the curious case of the inexpert witness
Post on 01-Nov-2014
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The Inexpert Witness
• Sir Roy Meadow, born 1933
• Distinguished paediatrician
• Famous for “Munchausen Syndrome by Proxy”
• Expert witness in cases of suspected child abuse and murder
• Notorious for high-profile miscarriage of justice in Sally Clark trial
The Case of Sally Clark• Solicitor Sally Clark was tried
in 1999 for the murder of two children (Christopher, 11 weeks), (Harry, 8 weeks).
• Medical testimony divided• Meadow’s evidence was
decisive, but flawed.• Appeal in autumn 2000 was
dismissed• Second appeal (for different
reasons) in 2003, but ruling cast doubt also on Meadow’s testimony; Clark released.
• Sally Clark died on16 March 2007 of alcohol poisoning
Publish and be damned• This case was mentioned
in my book “From Cosmos to Chaos”
• In 2005, Meadow appeared before a GMC tribunal and was struck off
• He appealed and pending the outcome my book was shelved by OUP
• His appeal succeeded, but was guilty of “serious professional misconduct” so it was published.
The Argument
• The frequency of natural cot-deaths (SIDS) in affluent non-smoking families is about 1 in 8500.
• Meadow argued that the probability of two such deaths in one family is this squared, or about 1 in 73,000,000.
• This was widely interpreted as meaning that these were the odds against Clark being innocent of murder.
• The Royal Statistical Society in 2001 issued a press release that summed up the two major flaws in Meadow’s argument.
Independence
• There is strong evidence that the SIDS does have genetic or environmental factors that may correlate within a family
• P(second death|first)=1/77, not 1 in 8500.• Changes the odds significantly
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unless X and Y are independent
The Prosecutor’s Fallacy• Even if the probability calculation were right, it is
the wrong probability. • P(Murder|Evidence) is not the same as
P(Evidence|Murder), although ordinary language can confuse the two.
• E.g. suppose a DNA sequence occurs in 1 in 10,000 people. Does this mean that if a suspect’s DNA matches that found at a crime scene,the probability he is guilty is 10,000:1?
• No!• E.g. in a city of a million people, there will be
about 100 other matches. In the absence of any other evidence, the DNA gives of odds of 100:1 against the suspect being guilty.
Inverse Reasoning
• If we calculate P(Deaths|SIDS) to be very small, that does not necessarily mean that P(Murder|Deaths) has to be close to unity!
• We need to invert the reasoning to produce P(SIDS|Deaths) and P(Murder|Deaths) both of which are small!
• The only fully consistent way to do this is by Bayes’ Theorem, although a (frequentist) likelihood ratio would also do…
A Load of Balls…
• Two urns A and B.• A has 999 white balls and 1 black one; B
has 1 white balls and 999 black ones.• P(white| urn A) = .999, etc. • Now shuffle the two urns, and pull out a
ball from one of them. Suppose it is white. What is the probability it came from urn A?
• P(Urn A| white) requires “inverse” reasoning: Bayes’ Theorem
Urn A Urn B
999 white
1 black
999 black
1 white
P(white ball | urn is A)=0.999, etc
Bayes’ Theorem: Inverse reasoning
• Rev. Thomas Bayes (1702-1761)
• Never published any papers during his lifetime
• The general form of Bayes’ theorem was actually given later (by Laplace).
Bayes’ Theorem
• In the toy example, X is “the urn is A” and Y is “the ball is white”.
• Everything is calculable, and the required posterior probability is 0.999
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Cot-Death Evidence
• Here M=Murder, D=Deaths, S=SIDS• P(M|D) is not obviously close to unity!!• Like DNA evidence statistical
arguments are not probative unless P(S) can be assigned.
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