32 Bayes’s theory
The early years of the Rev. Thomas Bayes are obscure. Born in the southeast of
England, probably in 1702, he became a nonconformist minister of religion, but also
gained a reputation as a mathematician and was elected to the Royal Society of London
in 1742. Bayes’s famous Essay towards solving a problem in the doctrine of chances was
published in 1763, two years after his death. It gave a formula for finding inverse
probability, the probability ‘the other way around’, and it helped create a concept
central to Bayesian philosophy – conditional probability.
Thomas Bayes has given his name to the Bayesians, the adherents of a brand
of statistics at variance with traditional statisticians or ‘frequentists’. The
frequentists adopt a view of probability based on hard numerical data. Bayesian
views are centred on the famous Bayes’s formula and the principle that subjective
degrees of belief can be treated as mathematical probability.
Conditional probability
Imagine that the dashing Dr Why has the task of diagnosing measles in his
patients. The appearance of spots is an indicator used for detection but diagnosis
is not straightforward. A patient may have measles without having spots and
some patients may have spots without having measles. The probability that a
patient has spots given that they have measles is a conditional probability.
Bayesians use a vertical line in their formulae to mean ‘given’, so if we write
prob(a patient has spots | the patient has measles)
it means the probability that a patient has spots given that they have measles.
The value of prob(a patient has spots|the patient has measles) is not the same as
prob(the patient has measles|the patient has spots). In relation to each other,
one is the probability the other way around. Bayes’s formula is the formula of
calculating one from the other. Mathematicians like nothing better than using
notation to stand for things. So let’s say the event of having measles is M and the
event of a patient having spots is S. The symbol is the event of a patient not
having spots and the event of not having measles. We can see this on a Venn
diagram.