Bayes’ Theorem in the Court – the Prosecutor’s Fallacy
source: https://academic.oup.com/aje/article/179/9/1125/103523
https://plus.maths.org/content/os/issue21/features/clark/index
Bayes’ Theorem is a fairly straightforward yet the most important statistic concept that is fundamental to a wide range of field like medical, machine learning and insurance. In the daily life, our understanding and application of the Bayes’ rule is closely associated with whether the objective and right decision could be made. Prosecutor’s fallacy is a statistical reasoning error often took place at the courtroom, which arose from the lack of understanding of the conditional probability and led to the Type I and Type II errors (the concept that we also have mentioned in the class.) in essence, the fallacy ignores the Bayes’ rule, assuming that the probability of A given B is equal to B given A. For example, the prosecutor could argue that the probability of the present of the evidence such as a DNA test match at the crime scene is high, if the accused defendant is guilty; therefore, with the evidence at hand, the probability that the defendant is guilty must be high. In other word, the argument is that the chance of matching an innocent man’s DNA with that at the crime scene is so low, so we could ignore the possibility that the accused is innocent.
The Sally Clark Case (1998) is one of the most famous cases involved the prosecutor’s fallacy. Clark was the victim of a miscarriage of justice as the court accused her for murdering her sons. In about a year, her two sons both died in a similar manner. In the court, a pediatrician presented false statistical evidence that the probability of two children in a wealthy family got sudden infant death syndrome was 1 in 73 million (he squared 1/8500– the chance for one child). This information was interpreted as there is 1 in 73 million chance that Clark is innocent therefore she must have murdered her sons. In term of Bayes’ rule, the court made fallacy when they put:
P(Evidence|innocence) = P(innocence | Evidence)
In which P (E|I)= prob that an innocent person match the evidence
P(I|E)= prob that the defendant matched the evidence is innocent.
The article offered the analysis include the Bayes reasoning for the Clark case, arguing the statistic in the was “wrong, irrelevant, biased and totally misleading.”) First, the testimony regarded the death of two children of Clark are independent which in fact are not.
According to the Confidential Enquiry for Stillbirths and Deaths in Infancy (CESDI, an authoritative and detailed study of deaths of babies in five regions of England between 1993 and 1996), there is about 1 in 1303 baby dying a cot death. The chance is reduced to 1/8500 if the baby lives in a relatively wealthy, nonsmoking family and with a mother over 26, which fits the Clark’s case. The “expert” at the court against Clark assumed there is no link between cot death of siblings by squaring 1/8500. However, the siblings of children died cot death is 10 and 22 times more likely to die in similar way than average kids. The chances of a second cot death in the same family could be between 1 in 60 and 1 in 130.
Therefore, how to calculate the chance that Clark was innocent, which in other words the chance her children died of cot death. The equation is written as the following:
H refers to cot death, and D means the baby death.
As we mentioned, the chance two babied died of cot death could be high as 1/130, and the number 1/73 million the testimony given was impossible. In this case we lower 1/130 to 1/100,000, still a very rare chance that essentially no much different from the impossible1/73million. P (D|H), chance of baby died given cot death is 1. A, referring to the alternate hypothesis that the children did not die of cot death(all other possibilities: for example someone else murdered both children, or Sally Clark murdered one of them etc., is equal to
. The Home Office statistic gives that fewer than 30 children are murdered by their mother each year in England and Wales where 650,000 are born each year. 30/650,000=0.000046. Since the chance two siblings are murdered is much more rare than single murder, we should use number much smaller than 0.000046, but in this case, we would just overestimate, using 0.0000046—number 10 times smaller. So, the chance Clark is innocent is that
Clark was released in 2003 after two appeals, however this prosecutor’s fallacy caused her death at home from alcohol poisoning after four years. The innocent mother was accused for killing her own sons because of the court’s neglect of the right statistical reasoning specifically the conditioned probability. The understanding of Bayes’ Rule is critical in the legal field.