In my first post I am going to look at a problem of Bayesian inference. This is a technique that can be used whenever we need to weigh up the evidence for or against some proposition. There are several good blogs explaining Bayes’ Theorem, but I have not seen one that addresses a real problem and shows how many different kinds of information can be combined to come to a conclusion expressed as a probability. In this case we are looking at a scientific question where Bayesian reasoning could be, but apparently hasn’t, been used.
On 4 February 2013 Leicester University announced that a skeleton discovered on the site of the former Greyfriars Church in Leicester was “beyond reasonable doubt” that of King Richard III of England, who had died at the battle of Bosworth Field on 22 August 1485, and whose resting place had been lost for centuries. “DNA confirms bones are king’s” said the BBC.
Now, I have an interest in forensic DNA, particularly familial matching, so I was all agog to learn the details. Dr Turi King said “The DNA is quite a rare type – just a few per cent of the population have it. But with the archaeological evidence it is very very very unlikely that it is just a coincidence”.
That interested me, because a DNA type (in this case mitochondrial DNA) shared by few percent of the population would clearly not be enough, by itself, to identify Richard III “beyond reasonable doubt”. Most skeletons from that period that matched, would not be Richard III. While it made sense not to make an announcement until the DNA results were in (since a negative DNA result could have refuted the identification), the other evidence in the case would have to be the clincher.
The evidence, in summary:
1) The skeleton was found beneath Greyfriars Church, a location suggested as the burial place by a historical source.
2) The skeleton was the right age at death, exhibited scoliosis (curvature of the spine) consistent with the tradition Richard III was a “hunchback”, but no withered arm (also said of Richard).
3) Carbon dating put the burial in the right period
4) Carbon dating also suggested a diet consistent with noble birth.
5) Injuries suggest death in battle
6) A “match” of mitochondrial DNA with two living maternal relatives.
The details can be found at the Leicester University web site.
Frustratingly, while a wealth of information was given about the skeletal and other evidence, the scientists at Leicester were not prepared to say which mitochondrial DNA haplotype had been identified, nor its exact frequency in the population. We would have to wait for the published papers. Nor did they say anything about the method used to combine all this evidence to come to an overall conclusion.
This led to some sceptical comments on the LU website, e.g. from geneticsman: “The readers of this thread should understand that this investigation appears to have thrown up a piece of controversial science: Clearly the results are not being accepted by a number of people (from various web sources, including a few respondents on this thread)”.
The first paper arising from this work was published on 25 May 2013. This is a detailed account of the archaeological dig, but unfortunately adds nothing to what we already knew about the identity of the skeleton.
The radiocarbon dates, evidence on the male skeleton of severe scoliosis, trauma consistent with injuries in battle and specific peri-mortem ‘humiliation injuries’, combined with the mtDNA match with two independent, well-verified matrilineal descendants, all point clearly to the identification of this individual as King Richard III. Indeed, it is difficult to explain
the combined evidence as anyone else.
Now, I have to say this strikes me as quite a lot of impressive evidence. But is it conclusive? I find it hard to say. There is no further detail of the DNA results (to be published separately at a later date), and no attempt to quantify the strength of the evidence. So I decided to do the calculation myself.
My conclusion (I won’t keep you in suspense) is that in this case, the DNA evidence is superfluous. Even without knowing the DNA match probability, we can estimate the probability that the skeleton is not that of Richard III as somewhere in the range 1 in 800,000 to 1 in 80 million.
How is that done? The method is fairly well known (though not always by those who ought to know it), and follows directly from Bayes’ Theorem. For a good exposition in the context of forensic genetics, see Weight-of-Evidence by David J. Balding. But the method is quite generally applicable.
I won’t give the derivation (perhaps in a follow-up post, if anyone asks) I will just state the notation and the conclusion. But if you want to skip the algebra, look past it and you may be pleasantly surprised at how easy it is to do this kind of calculation in practice.
