Simple Bayesian Problems: The Sally Clark Case
Starting in 1996, four British mothers were arrested for murder because two or more of their infants had died in their cribs. Three of the mothers were convicted and sentenced to life imprisonment. Yet by 2006, all four mothers were declared innocent of crimes that had never occurred. Bayes’ rule could have cleared them.
Sally Clark, a lawyer who lost her first son at 11 weeks and her second at 8 weeks, was convicted in 1999. A prominent pediatrician, Sir Roy Meadow, had testified for the prosecution about Sudden Infant Death Syndrome, known as SIDS in the U.S. and cot death in Britain. Citing a government study, Meadow said the incidence of one SIDS death was one in 8,500 in a family like Clark’s–stable, affluent, nonsmoking, with a mother more than 26 years old.
Then, despite the fact that some families are predisposed to SIDS, Meadow assumed erroneously that each sibling’s death occurred independently of the other. Multiplying 8,500 by 8,500, he calculated that the chance of two children dying in a family like Sally Clark’s was so rare–one in 73 million–that they must have been murdered.
A Bayesian analysis would have shown that the children most probably died of SIDS.
According to Bayes’ rule, a highly unlikely event can happen but it must be compared with other highly unlikely events. Thus, the question before the court should have been: Did the Clark babies more likely die of natural causes or murder?
Here’s how the Bayesian analysis was done.
First, we look at natural causes of sudden infant death. The chance of one random infant dying from SIDS was about 1 in 1,300 during this period in Britain. Meadow’s argument was flawed and produced a much slimmer chance of natural death. The estimated odds of a second SIDS death in the same family was much larger, perhaps one in 100, because family members can share a common environmental or genetic propensity for SIDS.
Second, we turn to the hypothesis that the babies were murdered. Only about 30 children out of 650,000 annual births in England, Scotland, and Wales were known to have been murdered by their mothers. The number of double murders must be much lower, estimated as 10 times less likely.
We take the hypothesis H to be updated to be: Sally Clark’s two children died of SIDS. The data D will be that both children died suddenly and unexpectedly. The starting probabilities that go into Bayes’ Rule are P(H), and its opposite, P(notH). As discussed above, P(H) is taken to have the value
P(H) = 1/1300 × 1/100 = 0.0000077
The alternative P(notH) is then given by
P(notH) = 1 — P(H) = 0.9999923
Now we look at the data. It appears in the probabilities P(D | H) and P(D | notH). The first of these equals one, since it is the probability that the children died suddenly and expectedly given that they died of SIDS. P(D | notH ) is the probability that a random pair of siblings dies suddenly and expectedly but not from SIDS. The prosecution will equate that with murder. The estimate made in the paragraph above was
P (D | notH) = 30/650000 × 1/10 = 0.0000046.
The goal is to estimate P(H | D), the probability that the cause of death was SIDS, given their unexplained deaths. Bayes’ Rule provides the formula
P(H | D) = P(H) P(D | H) / P(H) P(D | H)+P(notH) P(D | notH)
Inserting the numbers one finds
P(H | D) =
1 × 0.0000077 / 1 × 0.0000077 + 0.999992 × 0.0000046 ≈ 0.6
Thus, it is more than likely that the infants died of SIDS.
Clark spent three years in prison before medical evidence freed her. A pathologist had withheld evidence that her second son had a bacterial blood infection known to cause sudden infant death. With the case against Clark in shreds, the other mothers were eventually freed too. But broken by the deaths of two sons and her conviction and imprisonment, Sally Clark died a few years later of acute alcohol intoxication.
I am indebted to Ray Hill, Professor of Mathematics at Salford University, Manchester, England, for his help with this example. It is based on Helen Joyce’s PlusMagazine article http://plus.maths.org/content/beyond-reasonable-doubt and on Ray Hill, (2004) “Multiple sudden infant deaths–coincidence or beyond coincidence?” Paediatric and Perinatal Epidemiology (18) 320-326 and (2005) “Reflections on the cot death cases.” Significance (2:1). 2.
Click here for more simple Bayesian examples.
Several readers have asked for more introductory math problems using Bayes. So I asked some of the world’s leading statisticians to provide us with simple but exciting examples. Albert Madansky and Tony Richardson – familiar from the book – and Daniel Gianola and John Carlin kindly responded.