– or posterior odds: BAYESIAN APPROACH FREQUENTIST APPROACH and Find posterior probability of guilt: Look at & effect on decision rules
SALLY CLARK Sally and Stephen Clark’s sons Christopher and Harry died suddenly at ages 11 and 8 weeks, in Sally’s care The Clarks claimed that their children had died from natural causes (SIDS??) Contested prosecution medical evidence of maltreatment –SALLY CONVICTED OF MURDER
A paediatrician testified that, for a family like the Clarks, the probability of one child dying from SIDS is 1 in 8,543 At Trial: He was asked if the report calculated “the risk of two infants dying in that family by chance.” Answer: Yes, you have to multiply 1 in 8,543 times 1 in 8,543 …. [the CESDI study] points out that it’s approximately a chance of 1 in 73 million
WHAT TO THINK? Clear intuitive argument against independence (and thus calculation of “1 in 73 million”) BUT probability of 2 natural deaths remains very small HOW TO CONSIDER?
Prosecutor’s Fallacy = 1 in 73 million Probability of deaths arising from natural causes is 1 in 73 million = 1 in 73 million Probability of innocence is 1 in 73 million
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 = m ??
IDENTIFICATION EVIDENCE Assume “match probability” Individual i Criminal SuspectEvidence: Match
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
The Island Problem N+1 on island: N (100) innocent, 1 guilty Match, probability = P (0.004) Prosecution: Defence: (0.996) (0.714)
Other Arguments Let number of individuals i having I i = x be M – need distribution of M given Note: Initially So
Argument 1 Evidence tells us So (0.902)
Argument 2 Evidence tells us 1 (guilty) individual has x Our of remaining N innocents, number with x is ; while So (0.824)
Argument 3 Evidence E is equivalent to 2 successes on 2 Bernoulli trials with replacement So Then (0.714 – as for defence)
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
BAYES’S THEOREM POSTERIOR ODDS on guilt = LIKELIHOOD RATIO PRIOR ODDS = 2 million (1 / 200,000) = 10 (10:1) Posterior probability of guilt = 10/11 = 91% Reasonable doubt – ACQUIT
WHAT ABOUT OTHER EVIDENCE? Didn’t fit description Victim: “not him” Unshaken alibi LR = 0.1 / 0.9 = 1/9 LR = 0.25 / 0.5 = 1/2 Apply Bayes’s Theorem again: Final odds on guilt = 10 1/9 1/2 } = 5/9 (5:9) (probability of guilt = 5/14 = 35%)
Dependence on Match Probability Match probability1/200m1/20m1/2m Posterior probability of guilt 98%85%35% – number of noughts does matter!
DATABASE SEARCH Crime trace, frequency (match probability) 1 in 1 million Search Police DNA database (D) of size 10,000 Find unique match: “John Smith” (S) No other evidence
Defence Case Probability of finding a match in database if innocent ~ 10,000 (1/1,000,000) = 1/100 Match probability of 1/100 is not convincing evidence Evidence against John Smith is (significantly) weakened by virtue of database search – ACQUIT
Prosecution Case We have examined 10,000 individuals Of these, 9,999 found not to match This has reduced the pool of potential alternative culprits Evidence against John Smith is (marginally) strengthened by virtue of database search – CONVICT
Which likelihood ratio? Hypothesis H S : “John Smith did it” is data- dependent Replace by hypothesis H D : “Someone in database D did it” –equivalent after search identifies S (but not before) LR = 1/(match probability) is now only 100 –weak evidence? But H D is a priori 10,000 times more probable than H S –posterior odds the same! –agrees with prosecution argument
Multiple Stains 2 DNA stains –1 on sheet, 1 on pillow –assume 2 perpetrators, 1 stain from each John Smith (S) matches pillow stain –associated “match probability” P What are appropriate hypotheses, likelihoods, inferences?
Hypotheses S left one of 2 stains S left pillow stain S left neither stain S didn’t leave pillow stain ( = prior probability S is guilty)
What to present in Court? Hypotheses equivalent (only) after data Different prior odds Identical posterior odds
Mixed Stains Crime trace containing DNA from more than 1 contributor –Rape –Scuffle etc
O. J. SIMPSON Crime OJS RG A B C Marker DQ- Frequency 13% 20% 28% “MATCH” to OJS Allele
MATCH PROBABILITY? PROSECUTION: Frequency of OJS type AB: 5% DEFENCE: Combined frequency of all matching types AA, AB, AC, BB, BC, CC:39% LR approach assuming Goldman (AC) in mixture: AB, BB, BC:21% LR approach not assuming Goldman in mixture: (more complex calculation) ~ 21%
MISSING DNA DATA What if we can not obtain DNA from the suspect ? (or other relevant individual?) Sometimes we can obtain indirect information by DNA profiling of relatives But analysis is complex and subtle…
HANRATTY James Hanratty convicted and executed in 1962 DNA profile from crime items analysed in 1998 Population frequency less than 1 in 2.5 million DNA profiles from mother and brother – “consistent with” crime DNA being from Hanratty (“A6” murder and rape, 1961)
PRESS REPORTS “There is a 1 in 2.5 million chance that Hanratty was not the A6 killer” “The DNA is 2.5 million times more likely to belong to Hanratty than anyone else” Likelihood Ratio based on profiles of mother and brother (complex calculation): 440 –even though no direct match to Hanratty!
DISPUTED PATERNITY MOTHER (m1) of CHILD (c1) claims that PUTATIVE FATHER (pf) is its TRUE FATHER (tf) But DO have DNA profiles from: Two full BROTHERS (b1, b2) of PUTATIVE FATHER undisputed child disputed child brothers His UNDISPUTED CHILD (c2) and its MOTHER (m2) DNA profiles from MOTHER and CHILD No profile from PUTATIVE FATHER