We write P(A) for the probability that an event A happens; P(A,B) for the probability that both A and B happen; and P(A | B) for the probability that A will happen, given that we know B has happened (“A given B“). Mathematically we say:
P(A|B) = P(A,B)/P(B) … (1)
it follows that
P(A, B) = P(A|B).P(B) = P(B|A).P(A) … (2)
Bayes’ theorem follows directly from (2). An interesting case is when we have some evidence E and two alternative hypotheses H0 and H1. We would like to know the probability that each hypothesis is true, given the evidence. It easily follows from (2):
The term is called prior odds and is the relative likelihood of the two hypotheses before we gathered the evidence E. The term is the likelihood ratio and represents the strength of the evidence in favour of H1, relative to H0. The term we call the posterior odds and this is the updated odds in favour of H1.
For several pieces of evidence E1, E2, E3 … using the chain rule of probability we get
In words, the posterior odds of the hypothesis under test is the product of a series of terms: first is the prior odds, and each subsequent term measures the weight of one piece of evidence.
Note that each probability is conditional on previously included evidence. This can be important, as we will see, but if the individual items of evidence are independent, the conditionality vanishes and we get a simple product of likelihood ratios:
Don’t worry if you don’t understand all the algebra, in practice there is nothing more complicated to do than multiplication. Maybe the occasional division. No square roots, I promise. In what follows we are going to have to “guesstimate” probabilities for the events that make up the evidence. In several cases we can get good estimates of these from the UL website, in other cases we have to use our judgement. I will try to be “conservative” in the sense of underestimating the probability that the skeleton is Richard. Nevertheless, I am not an archaeologist, so if you think my estimates are wrong, you may well be right. Feel free to plug in your own numbers, and come up with a different answer. I am sure the experts at UL would be able to put in much better numbers and come up with a better answer, perhaps with a smaller range. The important point is that unless you put in very different numbers, in nearly all the steps, you are still going to come out with a big number at the end.
All we are doing here is asking, how likely would it be to see all this evidence from the skeleton of Richard III, and how likely would it be to see the same evidence from a skeleton that was not Richard? Combine that with the prior probability of Richard’s skeleton turning up in that particular place, and we will have the probability that the skeleton is his.
OK, lets go. define:
R: The skeleton is Richard III
N: The skeleton is NOT Richard III (N for Null Hypothesis, or N for Not Richard)
LR: Likelihood ratio
CPO: Cumulative Posterior Odds
Step 1: Prior odds
For the prior, we will take the odds that a skeleton (before we have examined it) found in the choir of Greyfriars Church would be that of Richard III.
i) We have two historical accounts placing the burial in Greyfriars, one giving the location as the choir.
ii) There is a further somewhat apocryphal account that the body was later disinterred and thrown in the river.
iii) Furthermore if the body had been buried, and not removed on that occasion, it could have been destroyed by robber trenches (as appears to have happened to the feet of the skeleton in question).
iv) Finally, if Richard’s bones were still under the choir, the skeleton found may still have been a different one, also buried in the choir, with the king’s, perhaps at a greater depth, nearby.
I am going to estimate those probabilities as 0.5, 0.5, 0.5 and 1/4.
P(R) = P(R3 was buried in GF) * P(R3 not thrown in river)
* P(R3 not destroyed by robber trenches)
* P(1/number of skeletons there may have been in the choir)
= 0.5 * 0.5 * 0.5 * 0.2 = 0.025 = 1/40
P(N) = 39/40
Prior ratio = 1/39
Step 2: Scoliosis (Sc)
Now, there is a tradition that Richard III was a hunchback, which is consistent with scoliosis. But we did not know for certain (prior to identifying this skeleton as Richard) that he had scoliosis. Many people doubted it. Let’s say the prior historical evidence supported a 50% chance Richard had scoliosis. Then
P(Sc | R) = 0.5
But we are told 1% of the general population has scoliosis, so
P(Sc | N) = 0.01
LR = 50
CPO = 50/39 = 1.28
Step 3: No withered arm (Nwa)
We must not ignore evidence that goes against the case. There is also a tradition that Richard had a withered arm, and the skeleton found does not show this. Again we will take the historical expectation of a withered arm as 50%, and the prevalance of withered arms in the general population (in the middle ages) as 1%.
