Bayes’s Theorem and the Weighing of Evidence by Juries

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

Prosecution Hypothesis INGREDIENTS • Defence Hypothesis • Evidence

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

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

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

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  = 73m ?? If prior odds = 1/2000 million, Posterior odds = 0.0365

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

IMPACT OF A LR OF 100 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 • DEFENCE argument OK if Only BAYES allows for explicit incorporation of B

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

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

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

DATABASE SEARCH • Rape, DNA sample • No suspect • Search police database, size 10,000 • Find single “match”, arrest • Match probability1/1 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 Defence switches hypotheses • Suspect is guilty • Some one in database is guilty – equivalent AFTER search – but NOT BEFORE Different priors Different 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

Bayes’s Theorem and the Weighing of Evidence by Juries