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Bayes’s Theorem and the Weighing of Evidence by Juries Philip Dawid University College London

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

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INGREDIENTS Prosecution Hypothesis Defence Hypothesis Evidence

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

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SALLY CLARK Sally Clark’s two babies died unexpectedly Sally Clark murdered them Cot deaths (SIDS)

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POSSIBLE DECISION RULE OCCURS Can we discount possibility of error? — if so, right to convict CONVICT whenever

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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

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BAYES: POSTERIOR ODDS = LIKELIHOOD RATIO PRIOR ODDS If prior odds = 1/2000 million, Posterior odds = 0.0365 73m ??

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IMPACT OF EVIDENCE By BAYES, this is carried by the LIKELIHOOD RATIO Appropriate subject of expert testimony? Instruct jury on how to combine LR with prior odds?

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IMPACT OF A LR OF 100 PRIOR.001.01.1.3.5.7.9 POSTERIOR.09.5.92.98.99.996.999 Probability of Guilt

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IDENTIFICATION EVIDENCE M = DNA match B = other background evidence Assume – “match probability” MP

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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

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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

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

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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 to link to crime Sexual assault DNA match

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Court presented with LR for match Instruction in Bayes’s theorem Suggested LR’s for defence evidence Suggested priors before any evidence

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PRIOR 150,000 males 18-60 in local area DEFENCE EVIDENCE B=D&A D: Doesn’t fit description/victim does not recognise A: Alibi

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POSTERIOR Match probability1/200m1/20m1/2m Posterior.98.85.35

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Trial –Appeal – Retrial – Appeal “usurps function of jury” “jury must apply its common sense” BAYES rejected – HOW? SALVAGE? 1.Use “Defence argument” 2.Apply other evidence

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DATABASE SEARCH Rape, DNA sample No suspect Search police database, size 10,000 Find single “match”, arrest Match probability 1/1 million EFFECT OF SEARCH??

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DEFENCE – (significantly) weakens impact of evidence PROSECUTION We have eliminated 9,999 potential culprits – (slightly) strengthens impact of evidence

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BAYES Prosecutor correct 1.Suspect is guilty 2.Some one in database is guilty Defence switches hypotheses – equivalent AFTER search – but NOT BEFORE Different priorsDifferent likelihood ratio – EFFECTS CANCEL!

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CONCLUSIONS Interpretation of evidence raises deep and subtle logical issues STATISTICS and PROBABILITY can address these BAYES’S THEOREM is the cornerstone Need much greater interaction between lawyers and statisticians

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