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

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

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

INGREDIENTS  Prosecution Hypothesis  Defence Hypothesis  Evidence

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

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

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

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

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

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?

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

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

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

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

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

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

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

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

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

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

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??

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

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!

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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