Bayes’s Theorem and the Weighing of Evidence by Juries

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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. Prosecution Hypothesis. INGREDIENTS. Defence Hypothesis. Evidence. BAYESIAN APPROACH.

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## Bayes’s Theorem and the Weighing of Evidence by Juries

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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
Prosecution HypothesisINGREDIENTS
• 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

• 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
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