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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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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