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