I use Bayes theorem: posterior odds equals prior odds times likelihood ratio. For an introduction, please read this nice blog post https://entropicthoughts.com/bayes-rule-odds-form
I use this rule, Bayes’ rule, repeatedly, each time taking account of another part of the evidence. It is named for Thomas Bayes, a presbyterian minister and mathematician, who was interested in using it to find a mathematical proof of the existence of God. https://en.wikipedia.org/wiki/Thomas_Bayes
The likelihood ratio for the question at hand, based on some part of the evidence, is the ratio of the probabilities of that part of the evidence under the two competing hypotheses. More precisely, one uses the conditional probabilities of that fact given previously incorporated evidence. We have to start somewhere and we start by describing two alternative hypotheses and our probabilities or degrees of belief or personal betting odds for those two hypotheses, before further evidence is taken into account.
Let’s start with the news reports of a police investigation of a possible killer nurse at a neonatology unit in the UK; the investigation being triggered by a disturbing spike in the death rate on that unit.
I think that in the last fifty years there simply hasn’t been been a case in the UK of a killer nurse on a neonatal ward, except possibly the case of Beverley Allitt. One might argue that there do exist doubts as to the safety of her conviction, or one might argue that there can have been serial killer nurses who completely evaded detection. Did Alittit work in an intensive care unit? I also think that in recent years, every year has seen a scandalous calamity in a UK neonatal ward, leading to avoidable deaths of quite a few babies. So a priori: the relative chances of a killer nurse being responsible for the spike, or simply poor care, is in my estimation 50:1 in favour of poor care in a failing hospital unit rather than activity of a killer. If you disagree, give me your arguments for both those rates and hence their ratio. If you would like to take a different starting point, try that. Eg, what is the chance a random nurse is a serial killer? At some point one will have to use the information that this was a neonatal unit and one will have to take account of the “normal” rate of deaths on the unit. I think my choice is reasonably specific. One could argue that the prior odds should be 10 to 1, or 100 to 1, instead of 50 to 1. I expect that most people will at least agree that killer nurses on neonatal units are very rare, disastrously poor care on a neonatal unit in the UK is not rare at all.
So we are back in 2017 and hear the news and rightly we should be sceptical that there really is a case here. But clearly there are grounds to investigate what is the cause of that spike, and maybe there is more information which the police already have.
Then, many years go by. A particular nurse is detained for questioning in two successive years; and finally arrested in a third year. Two more years go by (Corona). At last, a trial begins. It turns out that roughly seven years of police investigation has uncovered no direct evidence at all (neither medical evidence, toxicological evidence, witness testimony or CCTV recordings, finger prints or DNA) of unlawful action by the nurse who has been under intensive investigation all that time. And not just no evidence against that nurse – no direct strong evidence of malevolent activity by anyone.
One might want to argue that the insulin evidence is strong toxicological evidence. We could argue about that for a long time. Even if one or two babies were given unauthorised doses of insulin there is no direct proof that Lucy Letby did that herself. There is the possibility of accidental administration (twins in adjacent cots). The argument that Lucy did administer insulin seems to have been that we know at some point she carried out other murderous attacks and it is unlikely that there were two murderous nurses working in the unit. But why do we believe there are murderous nurses working on the unit? This argument can only be made after hearing all the other evidence in the case.
So we have to estimate the probability of a 7 year police hunt for evidence of murder by a particular nurse finding no direct evidence of any malevolent activity at all by anyone, if Lucy Letby actually was innocent, and if she truly was a serial killer. In my opinion ,what we actually observed is much more likely under the innocence hypothesis than under the guilty hypothesis. If she truly is innocent the chance of finding powerful directly incriminating evidence must be rather small; if she truly is a serial killer then it must be unlikely that that no baby can be definitely proven to have been murdered or attacked. I guess the two probabilities of no hard evidence to be 95% and 5% respectively. These are probabilities of 19/20 and 1/20 respectively, so a likelihood ratio of 19. I’ll be a bit more cautious and call it 10.
We already had odds of 50:1 in favour of innocence. We have a likelihood ratio of 10:1 in favour of innocence, having learnt that police investigation uncovered no strong and direct proof of malevolent harm to any baby. The odds on Lucy being innocent are therefore now 50 times 10, or 500 to 1.
Let’s now bring in the evidence from psychology. Are there reasons to believe Lucy is a psychopath? Which surely she must be, if she is a serial killer of babies in her care. It seems there is no reason at all to suspect she is a psychopath. I think that there very likely would be strong independent signs of psychopathy in her life history if she really is a serial killer, but obviously not so likely if she is completely innocent. [Clearly she could be both a psychopath but did not actually harm or try to harm any baby. I don’t think this is an interesting hypothesis to explore. I will also not pay attention to the Munchhausen by proxy idea, that she was trying to attract the attention of an older male doctor. All the evidence says that he was more romantically interested in her, than vice versa.]
Put the likelihood ratio at 2, ie twice as likely to see no evidence for psychopathy if innocent, than if a serial killer. Actually I think it should be closer to 10. We should ask some psychologists. Lucy Letby did not sadistically kill little animals when she was a child. By all accounts, she was a dedicated nurse and cared deeply for her work.
We were at 500 to 1 for innocence. Factor in a likelihood ratio of 2 for psychological evidence. Now it’s 1000 to 1. But we are not done yet.
Next, I would like to take account of the statistical evidence that the spike in deaths is quite adequately explained by the acuity of the patients being treated in those 18 months. I would say that this is exactly what we would expect if Lucy is innocent but very unlikely if she’s a serial killer. I think this hypothesis is very adequately supported by published MBRRACE-UK statistics, and what we know about the acuity of the babies in the case. We know why acuity went up in around 2014 and we know why it went down midway in 2017. The spike seems to have been caused by hospital policy which was being made and implemented by the consultants on that unit. They should have expected it.
Say a likelihood ratio of 10. That brings us to 10,000 to 1 she’s innocent; a posterior probability of 99.99%. I haven’t yet brought in the facts of an investigation driven by tunnel vision and coached by doctors who, as we now know, were making quite a few deadly mistakes themselves. I haven’t brought in yet the innocent explanation of the post-it note. In my opinion, the post-it note is powerful evidence for innocence; it makes absolutely no sense under the hypothesis of guilt. The irrelevance of the handover notes and the notations in her diary. Facebook searches? Her alleged lies (about what she was wearing when she was arrested). Anything else?
Anyway, I am now well above 99.99% sure that Lucy is innocent and since the press conference and the report of Shoo Lee and his colleagues, we can all be even more sure that that is the case.
