Bunhill Fields burial ground contains the relics of John Bunyan, William Blake and Daniel Defoe. Of import to statisticians is the tomb of the Rev Thomas Bayes, Presbyterian clergyman from Tunbridge Wells, who died 260 years ago. He is famous for his work on conditional probabilities, which concern the chance of one event occurring, given another event has happened.
Take lateral flow tests. The conditional probability of getting an (incorrect) positive result (call this A), given you are not infected (B), is less than one in 1,000. That rate is very low. Bayes’s theorem shows how to calculate what we really want: the conditional probability you are not infected (B), given you have a positive test (A). In that case, you would be isolating with no benefit. Unintuitively, when the virus is rare and there are very few “true positives”, this probability can be high. Currently in secondary schools, around three in 10 positive lateral flow tests turn out to be false.
In courtrooms, mixing up the probability of “A given B’” with “B given A” is known as the “prosecutor’s fallacy”. In 1999, a court convicted Sally Clark of the murder of her two sons, in part because a medical expert claimed the chance of two accidental cot deaths was one in 73m. Even if this number was right – which it isn’t – it did not reflect the chance she was innocent. A double murder was also very rare: the relative likelihood of the two explanations was key and with new evidence and better statistical reasoning, an appeal court quashed the conviction.
There was controversy after a recent Observer headline referred to Bayes’s theorem as “obscure”. His ideas may be little known by the public, but they are growing among scientists. Many complex analyses done during the pandemic have been “Bayesian”, including modelling lockdown effects, the ONS infection survey, and Pfizer-BioNTech’s vaccine trial. The term “credible interval”, rather than “confidence interval”, is the giveaway.
Last week, Cass Business School announced the renaming of its institution after Bayes and his theorem. The obscure tomb in nearby Bunhill Fields is worth a visit.
• David Spiegelhalter is chair of the Winton Centre for Risk and Evidence Communication at Cambridge. Anthony Masters is statistical ambassador for the Royal Statistical Society
