Prosecutor's Fallacy - The Sally Clark Case

The Sally Clark Case

Sally Clark, a British woman who was accused in 1998 of having killed her first child at 11 weeks of age, then conceived another child and allegedly killed it at 8 weeks of age. The prosecution had expert witness Sir Roy Meadow testify that the probability of two children in the same family dying from SIDS is about 1 in 73 million. That was much less frequent than the actual rate measured in historical data - Meadow estimated it from single-SIDS death data, and the assumption that the probability of such deaths should be uncorrelated between infants.

Meadow acknowledged that 1-in-73 million is not an impossibility, but argued that such accidents would happen "once every hundred years" and that, in a country of 15 million 2-child families, it is vastly more likely that the double-deaths are due to Münchausen syndrome by proxy than to such a rare accident. However, there is good reason to suppose that the likelihood of a death from SIDS in a family is significantly greater if a previous child has already died in these circumstances (a genetic predisposition to SIDS is likely to invalidate that assumed statistical independence) making some families more susceptible to SIDS and the error an outcome of the ecological fallacy. The likelihood of two SIDS deaths in the same family cannot be soundly estimated by squaring the likelihood of a single such death in all otherwise similar families.

1-in-73 million greatly underestimated the chance of two successive accidents, but, even if that assessment were accurate, the court seems to have missed the fact that the 1-in-73 million number meant nothing on its own. As an a priori probability, it should have been weighed against the a priori probabilities of the alternatives. Given that two deaths had occurred, one of the following explanations must be true, and all of them are a priori extremely improbable:

1. Two successive deaths in the same family, both by SIDS
2. Double homicide (the prosecution's case)
3. Other possibilities (including one homicide and one case of SIDS)

It's unclear that an estimate for the second possibility was ever proposed during the trial, or that the comparison of the first two probabilities was understood to be the key estimate to make in the statistical analysis assessing the prosecution's case against the case for innocence.

Mrs. Clark was convicted in 1999, resulting in a press release by the Royal Statistical Society which pointed out the mistakes.

In 2002, Ray Hill (Mathematics professor at Salford) attempted to accurately compare the chances of these two possible explanations; he concluded that successive accidents are between 4.5 and 9 times more likely than are successive murders, so that the a priori odds of Clark's guilt were between 4.5 to 1 and 9 to 1 against.

A higher court later quashed Sally Clark's conviction, on other grounds, on 29 January 2003. However, Sally Clark, a practising solicitor before the conviction, developed a number of serious psychiatric problems including serious alcohol dependency and died in 2007 from alcohol poisoning.