Now is the winter of our discontent, made glorious summer by this rediscovered son of York.
As you may have heard this week Richard III, King of England from 1483-1485, has been ‘rediscovered’ under a car park in Leicestershire. Or has he? How do we go about the process of deciding whether or not something is true given the evidence?
This was a question answered by the Reverend Thomas Bayes in his famous theorem of the 18th Century. He created a method to help us take a view of a piece of information, like what is the probability of a body found under a car park in Leicestershire being that Richard III. The key point about Bayes theorem is that it starts with an initial view of the likelihood of an event, and then determines the change in this likelihood given more evidence. This way you can build up a view of your belief in the probability of a particular event by applying more evidence.
So how does this work in practice? Let’s take this particular case of the unknown skeleton and its likelihood of being the famous Richard III or not, and apply Bayes Theorem.
The earliest surviving portrait of Richard III (c. 1520, after a lost original)
And seem a saint, When most I play the devil
We need to start with an estimate of how likely we think, given no information, that this was the body of Richard III. Since 1500 there have been around 588 million people who have lived in the UK . So given no knowledge at all we might suppose that the probability of this body being Richard III is 1 in 588 million – a pretty miniscule number.
And thus I clothe my naked villany, With old odd ends stolen out of holy writ
Let’s start with the fact that the body has been found to have scoliosis, a curvature of the spine. This is a pretty rare condition, but we do know (from Shakespeare and others) that it was one that Richard III suffered from. Doctors estimate that around 1 in 250 people suffer from this condition. Using Bayes theorem we can apply this to our prior view of 1 in 588 million to give us a revised probability of around 1 in 2.5 million – still incredibly small.
So wise so young, they say, do never live long
The body has also been carbon dated. Carbon dating has narrowed the bones to a period 1455-1540. Assuming we’re equally likely to dig up bones from any point in the period from 1200-1900 there is approx 1 in 8 chance of the body being from the right period. Applying this to our formula, this gives us a revised probability of around 1 in 300,000 – still small, but getting bigger.
Bloody thou art, bloody will be thy end
One thing that is clear from the body is that it died in battle. Of the around 3,000,000 people alive at the time of the Wars of the Roses, around 50,000 died in battle. This gives us a revised probability of 1 in 5000 – getting bigger all the time.
The king’s name is a tower of strength
We can use the known evidence for his age at death, and general build to increase the probability to around 1 in 50, before applying the final test of DNA testing. There are many varieties of this, with the one used in the case relating to DNA inherited down female descendents, and availability of living female relative descended from Richard’s sister. In general the probability of parentage identified by genetic markers is typically 99.99%.
By analysing the DNA of known relatives of Richard and comparing to the bone DNA, we calculate that the overall probability of the skeleton being a match to Richard III as around 99%. Given the usual test of something being ‘likely’ is at least 95% confidence, then we can say with confidence that the body found was that of Richard III.
An honest tale speeds best, being plainly told
Although classical probability theory is often taught in schools, the Bayesian approach is more often used by professionals in situations where there is considerable uncertainty, but a need to determine a future outcome for planning purposes. Examples include estimating natural gas reserves or fish stocks. It can also be successfully applied to pricing and product development costs, where you need to reduce uncertainty in making high risk decisions where there is little information, only probabilities.