LAW Interpretation of evidence Hypothesis testing Decision-making under uncertainty INGREDIENTS Prosecution Hypothesis Defence Hypothesis Evidence

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? or posterior odds: BAYESIAN APPROACH FREQUENTIST APPROACH in addition to Find posterior probability of guilt: Look at & effect on decision rules SALLY CLARK Sally in addition to Stephen Clark?s sons Christopher in addition to Harry died suddenly at ages 11 in addition to 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 At Trial: A paediatrician testified that, in consideration of 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 so that 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

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? 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 Alternatively? P(2 babies die of SIDS) = 1/73 million P(2 babies die of murder) = 1/2000 million BOTH figures are equally relevant so that the decision between the two possible causes

BAYES: POSTERIOR ODDS = LIKELIHOOD RATIO ? PRIOR ODDS If prior odds = 1/2000 million posterior odds = 0.0365 73m ?? IDENTIFICATION EVIDENCE Assume ?match probability? Individual i Criminal Suspect Evidence: Match 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 so that 1 ? CONVICT

DEFENCE ARGUMENT Absent other evidence, there are 30 million potential culprits 1 is GUILTY (and matches) ~3 are INNOCENT in addition to match Knowing only that the suspect matches, he could be any one of these 4 individuals So ?ACQUIT BAYES POSTERIOR ODDS = (10 MILLION) ? ?PRIOR? ODDS PROSECUTOR?S argument OK if Only BAYES allows in consideration of explicit incorporation of B DEFENCE argument OK if The Island Problem N+1 on island: N (100) innocent, 1 guilty Match, probability = P (0.004) Prosecution: Defence: (0.996) (0.714)

Other Arguments Let number of individuals i having Ii = x be M ? need distribution of M given Note: Initially So Argument 1 Evidence tells us So (0.902) Argument 2 Evidence tells us 1 (guilty) individual has x Our of remaining N innocents, number alongside x is ; while So (0.824)

Argument 3 Evidence E is equivalent so that 2 successes on 2 Bernoulli trials alongside replacement So So Then (0.714 ? as in consideration of defence) DENIS ADAMS Match probability = 1/200 million 1/20 million 1/2 million Doesn?t fit description Victim: ?not him? Unshaken alibi No other evidence so that link so that crime Sexual assault DNA match 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

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%) Dependence on Match Probability ? number of noughts does matter! 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

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 Prosecution Case We have examined 10,000 individuals Of these, 9,999 found not so that match This has reduced the pool of potential alternative culprits Evidence against John Smith is (marginally) strengthened by virtue of database search ? CONVICT 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 alongside prosecution argument

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? Hypotheses 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 (? = prior probability S is guilty) What so that present in Court? Hypotheses equivalent (only) after data Different prior odds Identical posterior odds

FURTHER COMPLEX DNA CASES Contamination Laboratory errors, mix-up, fraud Relatives ? EVIDENCE, INFERENCE AND ENQUIRY Statistics Law Crime Science Psychology Economics Philosophy of Science Geography Medicine Ancient History Computer Science Education ? evidencescience EVIDENCE SCIENCE Subject- in addition to 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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## Journal Ratings by Connecticut College

This Particular Journal got reviewed and rated by 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 Prosecution Case We have examined 10,000 individuals Of these, 9,999 found not so that match This has reduced the pool of potential alternative culprits Evidence against John Smith is (marginally) strengthened by virtue of database search ? CONVICT 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 alongside prosecution argument and short form of this particular Institution is US and gave this Journal an Excellent Rating.