Download - Bayes’s Theorem and the Weighing of Evidence by Juries Philip Dawid University College London
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 G
Defence Hypothesis G
Evidence E
– or posterior odds:
)|( EGP
)|(
)|(
E
E
GP
GP
BAYESIAN APPROACH
FREQUENTIST APPROACH
– and possibly
)|( GP E
)|( GP E
Find posterior probability of guilt:
Look at & effect on
decision rules
SALLY CLARK
E
G
G
1)|( GP E
Sally Clark’s two babies died unexpectedly
Sally Clark murdered them
Cot deaths (SIDS)
(??)million73/1)|( GP E
POSSIBLE DECISION RULE
E OCCURS
million73/1 ) |error (
0 ) |error (
GP
GP
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
)(
)(
)(
)(
)|(
)|(
GP
GP
GP
GP
GP
GP
|E
|E
E
E
=LIKELIHOOD
RATIO PRIOR
ODDS
If prior odds = 1/2000 million, Posterior odds = 0.0365
%5.3)|( EGP
73m ??
IMPACT OF EVIDENCE
By BAYES, this is carried by the
LIKELIHOOD RATIO
)|(
)|(
GP
GPLR
E
E
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),( BME
M = DNA matchB = other background evidence
Assume
million10/1)|(
1)|(
GMP
GMP
– “match probability”MP
PROSECUTOR’S ARGUMENT
The probability of a match having arisen by innocent means is 1/10 million.
So )|( MGP = 1/10 million
– i.e. )|( MGP 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 41)|( MGP
–ACQUIT
BAYES POSTERIOR ODDS = (10 MILLION) “PRIOR” ODDS
)|(
)|(
BGP
BGP
PROSECUTOR’S argument OK if
Only BAYES allows for explicit incorporation of B
2/1)|( BGP
DEFENCE argument OK if million 1/30)|( BGP
MPLR /1
DENIS ADAMS
– Match probability = 1/200 million
1/20 million
1/2 million
Doesn’t fit descriptionVictim: “not him”Unshaken alibiNo 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
?%80)|( EGP
PRIOR• 150,000 males 18-60 in local area
000,200/1)( GP
DEFENCE EVIDENCE B=D&A• D: Doesn’t fit description/victim does not
recognise 9/19.0/1.0 DLR
2/15.0/25.0 ALR
million36/1)|( BGP
• A: Alibi
POSTERIOR
Match probability 1/200m 1/20m 1/2m
Posterior .98 .85 .35)&|( BMGP
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
100
1)million1/1(000,10)|databaseinmatch( GP
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 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