P(Nwa | R) = 0.5
P(Nwa | N) = 0.99
LR = 0.5
CPO = 1.28 * 0.5 = 0.64
Step 4. late 20s – late 30s, consistent with 32 (Age)
I am going to guess 1/6 of skeletons in medieval graveyards are aged in their 30s. I am sure there must be actual data for this, do let me know if you can point me to it. So:
P(Age | R) = 1
P(Age | N) = 1/6
LR = 6
CPO = 0.64 * 6 = 3.84
5) Skeleton is male
Let’s not ignore the obvious. If 50% of skeletons in medieval graveyards are female, then:
P(Male | R) = 1
P(Male | N) = 0.5
LR = 2
CPO = 3.84 * 2 = 7.69
6) … but slender
How slender? Lets say:
P(Slender | R) = 0.8
P(Slender | N) = 0.2
LR = 4
CPO = 7.69 * 4 = 30.8
Step 7. Carbon: Date range 1455 – 1540
If the body was not Richard, but nevertheless associated with Greyfriars, then it must have been buried during the approximately 285 years between the building of the monastery and its dissolution. That gives 85/285 ~ 0.3 chance of falling in the measured date range.
P(Drange | R) = 1
P(Drange | N) = 0.3
LR = 3.3
CPO = 30.8 * 3.3 = 102
Step 8. Carbon: Diet
Carbon isotope analysis further suggests a high protein diet including seafood, indicative of a nobleman. Now, I don’t think we know for sure that Richard had a high protein diet including seafood. He may not have liked it. Nor do we know that a poor person would not have had such a diet. What about a fisherman? I am sure medieval historians and archaeologists could find a way to estimate these numbers properly, but I am just going to say that maybe 50% of noblemen had such a diet, and between 1% and 10% of the whole population.
P(diet | R) = 0.5
P(diet | N) = 0.1 ~ 0.01
LR = 5 ~ 50
CPO = 510 ~ 5100
Step 9: Died in Battle (DinB)
This is one of the most obvious facts about the skeleton. Most people, even in the middle ages, did not die in battle. And Richard did. But we must be careful. We have already considered evidence for the skeleton being that of a nobleman, so now we must consider the probability of a nobleman dying in battle, which was apparently quite high during the Wars of the Roses. (In fact, we considered the evidence of a diet typical of a nobleman, and that is the condition on the evidence. We allowed, cautiously, that the skeleton might not in fact have been that of a nobleman. But here, the cautious approach is to assume that he was). Let’s say between 0.1% and 1% of noblemen died in battle. Again, I would love to be corrected by someone with actual knowledge.
P(DinB | R) = 1
P(DinB | N, and a nobleman who died in Drange) = 0.01 – 0.001
LR = 100 ~ 1000
CPO = 5.1e4 ~ 5.1e6
Step 10: Nature of burial
To be buried in the choir you would have to be a high status person, but the burial was hurried: the grave was too small and there was no shroud or coffin. Each of these facts seems to fit King Richard rather better than another possible nobleman who had died in battle at around the same time. We could not have predicted with any certainty that Richard would be buried without a shroud, but it surprises us less, given the known facts of his death, than in a typical case.
P(Burial | R) = 0.5 * 0.5 * 0.5
P(Burial | N, nobleman died in battle in Drange) = 0.2 * 0.2 * 0.2
LR = 16
CPO = 8.1e5 ~ 8.1e8
In round numbers, that is odds of 800,000 – 80 million of favour of R.
Remember don’t take the actual numbers too seriously. I am really just trying to make two points:
1) The weight-of-evidence method is applicable and could be done much more thoroughly than I have done it here,
2) Even with my hand-wavy numbers, we can get a better feel for the problem than with no numbers at all.
We can expect the odds to improve by a factor of 10-100 when the DNA evidence is published, but we do not really need it.