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A criminal leaves fifty thousand blood cells at the scene of a crime.
It’s hardly enough to stain a handkerchief. A forensic scientist extracts
DNA from the sample to create a ‘DNA fingerprint’. Its pattern resembles
that of a suspect. The scientist calculates that the chance of a match bet-ween
the sample and a random member of the public is one in a million. How incriminating
is this evidence?

Key assumptions underlying the interpretation of DNA evidence were challenged
in December at the Court of Appeal in the case of Andrew Deen, who was convicted
of rape in 1990.

Typically, evidence is presented by declaring a ‘match’ between the
DNA fingerprint of the defendant and that of a sample taken from the scene
of the crime. The significance of this match is assessed by calculating
the ‘match probability’. This is the probability that the DNA profile of
a member of the population picked at random would match the crime sample.
Scientific debate about the match probability. has centred on the rarity
of combinations of DNA fragments within different racial subpopulations.
At the Deen appeal, Peter Donnelly, a professor of statistics at Queen Mary
and Westfield College, London, opened a new area of debate. His testimony
related to what match probabilities mean and how they should be presented
in court. He pointed out that forensic evidence answers the question: ‘What
is the probability that the defendant’s DNA profile matches that of the
crime sample, assuming that the defendant is innocent?’ But the jury must
try to answer the question ‘What is the probability that the defendant is
innocent, assuming that the DNA profiles of the defendant and the crime
sample match?’

At first sight only linguistic nuance seems to separate these two questions.
However, as Donnelly observed, they can lead to significantly different
answers. He suggested to Lord Chief Justice Taylor and his fellow judges
that they imagine themselves playing a game of poker with the Archbishop
of Canterbury. If the archbishop were to deal a royal flush on the first
hand, one might suspect him of cheating. The probability of the archbishop
dealing a royal flush on any one hand, assuming he is an honest card player,
is about 1 in 70 000. But if the judges were asked whether the archbishop
was honest, given that he had just dealt a royal flush, they would be likely
to quote a probability greater than 1 in 70 000.

The first probability is analogous to the answer of the forensic scientist’s
question, and the second probability analogous to the answer of the jury’s
question. ‘Hence,’ said Donnelly, ‘a very small answer to the first question
does not necessarily imply a very small answer to the second.’ In the card-playing
example, the answer to the second question requires an assessment of prior
belief in the honesty of the archbishop.

The foundations of statistical inference were laid in the 18th century
by Thomas Bayes, the English Presbyterian minister and mathematician. He
showed precisely how prior beliefs should be altered in the light of experimental
data. The central result of his work is a probability ‘chain rule’ known
as Bayes’ theorem.

The chain rule can be applied directly to the case of DNA evidence by
referring to the ‘odds’ of a defendant being innocent. (The odds of innocence
are the ratio of the probability of innocence to the probability of guilt.)
If the ‘prior odds’ are the odds of innocence before hearing DNA evidence,
and the ‘posterior odds’ are the odds of innocence after hearing the DNA
evidence, then the posterior odds are equal to the prior odds multiplied
by the DNA match probability. (This assumes that, if the defendant is guilty,
the probability of a fingerprint match is 1.)

Consider a hypothetical crime committed in Oxford by an unidentified
white man. The number of possible perpetrators could, at the upper limit,
be Oxford’s entire white male adult population, about 30 000. This implies
prior odds of 30 000 to 1 in favour of the defendant’s innocence. If the
probability of a random DNA match with a suspect were 1 in a million, then
the posterior odds of his innocence would be 33 to 1 against. (That is,
30 000 multiplied by 1 in a million.) It is the figure of 33 to 1 which
a jury should consider, not the figure of 1 in a million.

The ‘Bayesian’ approach shows that, even with match probabilities of
1 in a million, great care must be taken. Unless the numbers are put into
a precise context, they can be misinterpreted. ‘One of the biggest concerns
with the use of probabilities in connection with DNA fingerprinting is there
can be misunderstandings by juries, to the enormous disadvantage of defendants,’
observed Donnelly after the appeal.

This analysis emphasises that it is wholly wrong to imply that the DNA
match probability is the same as the probability of the defendant’s innocence.
‘A DNA test showed that the chances of the defendant not being the attacker
were 859 million to one’ is a typical newspaper error. This type of statement
has been dubbed ‘the prosecutor’s fallacy’. One of the central points of
the Deen appeal was that an expert witness had suggested that the match
probability was so small that the defendant had to be the source of the
semen sample. Deen’s barrister, Michael Mansfield, argued that the probable
origin of the sample could only be assessed by combining the DNA evidence
with all the other evidence available to the jury. It was not for the expert
witness to make this type of assessment.

The three judges quashed Deen’s conviction and ordered a retrial. They
accepted the concept of the prosecutor’s fallacy, and agreed that its effect
on the original trial had helped to render the verdict unsafe.

Lord Taylor stressed that the decision was ‘not to indicate that DNA
profiling was unsafe’. But it is likely that, at the very least, judges
and expert witnesses will in future have to tread warily on the linguistic
tightrope which the prosecutor’s fallacy has highlighted. In the meantime,
how many lawyers are assessing the appeal probabilities of their convicted
clients?

David Pringle writes from Oxford.